Ptychographic coherent x-ray diffractive imaging in the water window

Coherent x-ray diffractive microscopy enables full reconstruction of the complex transmission function of an isolated object to diffraction-limited resolution without relying on any optical elements between the sample and detector. In combination with ptychography, also specimens of unlimited lateral extension can be imaged. Here we report on an application of ptychographic coherent diffractive imaging (PCDI) in the soft x-ray regime, more precisely in the so-called water window of photon energies where the high scattering contrast between carbon and oxygen is well-suited to image biological samples. In particular, we have reconstructed the complex sample transmission function of a fossil diatom at a photon energy of 517 eV. In imaging a lithographically fabricated test sample a resolution on the order of 50 nm (half-period length) has been achieved. Along with this proof-of-principle for PCDI at soft x-ray wavelengths, we discuss the experimental and technical challenges which can occur especially for soft x-ray PCDI. © 2011 Optical Society of America OCIS codes: (340.7440) X-ray imaging; (180.7460) X-ray microscopy; (260.6048) Soft xrays. References and links 1. K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 1–99 (2010). 2. P. Thibault and V. Elser, “X-ray diffraction microscopy,” Annu. Rev. Condens. Matter Phys. 1, 237–255 (2010). 3. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999). 4. C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schoder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101, 090801 (2008). 5. J. Nelson, X. Huang, J. Steinbrener, D. 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Introduction
With the advent of third-generation synchrotron radiation sources and free electron lasers coherent x-ray diffractive imaging (CDI or CXDI) has emerged as a new tool for structure analysis on the nanoscale [1,2].In the classical CDI experiment [3] a coherent plane wave illuminates a sample of several microns in diameter, and the resulting diffracted intensity is recorded in the far field.In an iterative process the object transmission function is then recovered numerically from the measured intensity.This conceptually rather simple experimental scheme has been applied very successfully (see [1,2] and references therein) and has been proven to be extremely powerful in terms of resolution [4,5].Reconstruction is possible, if the diffraction pattern is band-limited and recorded on a fine enough grid to sample its smallest features [6].This restricts the application of the method to isolated specimens of a lateral extent much smaller than the beam diameter.In addition to slow convergence and uniqueness issues, this limitation has been a motivation for alternative approaches such as holography, based on deterministic single-step reconstruction [7][8][9][10].
Beyond an early non-iterative approach based on "Wigner-deconvolution" [11] Ptychographic Coherent (X-ray) Diffractive Imaging (PCDI) has emerged [12,13] as a generalization of conventional CDI suitable also for samples of unlimited lateral extent.Here a finite sample area is illuminated by a coherent beam, a diffraction pattern is recorded, and subsequently the sample is translated laterally before recording a new diffraction pattern and repeating the process until a desired field of view (FOV) has been scanned through the beam.The sampling condition is now obeyed by the finite size of the illuminated area.A certain degree of overlap [14] between neighboring illuminated areas allows for a high redundancy in the recorded data which strongly facilitates the reconstruction process.Importantly, there is no need for a planar illumination function any more, as in recent variants of PCDI [15][16][17] the complex illuminating wave field can be determined independently from the sample transmission function using the same experimental dataset.This allows for the routine application of many different (possibly distorted) illumination functions, such as the unfocused, Fresnel-diffracted beam of a circular pinhole [18,19] or highly-confined wave fields, either focused by Fresnel zone plates [15], compound refractive lenses [20] or focusing mirrors [21], or confined by X-ray waveguides [22].
Here we report on an application of the method using soft x-rays in the so-called water window energy range where the refractive index ratio of carbon and oxygen yields an especially high contrast of biological specimens against their natural aqueous environment [23,24].More specifically, we have applied ptychographic CDI at a photon energy of 517 eV to reconstruct the complex transmission function of a moderately absorbing fossil diatom.A pinhole was used to define the illumination on the sample.To assess the possible spatial resolution we have imaged a lithographically fabricated tantalum test pattern with essentially binary contrast (transmission values 1 and 0) at the given photon energy.In both experiments the complex illuminating wave field at the sample was reconstructed, allowing for back-propagation to the plane of the pinhole which was used to illuminate the specimens.

Setup
Experiments were carried out at the undulator beamline UE52 − SGM of the Berlin electron storage ring BESSY II, using the dedicated vacuum chamber HORST (holographic x-ray scattering chamber) developed at the University of Heidelberg for coherent imaging experiments with soft x rays [10].The incident beam was focused by mirrors and/or confined by slits to a size of about 17.4(h) × 100(v) μm 2 at a photon energy of 517 eV.After passing a pinhole (stainless Fig. 1.Experimental setup for ptychographic coherent x-ray diffractive imaging of a fossil diatom: The illuminating wave field is confined by a small pinhole with a diameter on the order of 2 μm, before it impinges onto the sample after propagating over a distance z 1 1 mm.The sample, a diatom on a silicon nitride membrane, is scanned laterally through the beam on a rectangular grid while at each scan point a diffraction pattern is collected on a two-dimensional CCD detector placed at a distance z 2 0.49 m away from the sample.The same setup and measurement principle was then used in a second experiment to image a test pattern structured by nano-lithography. steel, thickness ca. 13 μm, diameter on the order of 2 μm [25], Edmund Optics, Germany) positioned into the beam focus and free-space propagation over a distance of z 1 = 1 − 1.4 mm, the Fresnel-diffracted beam reached the sample, which was then scanned laterally through the beam on a rectangular grid with 800 nm step size in horizontal and vertical direction to allow for sufficient overlap between illuminated areas of adjacent scan points.A schematic of the experiment is depicted in Fig. 1.To assure high positioning accuracy, closed-loop piezo-electric positioning stages (Physik Instrumente, Germany) were used for scanning the sample through the beam.The resulting diffraction patterns were recorded at a distance z 2 = 0.49 m away from the sample on a back-illuminated, peltier-cooled CCD detector (DX436, Andor Technology, UK) with a pixel width of 13.5 μm in horizontal and vertical direction and a total number of 2048 × 2048 pixels.

Method
For the first experiment a suspension of fossil diatoms in water was dispersed onto a 100nm-thick silicon nitride membrane and air-dried.The diatom shown in Fig. 2 was translated through the beam at a distance of z 1 1 mm from the pinhole and diffraction patterns obtained at 14 × 24 scan points on a rectangular grid with a spacing of 800 nm in lateral and vertical direction were used for reconstruction.Each diffraction pattern was collected during an illumination time of 0.18 s, making use of the full dynamic range of the detector without using a beamstop to block the direct beam.The total exposure time was thus 60.48 s for a scanned area of ca.191 μm 2 .A dark image with the same illumination time was used for background correction.For reconstruction a region of 1920 × 1920 pixels was used on the detector, leading to a real-space pixel width of 45 nm in the sample plane.To reduce computational complexity the diffraction data was binned down by a factor of 2 along each dimension, yielding an effective detector pixel width of 27 μm.
For the second imaging experiment, a Siemens star test pattern (model ATN/XRESO-50HC, NTT-AT, Japan) consisting of a 500-nm-thick nanostructured tantalum layer on a transparent membrane (Ru(20 nm)/SiC(200 nm)/SiN(50 nm)) was translated at a distance of z 1 1.4 mm from the pinhole on a rectangular grid with the same spacing as used before.To minimize the effect of drift in the positioning stages only a subregion of the total scanned area was selected for reconstruction, consisting of 9 × 7 scan points.10 exposures with a duration of 0.22 s each were collected at every scan point, accumulated, and corrected by subtraction of an equivalent sum of dark images.The total exposure time here was thus 138.6 s for a scanned area of ca. 31 μm 2 .The combination of several exposures lead to an increased dynamic range of the diffraction patterns used for reconstruction.As for the first dataset, data from a detector region of 1920 × 1920 pixels was selected and binned by a factor of 2 along each dimension to make numerical calculations feasible within a duration of several hours.
An important step in the preparation of the data before reconstruction was the subtraction of a dark field from all CCD images, with identical total illumination time and exposure characteristics.A high dark current on the order of 795 counts/pixel/frame (with a standard deviation around 3 to 4 counts/pixel/frame) was inevitable and mostly generated by readout noise (independent of illumination time).The fast readout mode of the CCD had to be used to reduce thermal drift effects on the positioning stages in vacuum.The corrected intensity I corr in each pixel was calculated here as I corr = max{I meas − (1 + 2σ )I dark , 0} with I meas denoting the measured signal, I dark the dark signal and σ 0.01 denoting the standard deviation of the dark image, relative to its mean.Using this subtraction rule, remaining noise in I corr due to camera readout could be strongly suppressed.
For reconstruction, the algorithm first introduced in [15] was applied.The reconstruction process yields independently the complex illumination or probe function P(r) and the complex object transmission function O(r) in the exit plane directly behind the sample.r denotes the two-dimensional spacial coordinate in the sample plane.At each out of N P scan points r j the exit wave field ψ j (r) = P(r)O(r − r j ) is modelled as a product of the constant probe function and the laterally translated object transmission function.With the detector placed into the far field of the exit wave, propagation to the detector plane corresponds to a two-dimensional Fourier transform F [ψ j (r)] of the exit wave field with the measured intensity distribution I j given as where q denotes the two-dimensional reciprocal space coordinate.Starting with an initial guess {ψ (0) j } of exit waves the algorithm [15,26] then iteratively finds a solution {ψ j = P(r)O(r − r j )}, consisting of N P exit waves ψ j that each obey Eqs. ( 1) and (2).Most importantly, all these exit waves are formed by a product of the same probe P(r) and translated object function O(r − r j ), the two quantities one is usually interested in.
One important step during each iteration is the enforcement of consistency with the measured data.The simplest approach to enforce this consistency is to replace the Fourier modulus |F [ψ ( j) j (r)]| of the current iterate ψ ( j) j (r) (here for every j = 1, . . ., N P ) by the measured amplitude I j and retain its phase part ψ ( j) j (r)/|ψ ( j) j (r)|.A mere replacement operation of the Fourier amplitude by the measured amplitude does not take into account the experimental noise present in I j .To circumvent this problem we used the same modified projection operator as in [18], which allows a certain distance between the updated Fourier amplitude and the measured amplitude in the space of all possible exit waves.The parameter controlling the allowed distance was optimized towards best reconstruction results.
For reconstruction of the diatom dataset the algorithm was iterated for 200 iterations, starting with a nearly flat initial guess with small added random phase and amplitude distortions.To average out small fluctuations from one iteration to another the final object and probe reconstructions were obtained as a complex mean over the last 50 iterations.
For reconstruction of the Siemens star dataset the algorithm was iterated for 400 iterations, starting with an equally generated random complex field as before.Analogously, the final object transmission function was formed then as a complex mean over the last 50 iterations.Note that with ψ j = P• O j being a reconstructed solution at scan position r j also αP•α −1 O j with α ∈ R + is a solution [15].For this reason the object amplitude transmission |O| was forced to stay within T min < |O| < T max = 1.0 with T min = 0.0114 and T max = 1.0 being the minimum and maximum expected amplitude transmissions of the test sample.To suppress artifacts in the reconstruction due to complications discussed in more detail in section 4 an additional real space constraint had to be added which is more restrictive than the preceding one: Starting at iteration 152, at every fourth iteration the amplitude of the object function was set to the theoretical transmission value of T min = 0.0114 for 500 nm Ta at 517 eV photon energy, if |O| < 0.2.
A difficulty encountered in the reconstructions was the relatively large area at which residual scattering from the pinhole reached the sample at present experimental parameters (see section 4).Due to the scaling ambiguity between probe and object function the amplitude of the probe function was found to be amplified to unphysical high values in regions far away from the bright center.To prevent this, additional constraints on the probe function were introduced.For the diatom dataset, a window (or mask) function of the form Θ(r 0 − r) (with radial coordinate r and Θ(r) denoting the Heaviside step function) was multiplied with the probe during each iteration of probe retrieval, setting it to zero outside a circular region with a diameter 2r 0 = 0.9D 1 and leaving it unchanged within the circle.D 1 here denotes the width of the numerical FOV of the probe function in the sample plane and is related to the wavelength λ , the propagation distance z 2 between sample and detector and the detector pixel width Δ by D 1 = λ z 1 /Δ.The probe wave field for the Siemens star dataset was obtained with stronger 'guidance'.More specifically, the normalized probe amplitude was forced to stay below a "hat"-function of the form (ax 2 r + by 2 r ) −c with a, b, c ∈ R, c > 1 and (x r , y r ) denoting sample plane coordinates in a system rotated with respect to the one used for reconstruction.The free parameters of the hatfunction were adjusted by comparison with the probe reconstruction from the diatom dataset, keeping an effort to minimally restrict possible solutions.Furthermore, the probe wave field was restricted in the plane of the aperture by back propagation of the current probe reconstruction to the pinhole plane, multiplication with a binary elliptical mask outlining the support of the elliptical pinhole, and subsequent propagation to the sample plane.The binary mask in the pinhole plane was determined by a shrink-wrap mechanism with very loose constraints in order to not over-restrict the problem [27].

Diatom sample
An overview of the reconstructed complex object transmission function for the diatom sample is depicted in Fig. 2B.In contrast to the area covered by the center of the probe wave field which roughly corresponds to the extension of the scanning grid the reconstructed object transmission extends over a much larger area, even covering the edge of the silicon nitride window onto which the sample was placed.This is due to the relatively slow decay of the illumination field amplitude in the object plane (see also section 4).The reconstruction is consistent with an optical micrograph (Fig. 2A) of the same sample.
A magnified inset of the object reconstruction within the area that has been covered by the central part of the probe during the scan is shown in Fig. 3A.Details of the ornamental per- Note that also the edge of the silicon nitride window can be seen: Although the extension of the probe was on the order of 2 μm according to the full width at half maximum (FWHM) of the amplitude, the object is reconstructed far beyond the scanned area as marked by the scan positions.This is due to the relatively slow decay of the probe amplitude in the object plane (see section 4).
forations (holes several 100 nanometers in size) in the frustule (diatom shell) can be seen in subfigures 3B (phase) and 3D (amplitude).Note that in Fig. 2B and 3A phase values are shown modulo 2π, "wrapped" into an interval I P of length 2π.This wrapping of phase values is due to the multi-valued nature of the arg(z)-function of a complex number z and can lead to unphysical phase discontinuities.Unwrapping a discrete two-dimensional phase distribution, i.e. mapping of phase values from the interval I P into the space R of physical phase values, is a well-known mathematical problem which can be very difficult to solve due to phase aliasing, noise and physical discontinuities [28].For the small subregion shown in subfigure Fig. 3B unphysical phase discontinuities in horizontal direction have been removed using the Matlab [29] built-in one-dimensional phase unwrapping routine unwrap.m,leading to the true 'physical' phase.To a good approximation, the fossil diatom can be considered to be composed of silicon-dioxide with a uniform density.Assuming a silicon dioxide mass density of 2.2 g/cm 3 [30] one arrives at a phase shift of around 1π rad and amplitude transmission of T = 0.58 per 1 μm projected thickness [30], leading to a maximum thickness on the order of 2 − 3 μm.
An exact determination of the obtained resolution in direct space is difficult for biological objects which generally do not exhibit edges with a known sharpness.A rough estimate on the resolution can be given here based on the fit of an error-function to the sharp boundary of the diatom.The fit here yields an edge smoothness of 129 nm (FWHM).
Note that diffraction fringes are visible on the edges of the diatom in the reconstructed transmission function (see Fig. 3A), indicating a possible breakdown of the projection approxima- tion.A coarse criterion for the neglect of diffraction effects during propagation through the sample, i.e. for the validity of the projection approximation, has been given based on the angle of total external reflection due to a lateral refractive index gradient [31]: The approximation is valid as long as the lateral resolution r obeys r > a 1 = √ 2δ Δt where δ is the refractive index difference along a lateral resolution element and Δt the propagation distance through the sample.In addition, the resolution has to fulfill r > a 2 = √ λ Δt in order to avoid Fresnel diffraction effects within the sample [31].Assuming δ = 0.0012 and a thickness of 2 μm in the present example one arrives at a 1 100 nm and a 2 69 nm.With a resolution in the object reconstruction close to this value it is clear that the experimental configuration is at the validity limit of the projection approximation and the sample might extend the depth of focus at certain points.

Siemens star test object
The reconstructed amplitude of the Siemens star object transmission function is shown in Fig. 4A.The reconstruction shows details down to a half-period resolution on the order of 50 nm, the central angular width of the void stripes of the innermost ring in the test pattern.As visible in Fig. 4B the phase is only reconstructed uniformly in the void segments, exhibiting random fluctuations in the filled regions.Qualitatively this can be understood of the low intensity transmission of T 2 min = 1.3 • 10 −4 leading to insufficient transmission for a non-random phase reconstruction.With an average background-corrected accumulated count number of 2 • 10 9 analog-to-digital units ("counts") on the detector for each scan point the average "fluence" at the sample is around 1.2 • 10 8 counts/μm 2 or 2.6 • 10 5 counts per pixel.This allows for a relative error in intensity transmission of 1/ √ 2.6 • 10 5 = 2 • 10 −3 which is however insufficient to reliably detect a transmission of T 2 = 1.3 • 10 −4 as expected for the tantalum material.Instead, the average relative amplitude transmission between void and filled regions is around 22. Further reasons for the non-quantitative object reconstruction are discussed in section 4.

Probe reconstructions
The complex reconstructed probe functions, obtained from the diatom and Siemens star datasets are depicted in Fig. 5A and B, respectively, both back-propagated over the respective distance z 1 to the plane of the aperture.A comparison to the scanning electron micrograph of the pinhole exit surface indicates a considerable degree of similarity between the overall shapes of the reconstructed wave fields and the pinhole structure.Notably, the probe reconstruction obtained from the scan of the diatom sample, which scatters much less than the Siemens star, exhibits a flat central phase and amplitude within the elliptical pinhole area.On the other hand, the probe reconstruction from the Siemens star dataset is characterized by high-frequency distortions in amplitude and phase which are most likely artifacts of the reconstruction (see section 4).Note here that a typical diffraction pattern from the diatom dataset (see Fig. 5D) is dominated by the diffracted signal from the pinhole while the Siemens star diffraction pattern is very strongly dominated by the sample diffraction which totally suppresses the signal from the pinhole (see Fig. 5E).

Discussion
In the reconstruction of both, the diatom and especially of the Siemens star dataset certain artifacts remain, even though the reconstruction of the Siemens star dataset was restricted with rather strong additional real-space constraints.For a possible explanation we briefly discuss the geometry of diffraction pattern formation in both experiments.Conceptually, there are two extreme imaging regimes in which one could work using the present imaging setup (see Fig. 6A and B).The first situation is characterized by a very small distance z 1 between pinhole and sample and thus a large Fresnel number F = a 2 /(λ z 1 ) 1 (with pinhole diameter a), leading to an illumination that is sharply confined in amplitude and nearly flat in phase.In this configuration -which is closest to that of classical CDI where a plane wave is used to illuminate the isolated sample -the diffraction pattern at the detector is nearly given as the squared modulus of the Fourier transform of the object transmission function convolved with the Airy pattern of the pinhole.As a consequence, the diffraction pattern has no resemblance to the object.
On the other hand there is the limiting case of large distances z 1 between pinhole and sample, i.e. the limit of very small Fresnel numbers F 1, when the sample is illuminated by a pinhole beam already propagated into the far field.The illuminating wave field at the sample is then given as a product of a spherical phase term (with radius of curvature z 1 ) and the Fourier transform of the aperture function.Within the small-angle approximation the propagation of the exit wave over the distance z 2 to the detector can then be described in an equivalent planewave geometry (with the spherical part of the exit wave removed) over an effective distance z 2 /M with geometrical magnification M = (z 1 + z 2 )/z 1 [8].The FOV for a single exit wave in the sample plane is then given by D 1 = D 2 /M with the FOV width D 2 in the detection plane.In contrast to the case F 1 the diffraction pattern at the detector is now the modulus of the object transmission function propagated into the (effective) near field.Thus the recorded signal shows significant resemblance to the original object.Ptychographic CDI is usually performed with the sample illuminated by a relatively flat and sharply confined wavefront [15,[18][19][20][21], in the imaging regime represented by the setup shown in Fig. 6A.In fact, this can best be achieved by placing the sample into the focal plane of a strongly focused wave field [15,20,21] or very close to an opaque mask (e.g. a pinhole) [18,19].However, the geometry of the present experiment (F 1) leads to a situation where neither of both limiting cases illustrated in Fig. 6A and B is adequate to completely describe the process of diffraction pattern formation: Consider the vertical slices through the amplitude and phase of the probe function as reconstructed the Siemens star dataset in the plane of the object (see Fig. 6C and D): 78% of the total amplitude (98% of the intensity) is concentrated within the first side minima, a region with a relatively flat phase (variations up to 1.5π rad).This is the part of the beam that leads to the strongest diffraction effects which can be roughly interpreted as far-field patterns of the illuminated sample area.The remaining part of the beam, however, is subject to a phase curvature which is almost as high as that of a spherical wave with radius z 1 (see Fig. 6D) and leads to a weak, Fresnel-propagated image of a comparatively large part of the sample on the detector.Note that the increased amplitude in the probe reconstruction on the right in Fig. 6C is most probably an artifact due to the previously described product ambiguity between probe and object amplitude.These amplified outer components of the probe amplitude lead to the high-frequency artifacts appearing at the probe wave field back-propagated to the pinhole plane, as visible in Fig. 5B.They are much weaker in the probe reconstruction of the diatom dataset.
It has been shown previously that high-curvature beams can be used advantageously for CDI experiments [32] and also ptychography [33] in a mode that is called Fresnel CDI.Here the illumination is actually reconstructed independently from a diffraction pattern of the empty beam alone [34].
We now turn from the probe reconstruction towards a qualitative evaluation of typical observed diffraction patterns of the diatom and Siemens star (see Fig. 5D and E).They are both characterized by a Fresnel-propagated direct image of the sample as well as a far-field diffracted reciprocal-space signal (see Fig. 5), similar to diffraction patterns encountered in Fresnel-CDI [32].In contrast to the the diatom dataset the near-field propagated (direct) image of the Siemens star always covers the full detector area, as the sample laterally extends in all directions over a very large area (dozens of microns in diameter) which is still partly illuminated by the outer parts of the far-field propagated pinhole beam.Note that this leads to an extraordinary broad mix of length scales: The central flattest and strongest part of the beam with a diameter on the order of 2 μm hits the center of the Siemens star with smallest length scales down to 50 nm leading to diffracted signals at the edges of the detector.At the same time the weaker and highly curved outer parts of the probe illuminate a sample area with a maximum lateral extension of D 1 72 μm (assuming an ideal paraxial spherical wave emanating from the pinhole), leading to a direct image of very large structures towards the edges of the detector.Although the reciprocal signal from the center is stronger than the direct signal from the edges they are both in the same area of the detector.Thus, an additional source of artifacts in the reconstruction could be a possible miss-interpretation of direct low-frequency signal as reciprocal high-frequency signal in the reconstruction process.For the Siemens star such a failure is even more probable as there is a resemblance of both signals.The reconstruction from the diatom dataset, where the detected signal is dominated by the pinhole diffraction, showed a much better convergence compared to the Siemens star dataset.This can be mostly attributed to to the smaller overall extend of the sample, making the holographic direct image less dominant and extended in the diffraction pattern.
We note that binning of detector pixels can also lead to increased artifacts as the computationally relevant FOV in the sample area is given by D 1 rather than D 1 with the data being not translated into the effective plane-wave geometry for ptychographic reconstruction.For a binning factor B = 2, as used here, D 1 43 μm, so that some weak signal at the outer regions of the detector will be either not reconstructed or become a possible source of artifacts due to miss-interpretation as high-frequency signal.However, in tests with B = 1 (scaling up computation time roughly by a factor of 4) on the Siemens star dataset it was found that the direct signal from the sample at the outer parts of the detector is generally too weak to be reconstructed as a direct near-field propagated image.Therefore, the effect of shorter computation time was favored.

Summary, conclusion and outlook
In conclusion, we have demonstrated the successful application of ptychographic coherent diffractive imaging to a fossil diatom with a contrast largely comparable to that of unstained biological cells at the used photon energy in the water window (here E = 517 eV).As a consequence, also for soft x-ray energies the usual field-of-view restrictions of conventional CDI can be overcome.Imaging a heavy-element lithographic test pattern, a nearly full absorption object, allowed for reconstructions on the order of 50 nm (half-period) resolution.
For the semi-transparent diatom sample a full complex object wave reconstruction was obtained which is in good agreement with estimated phase and amplitude modifications due to such an object.Here the small depth of focus of diffraction microscopy at small x-ray wavelengths becomes apparent in the reconstruction.As a consequence, for large biological objects which are several microns thick along the direction of the beam, the projection approximation is likely to be violated at these wavelengths making it difficult to find a global focal plane for the whole object.In such cases, one has to be aware of the complications for future 3D tomographic reconstructions based on ptychographic CDI in the water window.
For both datasets consistent reconstructions of the complex illumination function were obtained, allowing for a complete characterization of the wave field exiting the pinhole used for illumination.While the two probe reconstructions show a reasonable agreement in view of the substantially different nature of the samples, we noticed that in this case the weaker scattering object, the diatom, led to the physically more meaningful probe and object reconstruction.This indicates that for probing a wave field using ptychographic CDI not necessarily the strongestscattering sample is to be preferred.
The observed challenges of pinhole-based ptychographic CDI at low wavelengths have been discussed, which are partly due to the fact that one easily enters imaging regimes where the propagated pinhole beam exhibits significant phase curvature and is not sharply confined in the sample plane.In such situations the observed diffraction patterns are characterized by an overlay of a direct 'holographic' image of a large fraction of the extended sample and a far-field diffraction pattern due to a very small sample region illuminated by the central part of the beam.Instead, a flat, sharply confined illumination is very desirable for ptychographic CDI also in the soft x-ray regime.A good way to achieve this experimentally can be the use of a zone plate focus as the probe.A current technical limitation to PCDI in the soft x-ray regime is the lack of detectors free of dark and readout noise and the relatively long readout times of present CCD systems.As a consequence, for standard fully quantitative PCDI also in the soft x-ray regime improvements on the detector side are very desirable.
Notwithstanding the technical challenge in the proper choice of experimental parameters and instrumentation as discussed above, we believe that the simple experimental concept of ptychographic coherent diffraction imaging will be a very valuable tool in particular in the soft-x-ray range, where the interaction with matter is comparably strong and the diffractive signal hence is high, up to high momentum transfer.The fact that PCDI allows for reconstruction of the illumination wave front is particularly important here, since short propagation distances between optics and sample already lead to strong propagation effects and simple assumptions on the probe function need to be considered carefully.In addition, wavefront reconstruction with resolutions in the nanometer range represents a truly unique tool for instrument characterization and development.

Fig. 2 .
Fig. 2. (A) Optical micrograph of the fossil diatom sample.The area scanned by the x-ray beam is marked by a black frame.The dark stripe on the left side of the image corresponds to the edge of the silicon-nitride window.(B) Complex-valued ptychographic reconstruction of the object transmission function from the same diatom sample as shown in subfigure A. Color encodes phase (modulo 2π), brightness the amplitude as indicated by the colorwheel on the lower right.The positions at which the illuminating wave field (the probe) was centered during the scan are marked by white dots covering an area of 10.4(h) × 18.4(v) μm 2 .Note that also the edge of the silicon nitride window can be seen: Although the extension of the probe was on the order of 2 μm according to the full width at half maximum (FWHM) of the amplitude, the object is reconstructed far beyond the scanned area as marked by the scan positions.This is due to the relatively slow decay of the probe amplitude in the object plane (see section 4).

Fig. 3 .
Fig. 3. (A) Complex-valued object reconstruction (phase values modulo 2π) within the area that has been covered by the central and most intensive part of the probe wave field during the scan, roughly corresponding to the extension of the grid of scan points shown in Fig. 2B.(B) and (D) Detailed view of the amplitude (B) and phase (D) of the reconstructed object transmission corresponding to the marked square (side length 6 μm) in subfigure A. For the subregion shown here the phase has been unwrapped, i.e. it is shown without non-physical phase jumps due to wrapping phase values into an interval of width 2π.(C) Line profile of the phase perpendicular to the edge of the sample as marked by the white line in subfigure B. The red line marks a fit to the phase step with an error-function.

Fig. 4 .
Fig. 4. (A) Reconstructed amplitude of the Siemens star object transmission function.Towards the center the void areas in the innermost ring of the test pattern reach a width of 50 nm in angular direction and many of them can be separated from the filled stripes along their whole radial extension.(B) Reconstructed phase (in radians) of the area indicated by a square (side length 3 μm) in subfigure A. (C) Scanning electron micrograph of roughly the same region as imaged in the experiment.On the innermost side of each segmented ring the void stripes reach an angular width of 0.2 μm (third ring from center), 0.1 μm (second ring from center) and 0.05 μm (innermost ring).

Fig. 5 .
Fig. 5. (A) Complex-valued reconstruction of the probe function as obtained from the diatom dataset.Phase is encoded as hue, amplitude as brightness.The reconstructed probe wave is shown here back-propagated over a distance of 1.08 mm with respect to the plane of reconstruction.(B) Reconstructed illumination function as obtained from the Siemens star dataset.Here the probe was back-propagated over a distance of 1.36 mm.(C) Scanning electron micrograph of the exit surface of the pinhole used for beam confinement.(D) Typical background-corrected diffracted intensity (arbitrary units) collected for one scan point of the diatom dataset.The diffraction pattern extends to a full spatial period of 11 μm −1 , or a corresponding real space pixel size of 45 nm.(E) Typical accumulated, backgroundcorrected diffracted intensity collected for a scan point of the Siemens star dataset, using the same pinhole to define the illumination function as in the first experiment.Scale bars in subfigures A, B, C indicate 1 μm.

Fig. 6 .
Fig. 6. (A) Far-field geometry: For Fresnel numbers F = a 2 /(λ z 1 ) 1 (with pinhole diameter a) the wavefront impinging on the sample is almost flat and the amplitude of the exit wave field well-confined.The field distribution at the detector in the far field shows no resemblance with the exit wave.There is no geometrical magnification of the exit wave field behind the sample and the width D 1 of the exit wave's FOV in the reconstruction plane is not related to the lateral size of the detector, but inversely proportional to its pixel width.(B) Effective near-field geometry: For Fresnel numbers F 1 the probe at the sample is nearly spherical and less well-confined in amplitude.This leads to a diffracted amplitude which can be interpreted as magnified near-field propagated object function (with the spherical part of the illumination removed).The extension D 1 = D 2 /M of the FOV in the sample plane is now proportional to the FOV width in the detector plane.(C) Vertical slice through the normalized reconstructed probe amplitude |P| for the Siemens star dataset.Red vertical lines mark the positions of the first side minima, within which 98 % of the intensity is located.(D) Vertical slice through the unwrapped reconstructed phase ϕ(P) of the probe function and the phase ϕ(S) of an ideal spherical wave S in paraxial approximation, emanating from the center of the pinhole and propagated over a distance z 1 .