Abstract

X-ray in-line holography is well suited for three-dimensional imaging, since it covers a large field of view without the necessity of scanning. However, its resolution does not extend to the range covered by coherent diffractive imaging or ptychography. In this work, we show full-field holographic x-ray imaging based on cone-beam illumination, beyond the resolution limit given by the cone-beam numerical aperture. Image information encoded in far-field diffraction and in holographic self-interference is treated in a common reconstruction scheme, without the usual empty beam correction step of in-line holography. An illumination profile tailored by waveguide optics and exactly known by prior probe retrieval is shown to be sufficient for solving the phase problem. The approach paves the way toward high-resolution and dose-efficient x-ray tomography, well suited for the current upgrades of synchrotron radiation sources to diffraction-limited storage rings.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Coherent x-ray optics have led to transformative progress in recent years [13], opening up novel opportunities to image matter at high resolution [46], with chemical sensitivity [7], and at ultrafast time scales [8,9]. For nondestructive imaging of three-dimensional (3D) bulk materials and biological specimens, hard x-ray in-line holography or propagation imaging is particularly suitable [1012], since it offers a phase-sensitive imaging scheme that can cover large specimens in a full-field approach without the need for scanning. Contrast formation is based on wave propagation and self-interference of the scattered and primary beam behind the object. By illumination with a parallel beam, macroscopic scales can be covered, while illumination with a spherical beam behind a nanofocus optic enables nanoscale resolution [1315]. In contrast to coherent diffractive imaging (CDI), the phase problem of holographic imaging is mathematically better posed [16] due to the near-field interference between a scattered wave and a reference wave. Further, coherence requirements in near-field propagation imaging can be relaxed with respect to CDI [17], and phase retrieval neither requires the object nor the illumination field to be compactly supported [18,19]. Based on its full-field nature and its dose efficiency, propagation imaging is well suited for 3D imaging of biological soft tissues [2023] as well as for dynamic imaging [24].

Unfortunately, the resolution of holographic imaging is limited by the source size of the cone-beam illumination, and it does not reach the values in the sub-20 nm regime that are routinely achieved by ptychography [4,6,25] or CDI [26,27]. While one can improve focusing to increase the numerical aperture (NA), the finite source size will always impose a limit. Further, image reconstruction is inconsistently based on an illumination by a perfect point source, which is tacitly assumed when applying the Fresnel scaling theorem [28]. The common procedure to deal with finite source size and wavefront aberrations is to divide the recorded hologram ${I_z}$ by the measured intensity image of the empty beam $I_z^E$ and then treat ${\bar I_z} = {I_z}/I_z^E = |{{\cal D}_z}\{O{\} |^2}$ as the propagation of the object function $O$ under the Fresnel propagator ${D_z}$. For sufficiently thin objects, $O$ is given by a projection integral $O({x,y}) = \exp [- i2\pi /\lambda \int_{- \Delta t}^0 [{\delta _\lambda}(x,y,z) - i{\beta _\lambda}(x,y,z)]{\rm{d}}z]$ for the object of thickness $\Delta t$ and refractive index $n(x,y,z) = 1 - {\delta _\lambda} (x,y,z) + i{\beta _\lambda}(x,y,z)$ at wavelength $\lambda$. Strictly speaking, this approach is valid only for the assumed idealized illumination conditions (ideal plane wave or ideal spherical wave). For general illumination (probe) functions $P$, the relation ${I_z}: = |{{\cal D}_z}\{P \cdot O{\} |^2} \simeq |{{\cal D}_z}\{P{\} |^2} \cdot |{{\cal D}_z}\{O{\} |^2}$ fails or is at best approximative. Instead of dividing intensities of the propagated wave in the detection plane (${z_{{\rm{De}}}}$), one should divide the object exit wave $\psi$ by the complex-valued probe $P$ in the object plane (${z_{{\rm{Ob}}}}$) to retrieve the object transmission function $O$ without aberrations introduced by the illumination field. Even if the empty-beam approximation is justified, as for a Gaussian beam, the information contained in the scattered radiation is not exploited, resulting in an NA reduced to the divergence of the illuminating beam. In general, the empty beam division compromises resolution and image quality, and it accounts for the fact that in-line holography often does not even reach the resolution given by the source size [29,30].

In this publication we extend full-field in-line holography to high resolutions and circumvent the spoiling effects of empty beam division. For this purpose, we adapt the principle of keyhole coherent diffractive imaging [31,32], which showed that iterative solutions to the phase problem are possible for extended objects with phase-curved incident illumination if the illumination function is well known. We show that both ptychographic and single-shot iterative probe reconstruction before recording of the object is greatly facilitated by the combination of high curvature in the object plane and high compactness of the probe in the source plane. At the same time, this waveguide illumination scheme also relaxes the constraints of a compact probe in the object plane to the benefit of combining high resolution and large field of view (FOV).

2. METHOD

A. Algorithm

The basic concept of the high-resolution x-ray holography approach is outlined in Fig. 1. The concept is based on the propagation of the full wave field $\psi ({x,y})$ between the object plane (${z_{{\rm{Ob}}}}$) and the detector plane (${z_{{\rm{De}}}}$). The approach uses the information provided by a single measurement of the holographic far-field intensities ${I_z}$ at ${z_{{\rm{De}}}}$ and the known illumination function $P$ at ${z_{{\rm{Ob}}}}$. $P$ is reconstructed by ptychography or by a single-frame inversion scheme, exploiting the high compactness in the waveguide exit plane (${z_{{\rm{WG}}}}$) due to the waveguide confinement. In the first case, a short ptychography scan needs to be recorded before (or after) the holographic imaging of the samples (or a holotomography scan). Importantly, we find that ${I_z}$ of a single distance recording and $P$ are sufficient inputs for objects with either (i) vanishing absorption (pure phase-contrast samples), (ii) phase-amplitude coupling (homogeneous object composition), or (iii) for objects that are compactly supported. Due to the high wavefront curvature at ${z_{{\rm{Ob}}}}$ and hence the holographic nature of the far-field pattern in the center of the detector image, this set is sufficient for reconstruction. This holds even if $P$ is more extended than the classical oversampling criterion [33] would require for CDI phase retrieval. As usual, compatibility with the measured data is assured by applying the magnitude constraint: the solution must satisfy the measured intensity distribution of the recorded diffraction pattern ${I_z}$ [see Fig. 1(a)]. To this end, the wave field $\psi (x,y)$ is modified such that

$${{\boldsymbol{\cal P}}_M}\left[{\psi (x,y)} \right] = {{\boldsymbol{\cal F}}^{- 1}}\left[{\frac{{{\boldsymbol{\cal F}}\left[{\psi (x,y)} \right]}}{{|{\boldsymbol{\cal F}}\left[{\psi (x,y)} \right]|}} \cdot \sqrt {{I_z}(x^\prime ,y^\prime)}} \right] .$$
In order to formulate phase retrieval only in terms of projections and to treat all constraints on equal footing, the propagation of the wave field between object and detector (and back) is incorporated into the projection operator. In contrast to typical x-ray holography experiments that describe image formation in an equivalent geometry after application of the Fresnel scaling theorem [28], we calculate propagation from the object exit plane to the detector in the direct geometry via a numerical fast Fourier transformation (FFT). The holographic intensity in the detector plane is then written as ${I_z}(x^\prime ,y^\prime): = |{\boldsymbol{\cal F}}\{\psi ({x,y}){\} |^2}$, where ${\boldsymbol{\cal F}}$ is the Fourier operator. This replaces the use of the Fresnel operator in conventional cone-beam x-ray holography ${{\cal D}_{{z_{{\rm{eff}}}}}}$ and the use of an effective distance ${z_{{\rm{eff}}}} = {z_{{\rm{De}}}}/M$, where $M = {z_{{\rm{De}}}}/{z_{{\rm{Ob}}}}$ is the magnification. Further, the assumption of an ideal point source illumination is made obsolete by the direct geometry so that the reconstructed illumination $P$ (at ${z_{{\rm{Ob}}}}$) can be properly fed into the constraint scheme of the object plane without the necessity of using an effective probe. Depending on the available constraints, the object plane projector ${{\boldsymbol{\cal P}}_O}$ is chosen:
$${{\boldsymbol{\cal P}}_O} \in \{{{\boldsymbol{\cal P}}_{{pp}}},{{\boldsymbol{\cal P}}_r},{{\boldsymbol{\cal P}}_S},{{\boldsymbol{\cal P}}_{{SH}}},{{\boldsymbol{\cal P}}_h}\} ,$$
where ${{\boldsymbol{\cal P}}_{{pp}}}$ denotes a projector imposing a pure phase constraint. Of course, the possible set of object projectors can be further extended by projectors implementing a range ${{\boldsymbol{\cal P}}_r}$, support ${{\boldsymbol{\cal P}}_S}$, shearlet (sparsity) ${{\boldsymbol{\cal P}}_{{SH}}}$, or homogeneous object ${{\boldsymbol{\cal P}}_h}$ constraints (see Supplement 1). In the present experimental demonstration, ${{\boldsymbol{\cal P}}_{{pp}}}$ has been used, well justified for hard x-rays and unstained biological cells. ${{\boldsymbol{\cal P}}_{{pp}}}$ acts on $\psi ({x,y})$ as
$${{\boldsymbol{\cal P}}_{{pp}}}\left[{\psi (x,y)} \right] = \frac{{\psi (x,y)}}{{|\psi (x,y)|}} \cdot |P\left({x,y} \right)| .$$
The iterative phase retrieval algorithm can be implemented via different update schemes. Here we used either the simple error reduction (ER) as shown in Fig. 1(a):
$${\psi _{j + 1}} = {{\boldsymbol{\cal P}}_O}{{\boldsymbol{\cal P}}_M}{\psi _j}$$
or the more sophisticated relaxed averaged alternating reflector (RAAR) [34] scheme:
$${\psi _{j + 1}} = \left[{\frac{1}{2}\beta \left({{{\boldsymbol{\cal R}}_O}{{\boldsymbol{\cal R}}_M} + {\cal I}} \right) + \left({1 - \beta} \right){{\boldsymbol{\cal P}}_M}} \right]{\psi _j} ,$$
employing not only projections but also reflection operators ${\boldsymbol{\cal R}}$, defined as
$${\boldsymbol{\cal R}} = 2{\boldsymbol{\cal P}} - {\cal I} ,$$
where ${\cal I}$ is the identity. The parameter $\beta$ controls the relaxation and can be optimized for convergence. In most cases, the differences in the resulting image quality between ER and RAAR were only minor.
 figure: Fig. 1.

Fig. 1. (a) Outline of the x-ray holography algorithm. ${{\boldsymbol{\cal P}}_{\rm{M}}}$ denotes the magnitude, and ${{\boldsymbol{\cal P}}_{{\rm{pp}}}}$ denotes the pure phase object constraint. ${\boldsymbol{\cal F}}$ and ${{\boldsymbol{\cal F}}^{- 1}}$ represent the forward and inverse FFT, respectively. As the initial ($j = 0$) wave field, the known illumination function $P$ is used. The phase object $O$ is calculated in the last iterative step ($j = N$) by subtracting the the phase of the probe $P$ from the exit wave field. The object phase (top, right), which has been used in the simulation as an example, is given by a freely sketched cardiomyocyte image, inspired by the real cardiomyocyte imaged below in Fig. 4. (b) Schematic of the experimental setup. A monochromatic hard x-ray beam is focused by KB mirrors onto a WG, which acts as a spatial and coherence filter. The object $O$ is positioned in the divergent beam at ${z_{{\rm{Ob}}}}$. The different radiation cones indicate the WG illumination and the scattered photons. The detector D is positioned at ${z_{{\rm{De}}}}$.

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In this way, we extend the classical work [31,32] by a few important modifications, which result in significant improvements of the image quality: (1) We exchange the Fresnel zone plate (FZP) by an x-ray waveguide (WG) optic, generating a compact source spot at the WG exit (${z_{{\rm{WG}}}}$) but a highly curved wavefront at ${z_{{\rm{Ob}}}}$. (2) We reconstruct the probe $P$ by ptychography before the single-frame acquisition of the object $O$. (3) We use a modified reconstruction scheme that is not based on subtraction of the probe. We show that the usual sampling constraints can be overcome, i.e., the FOV at ${z_{{\rm{Ob}}}}$ can be larger than the critical value calculated from the detector pixel size ${\Delta _{{\rm{px}}}}$ according to the classical oversampling criterion [33]. At the same time, we preserve the main advantage of the keyhole concept in achieving robust phase retrieval up to high spatial frequencies without restrictive object constraints. We demonstrate the method using a lithographic test pattern imaged at a (half-period) resolution of $\Delta = 11.2\;{\rm{nm}}$ and with FOV of $13 \times 13\;\unicode{x00B5} {{\rm{m}}^2}$, within a single frame acquisition of down to 0.2 s illumination time. The potential for biological samples at subcellular level is shown by imaging a cardiac cell (cardiomyocyte) with a FOV of $50 \times 50\;\unicode{x00B5} {{\rm{m}}^2}$. The dose efficiency of holographic imaging, which does not require any optical elements between object and detector, has already been stressed in a recent study [15]. Here we further improve the dose efficiency by using single photon counting pixel detectors with high quantum efficiency. The use of photon counting detectors with large pixel size becomes possible since reconstruction is neither limited by the oversampling criterion as in CDI or ptychography nor by the demagnified pixel size as in classical holography.

B. Experimental Setup

The experiments were performed using the GINIX instrument [35] at the coherence beamline P10 of the PETRA III storage ring (Hamburg, Germany, in Supplement 1). The undulator beam was monochromatized [$Si(111)$ channel cut] to a photon energy of $E = 8\;{\rm{keV}}$ and focused by Kirkpatrick–Baez (KB) mirrors to about 300 nm in horizontal and vertical direction. A two-dimensional x-ray silicon WG was placed in the focal plane of the mirrors [see Fig. 1(b)] in order to reduce the source spot size and filter the coherence [36]. Objects were placed into the divergent wave field exiting the WG at ${z_{{\rm{Ob}}}}$, and the coherent diffraction pattern with holographic and CDI components was recorded with a single photon counting pixel detector (Eiger 4M, Dectris Ltd. Switzerland) positioned at ${z_{{\rm{De}}}} = 5.1\;{\rm{m}}$. The detector has $2162 \times 2068$ pixels and a pixel size of ${\Delta _{{\rm{px}}}} = 75\;\unicode{x00B5}{\rm m}$. The WG was fabricated in silicon using electron-beam lithography and reactive ion etching (Eulitha, Switzerland) and was subsequently capped by wafer bonding in a clean room environment. The WG channels with a cross section of $89 \times 115\;{\rm{n}}{{\rm{m}}^2}$ (${\rm{horizontal}} \times {\rm{vertical}}$) were cut to an optical length of 1 mm. The measured WG exit flux was $1.3 \times {10^9}\;{\rm{photons/s}}$.

3. RESULTS AND DISCUSSION

First, we determine the complex-valued wave field of the probe $P$ at ${z_{{\rm{Ob}}}}$ by a ptychographic reconstruction using the ePIE algorithm [37,38]. A tantalum test object (NTT, Japan) including a Siemens star pattern with 50 nm smallest feature size was positioned and scanned in the WG beam. The ptychographic scan was performed with $16 \times 16$ scan points at ${z_{{\rm{Ob}}}} = 1.2\;{\rm{mm}}$, a step size of 0.2 µm, and 0.2 s acquisition time per frame. The resulting probe $P$ and phase objects are shown in Figs. 2(a) and 2(b). For comparison the probe is then backpropagated to its source plane at the WG exit [${z_{{\rm{WG}}}}$ Fig. 2(c)] and compared with the wave field reconstructed from a single empty beam measurement Fig. 2(d) using a support constraint defined by the shape of the WG exit. The resulting source sizes (FWHM) are Fig. 2(c) $29.1 \times 31.7\;{\rm{n}}{{\rm{m}}^2}$ and Fig. 2(d) $27.7 \times 27.3\;{\rm{n}}{{\rm{m}}^2}$, which is significantly smaller than the geometric channel dimensions of $89 \times 115\;{\rm{n}}{{\rm{m}}^2}$. The smaller WG source size results from multimodal interference and is in line with finite difference (FD) simulations [39] shown in Fig. 2(e). The FD simulations yield to a FWHM of $31.8 \times 34.6\;{\rm{n}}{{\rm{m}}^2}$ for the given parameters. For further details see Supplement 1.

 figure: Fig. 2.

Fig. 2. Ptychographic reconstruction of (a) the probe $P$ amplitude and phase and (b) the object’s $O$ phase with a pixel size ${\delta _{{\rm{px}}}} = 5.2\;{\rm{nm}}$. The inner radius of the Siemens star pattern corresponds to spokes with 50 nm feature size (see zoom in inset). (c)–(e) Intensity distributions at the WG exit, superimposed with a SEM image of the WG exit surface: (c) ptychographic reconstruction, (d) single-shot reconstruction by the RAAR algorithm using a support constraint [34], and (e) FD-simulated intensity distribution. The source sizes (FWHM) are (c) $29.1 \times 31.7\;{\rm{n}}{{\rm{m}}^2}$, (d) $27.7 \times 27.3\;{\rm{n}}{{\rm{m}}^2}$, and (e) $31.8 \times 34.6\;{\rm{n}}{{\rm{m}}^2}$. Scale bars: (a), (b) 1 µm; (c)–(e) 100 nm.

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Next we use the predetermined probe reconstructed by ptychography $P$ [shown in Fig. 2(a)] at ${z_{{\rm{Ob}}}}$ to phase single-shot acquisitions of an object $O$ illuminated by $P$. Here, no further scanning is required, and in particular there is no overlap constraint as in ptychography. Neither do we require compactness of the object function $O$ as in CDI. Instead, the only constraint that we use is the amplitude of the exit wave, which we constrain to $|PO| = |P|$ for a pure phase object. This is well justified for the objects of interest here (nanostructures, biological cells) at the given photon energy. More generally, other constraints such as homogeneous object (i.e., coupled phase and amplitude), sparsity, or simple range constraints (positive definiteness of absorption) can already be sufficient; see the simulation for the range constraint $|PO| \le |P|$ provided in the Supplement 1. Importantly, the highly curved illumination enables reconstruction beyond the classical oversampling criterion $o = {z_{{\rm{De}}}}\lambda /({\Delta _{{\rm{px}}}}L) \ge 2$ [33], i.e., the illuminated area on the object can be larger than $L$ calculated from $o = 2$. Note that this is not in contradiction to the Nyquist–Shannon theorem due to the holographic nature of the image formation.

 figure: Fig. 3.

Fig. 3. Inline x-ray holography reconstructions of a test pattern at ${z_{{\rm{Ob}}}} = 1.2\;{\rm{mm}}$ with the detector at ${z_{{\rm{De}}}} = 5.1\;{\rm{m}}$. (a) Reconstructed Siemens star with ${\delta _{{\rm{px}}}} = 5.2\;{\rm{nm}}$ and 1 s acquisition time. The inner radius of the Siemens star pattern corresponds to spokes with 50 nm feature size (see zoom in inset). (b) The corresponding detector image. Reconstruction of a 50 nm lines and spaces structure. (c) Split-view comparing reconstruction using the novel approach (${\delta _{{\rm{px}}}} = 5.2\;{\rm{nm}}$) with a reconstruction by empty beam division and CTF ($\delta _{{\rm{px}}}^{\rm{M}} = 17.3\;{\rm{nm}}$). Both use the same data, recorded with an acquisition time of 2 s. (d) Split view comparing the reconstruction from data with 2 to 0.2 s acquisition time, in a region of interest (ROI) of $0.5 \times 0.5\;\unicode{x00B5} {{\rm{m}}^2}$. (e) FRC analysis of this ROI, correlating reconstructions of two independent measurements with an acquisition time of 2 s each, indicating a resolution (half-period) of $\Delta = 11.2\;{\rm{nm}}$. Scale bars: (a) 1 µm; (b) $100{\Delta _{{\rm{px}}}}$ corresponding to a scattering vector $q = 0.06\;{\rm{n}}{{\rm{m}}^{- 1}}$; (c), (d) 250 nm.

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Figure 3(a) proves this claim by presenting the reconstruction of a Siemens star from a single frame acquisition with 1 s exposure time Fig. 3(b), recorded with a ${\rm{FOV}} = 13 \times 13\;\unicode{x00B5} {{\rm{m}}^2}$. The spokes of the Siemens star are fully resolved in a region of $8.2 \times 8.2\;\unicode{x00B5} {{\rm{m}}^2}$, corresponding to $o \simeq 1.3$. The high quality of the full-field image is demonstrated by the magnified region in the inset of Fig. 3(a). Note that the decay of the illumination function results in a decrease of the signal-to-noise ratio toward the image boundaries. Surprisingly, even in regions that receive very low flux, the object can still be recognized, albeit with some artefacts (for further details see Supplement 1). The smallest features of the pattern are the 50 nm stripes in the center, which are clearly resolved. The high resolution is achieved here by photons diffracted by the object to large angles and detected out of the central radiation cone. At the same time, a decisive advantage over standard far-field diffraction in (nearly) plane wave illumination as typical for CDI or ptychography is the fact that low and moderate spatial frequencies are phased from the in-line holographic signal. Further, we could show that a probe reconstruction of a single empty beam image is sufficient, given the compactness of the beam in the WG exit plane; see Supplement 1.

In order to further quantify the resolution, we have analyzed the reconstruction of a pattern with 50 nm (half-period) lines and spaces; see Figs. 3(c)–3(e). The reconstruction by the presented method shows higher resolution and image quality than the conventional reconstruction by the contrast-transfer-function (CTF) approach [18] after empty-beam division. Note that the pixel size ${\delta _{{\rm{px}}}} = 5.2\;{\rm{nm}}$ in the new method is determined by the detector NA and no longer by the geometrically magnified pixel size $\delta _{{\rm{px}}}^{\rm{M}} = {\Delta _{{\rm{px}}}}/M = 17.3\;{\rm{nm}}$ as in the conventional reconstruction. The resolution of the holographic reconstruction was determined by Fourier ring correlation (FRC) [40], correlating the reconstructions of two different data sets, each with independent ptychographic reconstructions of $P$. The FRC indicates a resolution (half-period) of $\Delta = 11.2\;{\rm{nm}}$ [Fig. 3(e)]. Further, we include a reconstruction from data with a shorter acquisition of 0.2 s, which still shows well-resolved lines and spaces but with slightly more noise [Fig. 3(d)].

Next, the method was used to image an adult murine cardiomyocyte. The cardiac cells were isolated by dissociation of healthy mouse hearts, chemically fixed, dispersed on a silicon nitride window, plunge frozen and freeze dried [41]. The cells were imaged using the same setup described before. $P$ was reconstructed from a ptychographic scan (of the same object) at ${z_{{\rm{Ob}}}} = 3.9\;{\rm{mm}}$ with $15 \times 12$ scan points of 1 s acquisition time and ${\delta _{{\rm{px}}}} = 10.2\;{\rm{nm}}$. Figure 4 presents reconstructions from single shot for two different defocus planes. The larger defocus distance (${z_{{\rm{Ob}}}} = 9.3\;{\rm{mm}}$) images shown in Fig. 4(a) result in a FOV of more than $50 \times 50\;\unicode{x00B5} {{\rm{m}}^2}$. The flux density in the object plane was $1.4 \times {10^6}\;{\rm{photons}}/\unicode{x00B5} {{\rm{m}}^2}$, corresponding to a dose of $1.6 \times {10^4}\;{\rm{Gy}}$. Subcellular structures such as the nuclei, mitochondria and microfibrils, and the sarcomeric architecture can be clearly identified. The brighter stripes perpendicular to the orientation of the cell’s main axis exhibit a smaller phase shift/lower electron density and can be associated with the M-lines of the sarcomere. In Fig. 4(b), the same cell was recorded closer to the WG at ${z_{{\rm{Ob}}}} = 3.9\;{\rm{mm}}$. As a result of the increased flux density ($8 \times {10^6}\;{\rm{photons}}/\unicode{x00B5} {{\rm{m}}^2}$), the subcellular structure is now more clearly resolved, and the electron dense z-discs within the individual sarcomeres become visible [Fig. 4(b), red square]. The thickness of the z-disc was determined to be 105 nm (FWHM), in agreement with [42].

 figure: Fig. 4.

Fig. 4. (a) In-line x-ray holography reconstruction using our novel approach of an entire cardiomyocyte at ${z_{{\rm{Ob}}}} = 9.3\;{\rm{mm}}$, covered by two exposures at different sample positions. The combined FOV is $103 \times 49\;{{\unicode{x00B5}}}{{\rm{m}}^2}$ with ${\delta _{{\rm{px}}}} = 10.2\;{\rm{nm}}$ (for comparison: $\delta _{{\rm{px}}}^{\rm{M}} = 136.5\;{\rm{nm}}$). The inset shows an optical microscope image of the cell. The x-ray FOV is marked by a dashed white rectangle. (b) Additional reconstruction measured at ${z_{{\rm{Ob}}}} = 3.9\;{\rm{mm}}$ with ${\delta _{{\rm{px}}}} = 10.2\;{\rm{nm}}$ (for comparison: $\delta _{{\rm{px}}}^{\rm{M}} = 57.3\;{\rm{nm}}$). Position marked in (a) by solid white squares. Subcellular structures can be identified, e.g., the z-disc (red square). The line profile through the z-disc has an FWHM of 105 nm. Scale bars: (a) 10 µm, (b) 2.5 µm, and insets 50 nm. A comparison of our novel in-line X-ray holography approach and ptychography is given in Supplement 1.

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4. CONCLUSION

In conclusion, we have presented a new approach to image extended specimen with full-field single-shot x-ray holography. In contrast to current x-ray holography, this approach offers super-resolution with respect to the illuminating NA. This was demonstrated by imaging a test structure with a resolution of 11.2 nm using a waveguide source size of 30.4 nm (FWHM) as shown in Fig. 3(e). In addition, the limitation of an effective pixel size by geometric magnification, as in in-line holography, does not apply. At the same time, the object’s FOV can be larger than in CDI. In other words, with respect to the usual in-line holography approach based on the Fresnel scaling theorem and empty beam division, we observe a super-resolution effect (resolution better than NA of illuminating beam). With respect to CDI, we achieve subsampling capability. This is a result of the particular design of the illumination, resulting in a diffraction pattern with the holographic contributions in the center and the coherent diffraction contributions in the annulus outside the primary beam. Note that the holographic pattern within the primary beam does not obey Friedel symmetry, i.e., even for weak phase objects the diffraction pattern is not centrosymmetric due to the curvature of the probe. In contrast to plane wave illumination, this “phase-structured” illumination results in enhanced “diversity” and less-redundant measurements. Altogether, this opens a brilliant perspective for high-resolution and dose-effective x-ray tomography, well suited for the current upgrades of synchrotron radiation sources to diffraction-limited storage rings.

Funding

Bundesministerium für Bildung und Forschung (BMBF) (05K19MG2); German Science Foundation (SFB1456/C03); Max Planck School of Photonics.

Acknowledgment

We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities. We thank Michael Sprung and Fabian Westermeier for assistance in using beamline P10 at PETRA III. We thank Mike Kanbach for WG fabrication and Bastian Hartmann for engineering help. We acknowledge financial support by: German Federal Ministry of Education and Research (BMBF) through grant No. 05K19MG2, the German Science Foundation through grand SFB1456/C03, and the Max Planck School of Photonics supported by BMBF, Max Planck Society, and Fraunhofer Society.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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6. M. Holler, M. Odstrcil, M. Guizar-Sicairos, M. Lebugle, E. Müller, S. Finizio, G. Tinti, C. David, J. Zusman, W. Unglaub, O. Bunk, J. Raabe, A. F. J. Levi, and G. Aeppli, “Three-dimensional imaging of integrated circuits with macro- to nanoscale zoom,” Nat. Electron. 2, 464–470 (2019). [CrossRef]  

7. J. Lim, Y. Li, D. H. Alsem, H. So, S. C. Lee, P. Bai, D. A. Cogswell, X. Liu, N. Jin, Y.-S. Yu, N. J. Salmon, D. A. Shapiro, M. Z. Bazant, T. Tyliszczak, and W. C. Chueh, “Origin and hysteresis of lithium compositional spatiodynamics within battery primary particles,” Science 353, 566–571 (2016). [CrossRef]  

8. K. Ayyer, P. L. Xavier, J. Bielecki, Z. Shen, B. J. Daurer, A. K. Samanta, S. Awel, R. Bean, A. Barty, M. Bergemann, T. Ekeberg, A. D. Estillore, H. Fangohr, K. Giewekemeyer, M. S. Hunter, M. Karnevskiy, R. A. Kirian, H. Kirkwood, Y. Kim, J. Koliyadu, H. Lange, R. Letrun, J. Lübke, T. Michelat, A. J. Morgan, N. Roth, T. Sato, M. Sikorski, F. Schulz, J. C. H. Spence, P. Vagovic, T. Wollweber, L. Worbs, O. Yefanov, Y. Zhuang, F. R. N. C. Maia, D. A. Horke, J. Küpper, N. D. Loh, A. P. Mancuso, and H. N. Chapman, “3D diffractive imaging of nanoparticle ensembles using an x-ray laser,” Optica 8, 15–23 (2021). [CrossRef]  

9. M. O. Wiedorn, D. Oberthür, and R. Bean, et al., “Megahertz serial crystallography,” Nat. Commun. 9, 1–11 (2018). [CrossRef]  

10. A. T. Kuan, J. S. Phelps, L. A. Thomas, T. M. Nguyen, J. Han, C.-L. Chen, A. W. Azevedo, J. C. Tuthill, J. Funke, P. Cloetens, A. Pacureanu, and W.-C. A. Lee, “Dense neuronal reconstruction through X-ray holographic nano-tomography,” Tech. Rep. (2020).

11. L. C. P. Croton, K. S. Morgan, D. M. Paganin, L. T. Kerr, M. J. Wallace, K. J. Crossley, S. L. Miller, N. Yagi, K. Uesugi, S. B. Hooper, and M. J. Kitchen, “In situ phase contrast X-ray brain CT,” Sci. Rep. 8, 1–12 (2018). [CrossRef]  

12. F. Garca-Moreno, P. H. Kamm, T. R. Neu, F. Bülk, R. Mokso, C. M. Schlepütz, M. Stampanoni, and J. Banhart, “Using X-ray tomoscopy to explore the dynamics of foaming metal,” Nat. Commun. 10, 1–9 (2019). [CrossRef]  

13. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996). [CrossRef]  

14. R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007). [CrossRef]  

15. M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015). [CrossRef]  

16. S. Maretzke, “A uniqueness result for propagation-based phase contrast imaging from a single measurement,” Inverse Prob. 31, 065003 (2015). [CrossRef]  

17. J. Hagemann and T. Salditt, “Coherence-resolution relationship in holographic and coherent diffractive imaging,” Opt. Express 26, 242–253 (2018). [CrossRef]  

18. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999). [CrossRef]  

19. J. Hagemann, M. Töpperwien, and T. Salditt, “Phase retrieval for near-field X-ray imaging beyond linearisation or compact support,” Appl. Phys. Lett. 113, 041109 (2018). [CrossRef]  

20. A. Khimchenko, H. Deyhle, G. Schulz, G. Schweighauser, J. Hench, N. Chicherova, C. Bikis, S. E. Hieber, and B. Müller, “Extending two-dimensional histology into the third dimension through conventional micro computed tomography,” Neuroimage 139, 26–36 (2016). [CrossRef]  

21. M. Töpperwien, F. van der Meer, C. Stadelmann, and T. Salditt, “Three-dimensional virtual histology of human cerebellum by X-ray phase-contrast tomography,” Proc. Natl. Acad. Sci. USA 115, 6940–6945 (2018). [CrossRef]  

22. A. Cedola, A. Bravin, I. Bukreeva, M. Fratini, A. Pacureanu, A. Mittone, L. Massimi, P. Cloetens, P. Coan, G. Campi, R. Spanoò, F. Brun, V. Grigoryev, V. Petrosino, C. Venturi, M. Mastrogiacomo, N. K. de Rosbo, and A. Uccelli, “X-Ray phase contrast tomography reveals early vascular alterations and neuronal loss in a multiple sclerosis model,” Sci. Rep. 7, 1–11 (2017). [CrossRef]  

23. A. Khimchenko, C. Bikis, A. Pacureanu, S. E. Hieber, P. Thalmann, H. Deyhle, G. Schweighauser, J. Hench, S. Frank, M. Müller-Gerbl, G. Schulz, P. Cloetens, and B. Müller, “Hard x-ray nanoholotomography: large-scale, label-free, 3D neuroimaging beyond optical limit,” Adv. Sci. 5, 1700694 (2018). [CrossRef]  

24. S. M. Walker, D. A. Schwyn, R. Mokso, M. Wicklein, T. Müller, M. Doube, M. Stampanoni, H. G. Krapp, and G. K. Taylor, “In vivo time-resolved microtomography reveals the mechanics of the blowfly flight motor,” PLoS Biol. 12, e1001823 (2014). [CrossRef]  

25. F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12, 9–17 (2018). [CrossRef]  

26. J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015). [CrossRef]  

27. Y. H. Lo, L. Zhao, M. Gallagher-Jones, A. Rana, J. J. Lodico, W. Xiao, B. C. Regan, and J. Miao, “In situ coherent diffractive imaging,” Nat. Commun. 9, 1–10 (2018). [CrossRef]  

28. D. Paganin, Coherent X-ray Optics (Oxford University Press on Demand, 2006), Vol. 6.

29. C. Homann, T. Hohage, J. Hagemann, A.-L. Robisch, and T. Salditt, “Validity of the empty-beam correction in near-field imaging,” Phys. Rev. A 91, 013821 (2015). [CrossRef]  

30. J. Hagemann, A.-L. Robisch, D. R. Luke, C. Homann, T. Hohage, P. Cloetens, H. Suhonen, and T. Salditt, “Reconstruction of wave front and object for inline holography from a set of detection planes,” Opt. Express 22, 11552–11569 (2014). [CrossRef]  

31. G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006). [CrossRef]  

32. B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008). [CrossRef]  

33. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998). [CrossRef]  

34. S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat. 19, 227–236 (2012). [CrossRef]  

35. S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen Holography Endstation of Beamline P10 at PETRA III/DESY,” in AIP Conference Proceedings (American Institute of Physics, 2011), pp. 96–99.

36. M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011). [CrossRef]  

37. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009). [CrossRef]  

38. A. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4, 736–745 (2017). [CrossRef]  

39. L. Melchior and T. Salditt, “Finite difference methods for stationary and time-dependent X-ray propagation,” Opt. Express 25, 32090–32109 (2017). [CrossRef]  

40. M. van Heel and M. Schatz, “Fourier shell correlation threshold criteria,” J. Struct. Biol. 151, 250–262 (2005). [CrossRef]  

41. M. Reichardt, C. Neuhaus, J.-D. Nicolas, M. Bernhardt, K. Toischer, and T. Salditt, “X-ray structural analysis of single adult cardiomyocytes: tomographic imaging and microdiffraction,” Biophys. J. 119, 1309–1323 (2020). [CrossRef]  

42. D. Frank and N. Frey, “Cardiac Z-disc signaling network*,” J. Biol. Chem. 286, 9897–9904 (2011). [CrossRef]  

References

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  7. J. Lim, Y. Li, D. H. Alsem, H. So, S. C. Lee, P. Bai, D. A. Cogswell, X. Liu, N. Jin, Y.-S. Yu, N. J. Salmon, D. A. Shapiro, M. Z. Bazant, T. Tyliszczak, and W. C. Chueh, “Origin and hysteresis of lithium compositional spatiodynamics within battery primary particles,” Science 353, 566–571 (2016).
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  9. M. O. Wiedorn, D. Oberthür, and R. Bean, et al., “Megahertz serial crystallography,” Nat. Commun. 9, 1–11 (2018).
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  10. A. T. Kuan, J. S. Phelps, L. A. Thomas, T. M. Nguyen, J. Han, C.-L. Chen, A. W. Azevedo, J. C. Tuthill, J. Funke, P. Cloetens, A. Pacureanu, and W.-C. A. Lee, “Dense neuronal reconstruction through X-ray holographic nano-tomography,” Tech. Rep. (2020).
  11. L. C. P. Croton, K. S. Morgan, D. M. Paganin, L. T. Kerr, M. J. Wallace, K. J. Crossley, S. L. Miller, N. Yagi, K. Uesugi, S. B. Hooper, and M. J. Kitchen, “In situ phase contrast X-ray brain CT,” Sci. Rep. 8, 1–12 (2018).
    [Crossref]
  12. F. Garca-Moreno, P. H. Kamm, T. R. Neu, F. Bülk, R. Mokso, C. M. Schlepütz, M. Stampanoni, and J. Banhart, “Using X-ray tomoscopy to explore the dynamics of foaming metal,” Nat. Commun. 10, 1–9 (2019).
    [Crossref]
  13. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
    [Crossref]
  14. R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007).
    [Crossref]
  15. M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
    [Crossref]
  16. S. Maretzke, “A uniqueness result for propagation-based phase contrast imaging from a single measurement,” Inverse Prob. 31, 065003 (2015).
    [Crossref]
  17. J. Hagemann and T. Salditt, “Coherence-resolution relationship in holographic and coherent diffractive imaging,” Opt. Express 26, 242–253 (2018).
    [Crossref]
  18. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
    [Crossref]
  19. J. Hagemann, M. Töpperwien, and T. Salditt, “Phase retrieval for near-field X-ray imaging beyond linearisation or compact support,” Appl. Phys. Lett. 113, 041109 (2018).
    [Crossref]
  20. A. Khimchenko, H. Deyhle, G. Schulz, G. Schweighauser, J. Hench, N. Chicherova, C. Bikis, S. E. Hieber, and B. Müller, “Extending two-dimensional histology into the third dimension through conventional micro computed tomography,” Neuroimage 139, 26–36 (2016).
    [Crossref]
  21. M. Töpperwien, F. van der Meer, C. Stadelmann, and T. Salditt, “Three-dimensional virtual histology of human cerebellum by X-ray phase-contrast tomography,” Proc. Natl. Acad. Sci. USA 115, 6940–6945 (2018).
    [Crossref]
  22. A. Cedola, A. Bravin, I. Bukreeva, M. Fratini, A. Pacureanu, A. Mittone, L. Massimi, P. Cloetens, P. Coan, G. Campi, R. Spanoò, F. Brun, V. Grigoryev, V. Petrosino, C. Venturi, M. Mastrogiacomo, N. K. de Rosbo, and A. Uccelli, “X-Ray phase contrast tomography reveals early vascular alterations and neuronal loss in a multiple sclerosis model,” Sci. Rep. 7, 1–11 (2017).
    [Crossref]
  23. A. Khimchenko, C. Bikis, A. Pacureanu, S. E. Hieber, P. Thalmann, H. Deyhle, G. Schweighauser, J. Hench, S. Frank, M. Müller-Gerbl, G. Schulz, P. Cloetens, and B. Müller, “Hard x-ray nanoholotomography: large-scale, label-free, 3D neuroimaging beyond optical limit,” Adv. Sci. 5, 1700694 (2018).
    [Crossref]
  24. S. M. Walker, D. A. Schwyn, R. Mokso, M. Wicklein, T. Müller, M. Doube, M. Stampanoni, H. G. Krapp, and G. K. Taylor, “In vivo time-resolved microtomography reveals the mechanics of the blowfly flight motor,” PLoS Biol. 12, e1001823 (2014).
    [Crossref]
  25. F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12, 9–17 (2018).
    [Crossref]
  26. J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
    [Crossref]
  27. Y. H. Lo, L. Zhao, M. Gallagher-Jones, A. Rana, J. J. Lodico, W. Xiao, B. C. Regan, and J. Miao, “In situ coherent diffractive imaging,” Nat. Commun. 9, 1–10 (2018).
    [Crossref]
  28. D. Paganin, Coherent X-ray Optics (Oxford University Press on Demand, 2006), Vol. 6.
  29. C. Homann, T. Hohage, J. Hagemann, A.-L. Robisch, and T. Salditt, “Validity of the empty-beam correction in near-field imaging,” Phys. Rev. A 91, 013821 (2015).
    [Crossref]
  30. J. Hagemann, A.-L. Robisch, D. R. Luke, C. Homann, T. Hohage, P. Cloetens, H. Suhonen, and T. Salditt, “Reconstruction of wave front and object for inline holography from a set of detection planes,” Opt. Express 22, 11552–11569 (2014).
    [Crossref]
  31. G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
    [Crossref]
  32. B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
    [Crossref]
  33. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
    [Crossref]
  34. S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat. 19, 227–236 (2012).
    [Crossref]
  35. S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen Holography Endstation of Beamline P10 at PETRA III/DESY,” in AIP Conference Proceedings (American Institute of Physics, 2011), pp. 96–99.
  36. M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
    [Crossref]
  37. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
    [Crossref]
  38. A. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4, 736–745 (2017).
    [Crossref]
  39. L. Melchior and T. Salditt, “Finite difference methods for stationary and time-dependent X-ray propagation,” Opt. Express 25, 32090–32109 (2017).
    [Crossref]
  40. M. van Heel and M. Schatz, “Fourier shell correlation threshold criteria,” J. Struct. Biol. 151, 250–262 (2005).
    [Crossref]
  41. M. Reichardt, C. Neuhaus, J.-D. Nicolas, M. Bernhardt, K. Toischer, and T. Salditt, “X-ray structural analysis of single adult cardiomyocytes: tomographic imaging and microdiffraction,” Biophys. J. 119, 1309–1323 (2020).
    [Crossref]
  42. D. Frank and N. Frey, “Cardiac Z-disc signaling network*,” J. Biol. Chem. 286, 9897–9904 (2011).
    [Crossref]

2021 (1)

2020 (1)

M. Reichardt, C. Neuhaus, J.-D. Nicolas, M. Bernhardt, K. Toischer, and T. Salditt, “X-ray structural analysis of single adult cardiomyocytes: tomographic imaging and microdiffraction,” Biophys. J. 119, 1309–1323 (2020).
[Crossref]

2019 (2)

M. Holler, M. Odstrcil, M. Guizar-Sicairos, M. Lebugle, E. Müller, S. Finizio, G. Tinti, C. David, J. Zusman, W. Unglaub, O. Bunk, J. Raabe, A. F. J. Levi, and G. Aeppli, “Three-dimensional imaging of integrated circuits with macro- to nanoscale zoom,” Nat. Electron. 2, 464–470 (2019).
[Crossref]

F. Garca-Moreno, P. H. Kamm, T. R. Neu, F. Bülk, R. Mokso, C. M. Schlepütz, M. Stampanoni, and J. Banhart, “Using X-ray tomoscopy to explore the dynamics of foaming metal,” Nat. Commun. 10, 1–9 (2019).
[Crossref]

2018 (11)

J. Hagemann and T. Salditt, “Coherence-resolution relationship in holographic and coherent diffractive imaging,” Opt. Express 26, 242–253 (2018).
[Crossref]

M. O. Wiedorn, D. Oberthür, and R. Bean, et al., “Megahertz serial crystallography,” Nat. Commun. 9, 1–11 (2018).
[Crossref]

L. C. P. Croton, K. S. Morgan, D. M. Paganin, L. T. Kerr, M. J. Wallace, K. J. Crossley, S. L. Miller, N. Yagi, K. Uesugi, S. B. Hooper, and M. J. Kitchen, “In situ phase contrast X-ray brain CT,” Sci. Rep. 8, 1–12 (2018).
[Crossref]

S. Bajt, M. Prasciolu, H. Fleckenstein, M. Domaracký, H. N. Chapman, A. J. Morgan, O. Yefanov, M. Messerschmidt, Y. Du, K. T. Murray, V. Mariani, M. Kuhn, S. Aplin, K. Pande, P. Villanueva-Perez, K. Stachnik, J. P. J. Chen, A. Andrejczuk, A. Meents, A. Burkhardt, D. Pennicard, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, and C. E. Hamm, “X-ray focusing with efficient high-NA multilayer Laue lenses,” Light Sci. Appl. 7, 17162 (2018).
[Crossref]

H. Oztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer Laue lenses,” Optica 5, 601–607 (2018).
[Crossref]

P. Villanueva-Perez, B. Pedrini, R. Mokso, P. Vagovic, V. A. Guzenko, S. J. Leake, P. R. Willmott, P. Oberta, C. David, H. N. Chapman, and M. Stampanoni, “Hard x-ray multi-projection imaging for single-shot approaches,” Optica 5, 1521–1524 (2018).
[Crossref]

J. Hagemann, M. Töpperwien, and T. Salditt, “Phase retrieval for near-field X-ray imaging beyond linearisation or compact support,” Appl. Phys. Lett. 113, 041109 (2018).
[Crossref]

M. Töpperwien, F. van der Meer, C. Stadelmann, and T. Salditt, “Three-dimensional virtual histology of human cerebellum by X-ray phase-contrast tomography,” Proc. Natl. Acad. Sci. USA 115, 6940–6945 (2018).
[Crossref]

A. Khimchenko, C. Bikis, A. Pacureanu, S. E. Hieber, P. Thalmann, H. Deyhle, G. Schweighauser, J. Hench, S. Frank, M. Müller-Gerbl, G. Schulz, P. Cloetens, and B. Müller, “Hard x-ray nanoholotomography: large-scale, label-free, 3D neuroimaging beyond optical limit,” Adv. Sci. 5, 1700694 (2018).
[Crossref]

F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12, 9–17 (2018).
[Crossref]

Y. H. Lo, L. Zhao, M. Gallagher-Jones, A. Rana, J. J. Lodico, W. Xiao, B. C. Regan, and J. Miao, “In situ coherent diffractive imaging,” Nat. Commun. 9, 1–10 (2018).
[Crossref]

2017 (5)

A. Cedola, A. Bravin, I. Bukreeva, M. Fratini, A. Pacureanu, A. Mittone, L. Massimi, P. Cloetens, P. Coan, G. Campi, R. Spanoò, F. Brun, V. Grigoryev, V. Petrosino, C. Venturi, M. Mastrogiacomo, N. K. de Rosbo, and A. Uccelli, “X-Ray phase contrast tomography reveals early vascular alterations and neuronal loss in a multiple sclerosis model,” Sci. Rep. 7, 1–11 (2017).
[Crossref]

I. Mohacsi, I. Vartiainen, B. Rösner, M. Guizar-Sicairos, V. A. Guzenko, I. McNulty, R. Winarski, M. V. Holt, and C. David, “Interlaced zone plate optics for hard X-ray imaging in the 10nm range,” Sci. Rep. 7, 1–10 (2017).
[Crossref]

J. Cesar da Silva, A. Pacureanu, Y. Yang, S. Bohic, C. Morawe, R. Barrett, and P. Cloetens, “Efficient concentration of high-energy x-rays for diffraction-limited imaging resolution,” Optica 4, 492–495 (2017).
[Crossref]

A. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4, 736–745 (2017).
[Crossref]

L. Melchior and T. Salditt, “Finite difference methods for stationary and time-dependent X-ray propagation,” Opt. Express 25, 32090–32109 (2017).
[Crossref]

2016 (2)

J. Lim, Y. Li, D. H. Alsem, H. So, S. C. Lee, P. Bai, D. A. Cogswell, X. Liu, N. Jin, Y.-S. Yu, N. J. Salmon, D. A. Shapiro, M. Z. Bazant, T. Tyliszczak, and W. C. Chueh, “Origin and hysteresis of lithium compositional spatiodynamics within battery primary particles,” Science 353, 566–571 (2016).
[Crossref]

A. Khimchenko, H. Deyhle, G. Schulz, G. Schweighauser, J. Hench, N. Chicherova, C. Bikis, S. E. Hieber, and B. Müller, “Extending two-dimensional histology into the third dimension through conventional micro computed tomography,” Neuroimage 139, 26–36 (2016).
[Crossref]

2015 (4)

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref]

C. Homann, T. Hohage, J. Hagemann, A.-L. Robisch, and T. Salditt, “Validity of the empty-beam correction in near-field imaging,” Phys. Rev. A 91, 013821 (2015).
[Crossref]

M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref]

S. Maretzke, “A uniqueness result for propagation-based phase contrast imaging from a single measurement,” Inverse Prob. 31, 065003 (2015).
[Crossref]

2014 (2)

J. Hagemann, A.-L. Robisch, D. R. Luke, C. Homann, T. Hohage, P. Cloetens, H. Suhonen, and T. Salditt, “Reconstruction of wave front and object for inline holography from a set of detection planes,” Opt. Express 22, 11552–11569 (2014).
[Crossref]

S. M. Walker, D. A. Schwyn, R. Mokso, M. Wicklein, T. Müller, M. Doube, M. Stampanoni, H. G. Krapp, and G. K. Taylor, “In vivo time-resolved microtomography reveals the mechanics of the blowfly flight motor,” PLoS Biol. 12, e1001823 (2014).
[Crossref]

2012 (1)

S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat. 19, 227–236 (2012).
[Crossref]

2011 (2)

M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
[Crossref]

D. Frank and N. Frey, “Cardiac Z-disc signaling network*,” J. Biol. Chem. 286, 9897–9904 (2011).
[Crossref]

2009 (1)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref]

2008 (1)

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

2007 (1)

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007).
[Crossref]

2006 (1)

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[Crossref]

2005 (1)

M. van Heel and M. Schatz, “Fourier shell correlation threshold criteria,” J. Struct. Biol. 151, 250–262 (2005).
[Crossref]

1999 (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

1998 (1)

1996 (1)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
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Abbey, B.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Aeppli, G.

M. Holler, M. Odstrcil, M. Guizar-Sicairos, M. Lebugle, E. Müller, S. Finizio, G. Tinti, C. David, J. Zusman, W. Unglaub, O. Bunk, J. Raabe, A. F. J. Levi, and G. Aeppli, “Three-dimensional imaging of integrated circuits with macro- to nanoscale zoom,” Nat. Electron. 2, 464–470 (2019).
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Alsem, D. H.

J. Lim, Y. Li, D. H. Alsem, H. So, S. C. Lee, P. Bai, D. A. Cogswell, X. Liu, N. Jin, Y.-S. Yu, N. J. Salmon, D. A. Shapiro, M. Z. Bazant, T. Tyliszczak, and W. C. Chueh, “Origin and hysteresis of lithium compositional spatiodynamics within battery primary particles,” Science 353, 566–571 (2016).
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Andrejczuk, A.

S. Bajt, M. Prasciolu, H. Fleckenstein, M. Domaracký, H. N. Chapman, A. J. Morgan, O. Yefanov, M. Messerschmidt, Y. Du, K. T. Murray, V. Mariani, M. Kuhn, S. Aplin, K. Pande, P. Villanueva-Perez, K. Stachnik, J. P. J. Chen, A. Andrejczuk, A. Meents, A. Burkhardt, D. Pennicard, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, and C. E. Hamm, “X-ray focusing with efficient high-NA multilayer Laue lenses,” Light Sci. Appl. 7, 17162 (2018).
[Crossref]

Aplin, S.

S. Bajt, M. Prasciolu, H. Fleckenstein, M. Domaracký, H. N. Chapman, A. J. Morgan, O. Yefanov, M. Messerschmidt, Y. Du, K. T. Murray, V. Mariani, M. Kuhn, S. Aplin, K. Pande, P. Villanueva-Perez, K. Stachnik, J. P. J. Chen, A. Andrejczuk, A. Meents, A. Burkhardt, D. Pennicard, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, and C. E. Hamm, “X-ray focusing with efficient high-NA multilayer Laue lenses,” Light Sci. Appl. 7, 17162 (2018).
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Awel, S.

Ayyer, K.

Azevedo, A. W.

A. T. Kuan, J. S. Phelps, L. A. Thomas, T. M. Nguyen, J. Han, C.-L. Chen, A. W. Azevedo, J. C. Tuthill, J. Funke, P. Cloetens, A. Pacureanu, and W.-C. A. Lee, “Dense neuronal reconstruction through X-ray holographic nano-tomography,” Tech. Rep. (2020).

Bai, P.

J. Lim, Y. Li, D. H. Alsem, H. So, S. C. Lee, P. Bai, D. A. Cogswell, X. Liu, N. Jin, Y.-S. Yu, N. J. Salmon, D. A. Shapiro, M. Z. Bazant, T. Tyliszczak, and W. C. Chueh, “Origin and hysteresis of lithium compositional spatiodynamics within battery primary particles,” Science 353, 566–571 (2016).
[Crossref]

Bajt, S.

S. Bajt, M. Prasciolu, H. Fleckenstein, M. Domaracký, H. N. Chapman, A. J. Morgan, O. Yefanov, M. Messerschmidt, Y. Du, K. T. Murray, V. Mariani, M. Kuhn, S. Aplin, K. Pande, P. Villanueva-Perez, K. Stachnik, J. P. J. Chen, A. Andrejczuk, A. Meents, A. Burkhardt, D. Pennicard, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, and C. E. Hamm, “X-ray focusing with efficient high-NA multilayer Laue lenses,” Light Sci. Appl. 7, 17162 (2018).
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Banhart, J.

F. Garca-Moreno, P. H. Kamm, T. R. Neu, F. Bülk, R. Mokso, C. M. Schlepütz, M. Stampanoni, and J. Banhart, “Using X-ray tomoscopy to explore the dynamics of foaming metal,” Nat. Commun. 10, 1–9 (2019).
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Barrett, R.

Bartels, M.

M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref]

S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat. 19, 227–236 (2012).
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S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen Holography Endstation of Beamline P10 at PETRA III/DESY,” in AIP Conference Proceedings (American Institute of Physics, 2011), pp. 96–99.

Barty, A.

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

Bazant, M. Z.

J. Lim, Y. Li, D. H. Alsem, H. So, S. C. Lee, P. Bai, D. A. Cogswell, X. Liu, N. Jin, Y.-S. Yu, N. J. Salmon, D. A. Shapiro, M. Z. Bazant, T. Tyliszczak, and W. C. Chueh, “Origin and hysteresis of lithium compositional spatiodynamics within battery primary particles,” Science 353, 566–571 (2016).
[Crossref]

Bean, R.

Bergemann, M.

Bernhardt, M.

M. Reichardt, C. Neuhaus, J.-D. Nicolas, M. Bernhardt, K. Toischer, and T. Salditt, “X-ray structural analysis of single adult cardiomyocytes: tomographic imaging and microdiffraction,” Biophys. J. 119, 1309–1323 (2020).
[Crossref]

Bielecki, J.

Bikis, C.

A. Khimchenko, C. Bikis, A. Pacureanu, S. E. Hieber, P. Thalmann, H. Deyhle, G. Schweighauser, J. Hench, S. Frank, M. Müller-Gerbl, G. Schulz, P. Cloetens, and B. Müller, “Hard x-ray nanoholotomography: large-scale, label-free, 3D neuroimaging beyond optical limit,” Adv. Sci. 5, 1700694 (2018).
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A. Khimchenko, H. Deyhle, G. Schulz, G. Schweighauser, J. Hench, N. Chicherova, C. Bikis, S. E. Hieber, and B. Müller, “Extending two-dimensional histology into the third dimension through conventional micro computed tomography,” Neuroimage 139, 26–36 (2016).
[Crossref]

Bohic, S.

Bravin, A.

A. Cedola, A. Bravin, I. Bukreeva, M. Fratini, A. Pacureanu, A. Mittone, L. Massimi, P. Cloetens, P. Coan, G. Campi, R. Spanoò, F. Brun, V. Grigoryev, V. Petrosino, C. Venturi, M. Mastrogiacomo, N. K. de Rosbo, and A. Uccelli, “X-Ray phase contrast tomography reveals early vascular alterations and neuronal loss in a multiple sclerosis model,” Sci. Rep. 7, 1–11 (2017).
[Crossref]

Brun, F.

A. Cedola, A. Bravin, I. Bukreeva, M. Fratini, A. Pacureanu, A. Mittone, L. Massimi, P. Cloetens, P. Coan, G. Campi, R. Spanoò, F. Brun, V. Grigoryev, V. Petrosino, C. Venturi, M. Mastrogiacomo, N. K. de Rosbo, and A. Uccelli, “X-Ray phase contrast tomography reveals early vascular alterations and neuronal loss in a multiple sclerosis model,” Sci. Rep. 7, 1–11 (2017).
[Crossref]

Buffière, J.-Y.

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007).
[Crossref]

Bukreeva, I.

A. Cedola, A. Bravin, I. Bukreeva, M. Fratini, A. Pacureanu, A. Mittone, L. Massimi, P. Cloetens, P. Coan, G. Campi, R. Spanoò, F. Brun, V. Grigoryev, V. Petrosino, C. Venturi, M. Mastrogiacomo, N. K. de Rosbo, and A. Uccelli, “X-Ray phase contrast tomography reveals early vascular alterations and neuronal loss in a multiple sclerosis model,” Sci. Rep. 7, 1–11 (2017).
[Crossref]

Bülk, F.

F. Garca-Moreno, P. H. Kamm, T. R. Neu, F. Bülk, R. Mokso, C. M. Schlepütz, M. Stampanoni, and J. Banhart, “Using X-ray tomoscopy to explore the dynamics of foaming metal,” Nat. Commun. 10, 1–9 (2019).
[Crossref]

Bunk, O.

M. Holler, M. Odstrcil, M. Guizar-Sicairos, M. Lebugle, E. Müller, S. Finizio, G. Tinti, C. David, J. Zusman, W. Unglaub, O. Bunk, J. Raabe, A. F. J. Levi, and G. Aeppli, “Three-dimensional imaging of integrated circuits with macro- to nanoscale zoom,” Nat. Electron. 2, 464–470 (2019).
[Crossref]

Burkhardt, A.

S. Bajt, M. Prasciolu, H. Fleckenstein, M. Domaracký, H. N. Chapman, A. J. Morgan, O. Yefanov, M. Messerschmidt, Y. Du, K. T. Murray, V. Mariani, M. Kuhn, S. Aplin, K. Pande, P. Villanueva-Perez, K. Stachnik, J. P. J. Chen, A. Andrejczuk, A. Meents, A. Burkhardt, D. Pennicard, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, and C. E. Hamm, “X-ray focusing with efficient high-NA multilayer Laue lenses,” Light Sci. Appl. 7, 17162 (2018).
[Crossref]

Campi, G.

A. Cedola, A. Bravin, I. Bukreeva, M. Fratini, A. Pacureanu, A. Mittone, L. Massimi, P. Cloetens, P. Coan, G. Campi, R. Spanoò, F. Brun, V. Grigoryev, V. Petrosino, C. Venturi, M. Mastrogiacomo, N. K. de Rosbo, and A. Uccelli, “X-Ray phase contrast tomography reveals early vascular alterations and neuronal loss in a multiple sclerosis model,” Sci. Rep. 7, 1–11 (2017).
[Crossref]

Cedola, A.

A. Cedola, A. Bravin, I. Bukreeva, M. Fratini, A. Pacureanu, A. Mittone, L. Massimi, P. Cloetens, P. Coan, G. Campi, R. Spanoò, F. Brun, V. Grigoryev, V. Petrosino, C. Venturi, M. Mastrogiacomo, N. K. de Rosbo, and A. Uccelli, “X-Ray phase contrast tomography reveals early vascular alterations and neuronal loss in a multiple sclerosis model,” Sci. Rep. 7, 1–11 (2017).
[Crossref]

Cesar da Silva, J.

Chapman, H. N.

K. Ayyer, P. L. Xavier, J. Bielecki, Z. Shen, B. J. Daurer, A. K. Samanta, S. Awel, R. Bean, A. Barty, M. Bergemann, T. Ekeberg, A. D. Estillore, H. Fangohr, K. Giewekemeyer, M. S. Hunter, M. Karnevskiy, R. A. Kirian, H. Kirkwood, Y. Kim, J. Koliyadu, H. Lange, R. Letrun, J. Lübke, T. Michelat, A. J. Morgan, N. Roth, T. Sato, M. Sikorski, F. Schulz, J. C. H. Spence, P. Vagovic, T. Wollweber, L. Worbs, O. Yefanov, Y. Zhuang, F. R. N. C. Maia, D. A. Horke, J. Küpper, N. D. Loh, A. P. Mancuso, and H. N. Chapman, “3D diffractive imaging of nanoparticle ensembles using an x-ray laser,” Optica 8, 15–23 (2021).
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S. Bajt, M. Prasciolu, H. Fleckenstein, M. Domaracký, H. N. Chapman, A. J. Morgan, O. Yefanov, M. Messerschmidt, Y. Du, K. T. Murray, V. Mariani, M. Kuhn, S. Aplin, K. Pande, P. Villanueva-Perez, K. Stachnik, J. P. J. Chen, A. Andrejczuk, A. Meents, A. Burkhardt, D. Pennicard, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, and C. E. Hamm, “X-ray focusing with efficient high-NA multilayer Laue lenses,” Light Sci. Appl. 7, 17162 (2018).
[Crossref]

P. Villanueva-Perez, B. Pedrini, R. Mokso, P. Vagovic, V. A. Guzenko, S. J. Leake, P. R. Willmott, P. Oberta, C. David, H. N. Chapman, and M. Stampanoni, “Hard x-ray multi-projection imaging for single-shot approaches,” Optica 5, 1521–1524 (2018).
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J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
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Chen, C.-L.

A. T. Kuan, J. S. Phelps, L. A. Thomas, T. M. Nguyen, J. Han, C.-L. Chen, A. W. Azevedo, J. C. Tuthill, J. Funke, P. Cloetens, A. Pacureanu, and W.-C. A. Lee, “Dense neuronal reconstruction through X-ray holographic nano-tomography,” Tech. Rep. (2020).

Chen, J. P. J.

S. Bajt, M. Prasciolu, H. Fleckenstein, M. Domaracký, H. N. Chapman, A. J. Morgan, O. Yefanov, M. Messerschmidt, Y. Du, K. T. Murray, V. Mariani, M. Kuhn, S. Aplin, K. Pande, P. Villanueva-Perez, K. Stachnik, J. P. J. Chen, A. Andrejczuk, A. Meents, A. Burkhardt, D. Pennicard, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, and C. E. Hamm, “X-ray focusing with efficient high-NA multilayer Laue lenses,” Light Sci. Appl. 7, 17162 (2018).
[Crossref]

Chicherova, N.

A. Khimchenko, H. Deyhle, G. Schulz, G. Schweighauser, J. Hench, N. Chicherova, C. Bikis, S. E. Hieber, and B. Müller, “Extending two-dimensional histology into the third dimension through conventional micro computed tomography,” Neuroimage 139, 26–36 (2016).
[Crossref]

Chu, Y. S.

S. Bajt, M. Prasciolu, H. Fleckenstein, M. Domaracký, H. N. Chapman, A. J. Morgan, O. Yefanov, M. Messerschmidt, Y. Du, K. T. Murray, V. Mariani, M. Kuhn, S. Aplin, K. Pande, P. Villanueva-Perez, K. Stachnik, J. P. J. Chen, A. Andrejczuk, A. Meents, A. Burkhardt, D. Pennicard, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, and C. E. Hamm, “X-ray focusing with efficient high-NA multilayer Laue lenses,” Light Sci. Appl. 7, 17162 (2018).
[Crossref]

H. Oztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer Laue lenses,” Optica 5, 601–607 (2018).
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Chueh, W. C.

J. Lim, Y. Li, D. H. Alsem, H. So, S. C. Lee, P. Bai, D. A. Cogswell, X. Liu, N. Jin, Y.-S. Yu, N. J. Salmon, D. A. Shapiro, M. Z. Bazant, T. Tyliszczak, and W. C. Chueh, “Origin and hysteresis of lithium compositional spatiodynamics within battery primary particles,” Science 353, 566–571 (2016).
[Crossref]

Clark, J. N.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Cloetens, P.

A. Khimchenko, C. Bikis, A. Pacureanu, S. E. Hieber, P. Thalmann, H. Deyhle, G. Schweighauser, J. Hench, S. Frank, M. Müller-Gerbl, G. Schulz, P. Cloetens, and B. Müller, “Hard x-ray nanoholotomography: large-scale, label-free, 3D neuroimaging beyond optical limit,” Adv. Sci. 5, 1700694 (2018).
[Crossref]

A. Cedola, A. Bravin, I. Bukreeva, M. Fratini, A. Pacureanu, A. Mittone, L. Massimi, P. Cloetens, P. Coan, G. Campi, R. Spanoò, F. Brun, V. Grigoryev, V. Petrosino, C. Venturi, M. Mastrogiacomo, N. K. de Rosbo, and A. Uccelli, “X-Ray phase contrast tomography reveals early vascular alterations and neuronal loss in a multiple sclerosis model,” Sci. Rep. 7, 1–11 (2017).
[Crossref]

J. Cesar da Silva, A. Pacureanu, Y. Yang, S. Bohic, C. Morawe, R. Barrett, and P. Cloetens, “Efficient concentration of high-energy x-rays for diffraction-limited imaging resolution,” Optica 4, 492–495 (2017).
[Crossref]

J. Hagemann, A.-L. Robisch, D. R. Luke, C. Homann, T. Hohage, P. Cloetens, H. Suhonen, and T. Salditt, “Reconstruction of wave front and object for inline holography from a set of detection planes,” Opt. Express 22, 11552–11569 (2014).
[Crossref]

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007).
[Crossref]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

A. T. Kuan, J. S. Phelps, L. A. Thomas, T. M. Nguyen, J. Han, C.-L. Chen, A. W. Azevedo, J. C. Tuthill, J. Funke, P. Cloetens, A. Pacureanu, and W.-C. A. Lee, “Dense neuronal reconstruction through X-ray holographic nano-tomography,” Tech. Rep. (2020).

Coan, P.

A. Cedola, A. Bravin, I. Bukreeva, M. Fratini, A. Pacureanu, A. Mittone, L. Massimi, P. Cloetens, P. Coan, G. Campi, R. Spanoò, F. Brun, V. Grigoryev, V. Petrosino, C. Venturi, M. Mastrogiacomo, N. K. de Rosbo, and A. Uccelli, “X-Ray phase contrast tomography reveals early vascular alterations and neuronal loss in a multiple sclerosis model,” Sci. Rep. 7, 1–11 (2017).
[Crossref]

Cogswell, D. A.

J. Lim, Y. Li, D. H. Alsem, H. So, S. C. Lee, P. Bai, D. A. Cogswell, X. Liu, N. Jin, Y.-S. Yu, N. J. Salmon, D. A. Shapiro, M. Z. Bazant, T. Tyliszczak, and W. C. Chueh, “Origin and hysteresis of lithium compositional spatiodynamics within battery primary particles,” Science 353, 566–571 (2016).
[Crossref]

Crossley, K. J.

L. C. P. Croton, K. S. Morgan, D. M. Paganin, L. T. Kerr, M. J. Wallace, K. J. Crossley, S. L. Miller, N. Yagi, K. Uesugi, S. B. Hooper, and M. J. Kitchen, “In situ phase contrast X-ray brain CT,” Sci. Rep. 8, 1–12 (2018).
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Croton, L. C. P.

L. C. P. Croton, K. S. Morgan, D. M. Paganin, L. T. Kerr, M. J. Wallace, K. J. Crossley, S. L. Miller, N. Yagi, K. Uesugi, S. B. Hooper, and M. J. Kitchen, “In situ phase contrast X-ray brain CT,” Sci. Rep. 8, 1–12 (2018).
[Crossref]

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Supplementary Material (1)

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» Supplement 1       Supplemental information

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (4)

Fig. 1.
Fig. 1. (a) Outline of the x-ray holography algorithm. ${{\boldsymbol{\cal P}}_{\rm{M}}}$ denotes the magnitude, and ${{\boldsymbol{\cal P}}_{{\rm{pp}}}}$ denotes the pure phase object constraint. ${\boldsymbol{\cal F}}$ and ${{\boldsymbol{\cal F}}^{- 1}}$ represent the forward and inverse FFT, respectively. As the initial ( $j = 0$ ) wave field, the known illumination function $P$ is used. The phase object $O$ is calculated in the last iterative step ( $j = N$ ) by subtracting the the phase of the probe $P$ from the exit wave field. The object phase (top, right), which has been used in the simulation as an example, is given by a freely sketched cardiomyocyte image, inspired by the real cardiomyocyte imaged below in Fig. 4. (b) Schematic of the experimental setup. A monochromatic hard x-ray beam is focused by KB mirrors onto a WG, which acts as a spatial and coherence filter. The object $O$ is positioned in the divergent beam at ${z_{{\rm{Ob}}}}$ . The different radiation cones indicate the WG illumination and the scattered photons. The detector D is positioned at  ${z_{{\rm{De}}}}$ .
Fig. 2.
Fig. 2. Ptychographic reconstruction of (a) the probe $P$ amplitude and phase and (b) the object’s $O$ phase with a pixel size ${\delta _{{\rm{px}}}} = 5.2\;{\rm{nm}}$ . The inner radius of the Siemens star pattern corresponds to spokes with 50 nm feature size (see zoom in inset). (c)–(e) Intensity distributions at the WG exit, superimposed with a SEM image of the WG exit surface: (c) ptychographic reconstruction, (d) single-shot reconstruction by the RAAR algorithm using a support constraint [34], and (e) FD-simulated intensity distribution. The source sizes (FWHM) are (c)  $29.1 \times 31.7\;{\rm{n}}{{\rm{m}}^2}$ , (d)  $27.7 \times 27.3\;{\rm{n}}{{\rm{m}}^2}$ , and (e)  $31.8 \times 34.6\;{\rm{n}}{{\rm{m}}^2}$ . Scale bars: (a), (b) 1 µm; (c)–(e) 100 nm.
Fig. 3.
Fig. 3. Inline x-ray holography reconstructions of a test pattern at ${z_{{\rm{Ob}}}} = 1.2\;{\rm{mm}}$ with the detector at ${z_{{\rm{De}}}} = 5.1\;{\rm{m}}$ . (a) Reconstructed Siemens star with ${\delta _{{\rm{px}}}} = 5.2\;{\rm{nm}}$ and 1 s acquisition time. The inner radius of the Siemens star pattern corresponds to spokes with 50 nm feature size (see zoom in inset). (b) The corresponding detector image. Reconstruction of a 50 nm lines and spaces structure. (c) Split-view comparing reconstruction using the novel approach ( ${\delta _{{\rm{px}}}} = 5.2\;{\rm{nm}}$ ) with a reconstruction by empty beam division and CTF ( $\delta _{{\rm{px}}}^{\rm{M}} = 17.3\;{\rm{nm}}$ ). Both use the same data, recorded with an acquisition time of 2 s. (d) Split view comparing the reconstruction from data with 2 to 0.2 s acquisition time, in a region of interest (ROI) of $0.5 \times 0.5\;\unicode{x00B5} {{\rm{m}}^2}$ . (e) FRC analysis of this ROI, correlating reconstructions of two independent measurements with an acquisition time of 2 s each, indicating a resolution (half-period) of $\Delta = 11.2\;{\rm{nm}}$ . Scale bars: (a) 1 µm; (b)  $100{\Delta _{{\rm{px}}}}$ corresponding to a scattering vector $q = 0.06\;{\rm{n}}{{\rm{m}}^{- 1}}$ ; (c), (d) 250 nm.
Fig. 4.
Fig. 4. (a) In-line x-ray holography reconstruction using our novel approach of an entire cardiomyocyte at ${z_{{\rm{Ob}}}} = 9.3\;{\rm{mm}}$ , covered by two exposures at different sample positions. The combined FOV is $103 \times 49\;{{\unicode{x00B5}}}{{\rm{m}}^2}$ with ${\delta _{{\rm{px}}}} = 10.2\;{\rm{nm}}$ (for comparison: $\delta _{{\rm{px}}}^{\rm{M}} = 136.5\;{\rm{nm}}$ ). The inset shows an optical microscope image of the cell. The x-ray FOV is marked by a dashed white rectangle. (b) Additional reconstruction measured at ${z_{{\rm{Ob}}}} = 3.9\;{\rm{mm}}$ with ${\delta _{{\rm{px}}}} = 10.2\;{\rm{nm}}$ (for comparison: $\delta _{{\rm{px}}}^{\rm{M}} = 57.3\;{\rm{nm}}$ ). Position marked in (a) by solid white squares. Subcellular structures can be identified, e.g., the z-disc (red square). The line profile through the z-disc has an FWHM of 105 nm. Scale bars: (a) 10 µm, (b) 2.5 µm, and insets 50 nm. A comparison of our novel in-line X-ray holography approach and ptychography is given in Supplement 1.

Equations (6)

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P M [ ψ ( x , y ) ] = F 1 [ F [ ψ ( x , y ) ] | F [ ψ ( x , y ) ] | I z ( x , y ) ] .
P O { P p p , P r , P S , P S H , P h } ,
P p p [ ψ ( x , y ) ] = ψ ( x , y ) | ψ ( x , y ) | | P ( x , y ) | .
ψ j + 1 = P O P M ψ j
ψ j + 1 = [ 1 2 β ( R O R M + I ) + ( 1 β ) P M ] ψ j ,
R = 2 P I ,