ptychographic coherent x-ray diffractive imaging in the water

$: Ptychographic coherent x-ray diffractive imaging in the water$
Ptychographic coherent x-ray diffractiveimaging in the water window

K. Giewekemeyer,1 M. Beckers,2 T. Gorniak,2 M. Grunze,2,3

T. Salditt,1 and A. Rosenhahn2,3,∗1Institut fur Rontgenphysik, Georg-August-Universitat Gottingen, Friedrich-Hund-Platz 1,

37077 Gottingen, Germany2Angewandte Physikalische Chemie, Ruprecht-Karls-Universitat Heidelberg, Im Neuenheimer

Feld 253, 69120 Heidelberg, Germany3Institut fur funktionelle Grenzflachen, Karlsruher Institut fur Technologie, P. O. Box 3640,

76021 Karlsruhe, Germany

*[email protected]

Abstract: Coherent x-ray diffractive microscopy enables full recon-struction of the complex transmission function of an isolated object todiffraction-limited resolution without relying on any optical elementsbetween the sample and detector. In combination with ptychography, alsospecimens of unlimited lateral extension can be imaged. Here we reporton an application of ptychographic coherent diffractive imaging (PCDI)in the soft x-ray regime, more precisely in the so-called water window ofphoton energies where the high scattering contrast between carbon andoxygen is well-suited to image biological samples. In particular, we havereconstructed the complex sample transmission function of a fossil diatomat a photon energy of 517 eV. In imaging a lithographically fabricatedtest sample a resolution on the order of 50 nm (half-period length) hasbeen achieved. Along with this proof-of-principle for PCDI at soft x-raywavelengths, we discuss the experimental and technical challenges whichcan 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 x-rays.

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1. Introduction

With the advent of third-generation synchrotron radiation sources and free electron lasers co-herent x-ray diffractive imaging (CDI or CXDI) has emerged as a new tool for structure analysison the nanoscale [1,2]. In the classical CDI experiment [3] a coherent plane wave illuminates asample of several microns in diameter, and the resulting diffracted intensity is recorded in thefar field. In an iterative process the object transmission function is then recovered numericallyfrom the measured intensity. This conceptually rather simple experimental scheme has been ap-plied very successfully (see [1, 2] and references therein) and has been proven to be extremelypowerful in terms of resolution [4, 5]. Reconstruction is possible, if the diffraction pattern isband-limited and recorded on a fine enough grid to sample its smallest features [6]. This re-stricts the application of the method to isolated specimens of a lateral extent much smallerthan the beam diameter. In addition to slow convergence and uniqueness issues, this limitationhas been a motivation for alternative approaches such as holography, based on deterministicsingle-step reconstruction [7–10].

Beyond an early non-iterative approach based on “Wigner-deconvolution” [11] Ptycho-graphic Coherent (X-ray) Diffractive Imaging (PCDI) has emerged [12, 13] as a generalizationof conventional CDI suitable also for samples of unlimited lateral extent. Here a finite sam-ple area is illuminated by a coherent beam, a diffraction pattern is recorded, and subsequentlythe sample is translated laterally before recording a new diffraction pattern and repeating theprocess until a desired field of view (FOV) has been scanned through the beam. The samplingcondition is now obeyed by the finite size of the illuminated area. A certain degree of over-lap [14] between neighboring illuminated areas allows for a high redundancy in the recordeddata which strongly facilitates the reconstruction process. Importantly, there is no need for aplanar illumination function any more, as in recent variants of PCDI [15–17] the complex il-luminating wave field can be determined independently from the sample transmission functionusing the same experimental dataset. This allows for the routine application of many differ-ent (possibly distorted) illumination functions, such as the unfocused, Fresnel-diffracted beamof a circular pinhole [18, 19] or highly-confined wave fields, either focused by Fresnel zoneplates [15], compound refractive lenses [20] or focusing mirrors [21], or confined by X-raywaveguides [22].

Here we report on an application of the method using soft x-rays in the so-called waterwindow energy range where the refractive index ratio of carbon and oxygen yields an especiallyhigh contrast of biological specimens against their natural aqueous environment [23,24]. Morespecifically, we have applied ptychographic CDI at a photon energy of 517 eV to reconstruct thecomplex transmission function of a moderately absorbing fossil diatom. A pinhole was used todefine the illumination on the sample. To assess the possible spatial resolution we have imageda lithographically fabricated tantalum test pattern with essentially binary contrast (transmissionvalues 1 and 0) at the given photon energy. In both experiments the complex illuminating wavefield at the sample was reconstructed, allowing for back-propagation to the plane of the pinholewhich was used to illuminate the specimens.

2. Experiments

2.1. Setup

Experiments were carried out at the undulator beamline UE52−SGM of the Berlin electronstorage ring BESSY II, using the dedicated vacuum chamber HORST (holographic x-ray scat-tering chamber) developed at the University of Heidelberg for coherent imaging experimentswith soft x rays [10]. The incident beam was focused by mirrors and/or confined by slits to a sizeof about 17.4(h)×100(v) μm2 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 fossildiatom: The illuminating wave field is confined by a small pinhole with a diameter on theorder of 2 μm, before it impinges onto the sample after propagating over a distance z1 �1 mm. The sample, a diatom on a silicon nitride membrane, is scanned laterally throughthe beam on a rectangular grid while at each scan point a diffraction pattern is collected ona two-dimensional CCD detector placed at a distance z2 � 0.49 m away from the sample.The same setup and measurement principle was then used in a second experiment to imagea 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 z1 = 1−1.4 mm,the Fresnel-diffracted beam reached the sample, which was then scanned laterally through thebeam on a rectangular grid with 800 nm step size in horizontal and vertical direction to allowfor sufficient overlap between illuminated areas of adjacent scan points. A schematic of the ex-periment is depicted in Fig. 1. To assure high positioning accuracy, closed-loop piezo-electricpositioning stages (Physik Instrumente, Germany) were used for scanning the sample throughthe beam. The resulting diffraction patterns were recorded at a distance z2 = 0.49 m away fromthe 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 of2048×2048 pixels.

2.2. Method

For the first experiment a suspension of fossil diatoms in water was dispersed onto a 100-nm-thick silicon nitride membrane and air-dried. The diatom shown in Fig. 2 was translatedthrough the beam at a distance of z1 � 1 mm from the pinhole and diffraction patterns obtainedat 14× 24 scan points on a rectangular grid with a spacing of 800 nm in lateral and verticaldirection were used for reconstruction. Each diffraction pattern was collected during an illu-mination time of 0.18 s, making use of the full dynamic range of the detector without using abeamstop to block the direct beam. The total exposure time was thus 60.48 s for a scanned areaof ca. 191 μm2. A dark image with the same illumination time was used for background cor-rection. For reconstruction a region of 1920×1920 pixels was used on the detector, leading toa real-space pixel width of 45 nm in the sample plane. To reduce computational complexity thediffraction data was binned down by a factor of 2 along each dimension, yielding an effectivedetector 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 transparentmembrane (Ru(20 nm)/SiC(200 nm)/SiN(50 nm)) was translated at a distance of z1 � 1.4 mmfrom the pinhole on a rectangular grid with the same spacing as used before. To minimize theeffect of drift in the positioning stages only a subregion of the total scanned area was selectedfor reconstruction, consisting of 9×7 scan points. 10 exposures with a duration of 0.22 s eachwere collected at every scan point, accumulated, and corrected by subtraction of an equivalentsum of dark images. The total exposure time here was thus 138.6 s for a scanned area ofca. 31 μm2. The combination of several exposures lead to an increased dynamic range of thediffraction patterns used for reconstruction. As for the first dataset, data from a detector regionof 1920×1920 pixels was selected and binned by a factor of 2 along each dimension to makenumerical calculations feasible within a duration of several hours.

An important step in the preparation of the data before reconstruction was the subtraction ofa dark field from all CCD images, with identical total illumination time and exposure charac-teristics. A high dark current on the order of 795 counts/pixel/frame (with a standard deviationaround 3 to 4 counts/pixel/frame) was inevitable and mostly generated by readout noise (in-dependent of illumination time). The fast readout mode of the CCD had to be used to reducethermal drift effects on the positioning stages in vacuum. The corrected intensity Icorr in eachpixel was calculated here as Icorr = max{Imeas − (1+2σ)Idark,0} with Imeas denoting the meas-ured signal, Idark the dark signal and σ � 0.01 denoting the standard deviation of the darkimage, relative to its mean. Using this subtraction rule, remaining noise in Icorr due to camerareadout could be strongly suppressed.

For reconstruction, the algorithm first introduced in [15] was applied. The reconstructionprocess yields independently the complex illumination or probe function P(r) and the complexobject transmission function O(r) in the exit plane directly behind the sample. r denotes thetwo-dimensional spacial coordinate in the sample plane. At each out of NP scan points r j theexit wave field

ψ j(r) = P(r)O(r− r j) (1)

is modelled as a product of the constant probe function and the laterally translated object trans-mission function. With the detector placed into the far field of the exit wave, propagation to thedetector plane corresponds to a two-dimensional Fourier transform F [ψ j(r)] of the exit wavefield with the measured intensity distribution I j given as

I j(q) = |F [ψ j(r)]|2, (2)

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 NP exit waves ψ j that each obey Eqs. (1) and (2). Most importantly, allthese exit waves are formed by a product of the same probe P(r) and translated object functionO(r− r j), the two quantities one is usually interested in.

One important step during each iteration is the enforcement of consistency with the meas-ured 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, . . . ,NP) by the measured am-

plitude√

I j and retain its phase part ψ( j)j (r)/|ψ( j)

j (r)|. A mere replacement operation of theFourier amplitude by the measured amplitude does not take into account the experimental noisepresent 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 meas-ured amplitude in the space of all possible exit waves. The parameter controlling the alloweddistance was optimized towards best reconstruction results.


For reconstruction of the diatom dataset the algorithm was iterated for 200 iterations, start-ing 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 recon-structions 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 objecttransmission function was formed then as a complex mean over the last 50 iterations. Note thatwith ψ j =P ·O j being a reconstructed solution at scan position r j also αP ·α−1O j with α ∈R

+

is a solution [15]. For this reason the object amplitude transmission |O| was forced to stay withinTmin < |O|< Tmax = 1.0 with Tmin = 0.0114 and Tmax = 1.0 being the minimum and maximumexpected amplitude transmissions of the test sample. To suppress artifacts in the reconstructiondue to complications discussed in more detail in section 4 an additional real space constrainthad to be added which is more restrictive than the preceding one: Starting at iteration 152, atevery fourth iteration the amplitude of the object function was set to the theoretical transmissionvalue of Tmin = 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 residualscattering from the pinhole reached the sample at present experimental parameters (see section4). Due to the scaling ambiguity between probe and object function the amplitude of the probefunction was found to be amplified to unphysical high values in regions far away from the brightcenter. To prevent this, additional constraints on the probe function were introduced. For thediatom dataset, a window (or mask) function of the form Θ(r0−r) (with radial coordinate r andΘ(r) denoting the Heaviside step function) was multiplied with the probe during each iterationof probe retrieval, setting it to zero outside a circular region with a diameter 2r0 = 0.9D1 andleaving it unchanged within the circle. D1 here denotes the width of the numerical FOV ofthe probe function in the sample plane and is related to the wavelength λ , the propagationdistance z2 between sample and detector and the detector pixel width Δ by D1 = λ z1/Δ. Theprobe wave field for the Siemens star dataset was obtained with stronger ’guidance’. Morespecifically, the normalized probe amplitude was forced to stay below a ”hat”-function of theform (ax2

r + by2r )

−c with a,b,c ∈ R, c > 1 and (xr,yr) denoting sample plane coordinates in asystem rotated with respect to the one used for reconstruction. The free parameters of the hat-function 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 wasrestricted in the plane of the aperture by back propagation of the current probe reconstructionto the pinhole plane, multiplication with a binary elliptical mask outlining the support of theelliptical pinhole, and subsequent propagation to the sample plane. The binary mask in thepinhole plane was determined by a shrink-wrap mechanism with very loose constraints in orderto not over-restrict the problem [27].

3. Results

3.1. Diatom sample

An overview of the reconstructed complex object transmission function for the diatom sampleis depicted in Fig. 2B. In contrast to the area covered by the center of the probe wave field whichroughly corresponds to the extension of the scanning grid the reconstructed object transmissionextends over a much larger area, even covering the edge of the silicon nitride window ontowhich the sample was placed. This is due to the relatively slow decay of the illumination fieldamplitude in the object plane (see also section 4). The reconstruction is consistent with anoptical micrograph (Fig. 2A) of the same sample.

A magnified inset of the object reconstruction within the area that has been covered by thecentral part of the probe during the scan is shown in Fig. 3A. Details of the ornamental per-


Fig. 2. (A) Optical micrograph of the fossil diatom sample. The area scanned by the x-raybeam is marked by a black frame. The dark stripe on the left side of the image corre-sponds to the edge of the silicon-nitride window. (B) Complex-valued ptychographic re-construction of the object transmission function from the same diatom sample as shownin subfigure A. Color encodes phase (modulo 2π), brightness the amplitude as indicatedby 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 of10.4(h)×18.4(v) μm2. 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 widthat half maximum (FWHM) of the amplitude, the object is reconstructed far beyond thescanned area as marked by the scan positions. This is due to the relatively slow decay ofthe probe amplitude in the object plane (see section 4).

forations (holes several 100 nanometers in size) in the frustule (diatom shell) can be seen insubfigures 3B (phase) and 3D (amplitude). Note that in Fig. 2B and 3A phase values are shownmodulo 2π , ”wrapped” into an interval IP of length 2π . This wrapping of phase values is due tothe multi-valued nature of the arg(z)-function of a complex number z and can lead to unphysicalphase discontinuities. Unwrapping a discrete two-dimensional phase distribution, i.e. mappingof phase values from the interval IP into the space R of physical phase values, is a well-knownmathematical problem which can be very difficult to solve due to phase aliasing, noise andphysical discontinuities [28]. For the small subregion shown in subfigure Fig. 3B unphysicalphase discontinuities in horizontal direction have been removed using the Matlab [29] built-inone-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-dioxidewith a uniform density. Assuming a silicon dioxide mass density of 2.2 g/cm3 [30] one arrivesat a phase shift of around 1π rad and amplitude transmission of T = 0.58 per 1 μm projectedthickness [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 biologicalobjects which generally do not exhibit edges with a known sharpness. A rough estimate on theresolution can be given here based on the fit of an error-function to the sharp boundary of thediatom. 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 trans-mission function (see Fig. 3A), indicating a possible breakdown of the projection approxima-


Fig. 3. (A) Complex-valued object reconstruction (phase values modulo 2π) within the areathat has been covered by the central and most intensive part of the probe wave field duringthe 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 reconstructedobject transmission corresponding to the marked square (side length 6 μm) in subfigureA. For the subregion shown here the phase has been unwrapped, i.e. it is shown withoutnon-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 whiteline in subfigure B. The red line marks a fit to the phase step with an error-function.

tion. A coarse criterion for the neglect of diffraction effects during propagation through thesample, i.e. for the validity of the projection approximation, has been given based on the angleof total external reflection due to a lateral refractive index gradient [31]: The approximation isvalid as long as the lateral resolution r obeys r > a1 =

√2δΔt where δ is the refractive index

difference along a lateral resolution element and Δt the propagation distance through the sam-ple. In addition, the resolution has to fulfill r > a2 =

√λΔt in order to avoid Fresnel diffraction

effects within the sample [31]. Assuming δ = 0.0012 and a thickness of 2 μm in the presentexample one arrives at a1 � 100 nm and a2 � 69 nm. With a resolution in the object reconstruc-tion close to this value it is clear that the experimental configuration is at the validity limit ofthe projection approximation and the sample might extend the depth of focus at certain points.

3.2. Siemens star test object

The reconstructed amplitude of the Siemens star object transmission function is shown inFig. 4A. The reconstruction shows details down to a half-period resolution on the order of50 nm, the central angular width of the void stripes of the innermost ring in the test pattern.


Fig. 4. (A) Reconstructed amplitude of the Siemens star object transmission function. To-wards the center the void areas in the innermost ring of the test pattern reach a width of50 nm in angular direction and many of them can be separated from the filled stripes alongtheir whole radial extension. (B) Reconstructed phase (in radians) of the area indicated bya square (side length 3 μm) in subfigure A. (C) Scanning electron micrograph of roughlythe same region as imaged in the experiment. On the innermost side of each segmented ringthe void stripes reach an angular width of 0.2 μm (third ring from center), 0.1 μm (secondring from center) and 0.05 μm (innermost ring).

As visible in Fig. 4B the phase is only reconstructed uniformly in the void segments, exhibit-ing random fluctuations in the filled regions. Qualitatively this can be understood in view of thelow minimum intensity transmission of T 2

min = 1.3 ·10−4 leading to insufficient transmission fora non-random phase reconstruction. With an average background-corrected accumulated countnumber of 2 ·109 analog-to-digital units (”counts”) on the detector for each scan point the av-erage ”fluence” at the sample is around 1.2 ·108 counts/μm2 or 2.6 ·105 counts per pixel. Thisallows for a relative error in intensity transmission of 1/

√2.6 ·105 = 2 ·10−3 which is however

insufficient to reliably detect a transmission of T 2 = 1.3 · 10−4 as expected for the tantalummaterial. Instead, the average relative amplitude transmission between void and filled regionsis around 22. Further reasons for the non-quantitative object reconstruction are discussed insection 4.

3.3. Probe reconstructions

The complex reconstructed probe functions, obtained from the diatom and Siemens star datasetsare depicted in Fig. 5A and B, respectively, both back-propagated over the respective distance z1

to the plane of the aperture. A comparison to the scanning electron micrograph of the pinholeexit surface indicates a considerable degree of similarity between the overall shapes of thereconstructed wave fields and the pinhole structure. Notably, the probe reconstruction obtainedfrom the scan of the diatom sample, which scatters much less than the Siemens star, exhibits aflat central phase and amplitude within the elliptical pinhole area. On the other hand, the probereconstruction from the Siemens star dataset is characterized by high-frequency distortions inamplitude and phase which are most likely artifacts of the reconstruction (see section 4). Note


Fig. 5. (A) Complex-valued reconstruction of the probe function as obtained from the di-atom dataset. Phase is encoded as hue, amplitude as brightness. The reconstructed probewave is shown here back-propagated over a distance of 1.08 mm with respect to the planeof reconstruction. (B) Reconstructed illumination function as obtained from the Siemensstar dataset. Here the probe was back-propagated over a distance of 1.36 mm. (C) Scanningelectron micrograph of the exit surface of the pinhole used for beam confinement. (D) Typ-ical background-corrected diffracted intensity (arbitrary units) collected for one scan pointof 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, background-corrected diffracted intensity collected for a scan point of the Siemens star dataset, usingthe same pinhole to define the illumination function as in the first experiment. Scale bars insubfigures A, B, C indicate 1 μm.

here that a typical diffraction pattern from the diatom dataset (see Fig. 5D) is dominated bythe diffracted signal from the pinhole while the Siemens star diffraction pattern is very stronglydominated by the sample diffraction which totally suppresses the signal from the pinhole (seeFig. 5E).

4. Discussion

In the reconstruction of both, the diatom and especially of the Siemens star dataset certainartifacts remain, even though the reconstruction of the Siemens star dataset was restricted withrather strong additional real-space constraints. For a possible explanation we briefly discussthe geometry of diffraction pattern formation in both experiments. Conceptually, there are twoextreme 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 z1 between pinholeand sample and thus a large Fresnel number F = a2/(λ z1) � 1 (with pinhole diameter a),leading to an illumination that is sharply confined in amplitude and nearly flat in phase. Inthis configuration — which is closest to that of classical CDI where a plane wave is used toilluminate the isolated sample — the diffraction pattern at the detector is nearly given as the


Fig. 6. (A) Far-field geometry: For Fresnel numbers F = a2/(λ z1)� 1 (with pinhole di-ameter a) the wavefront impinging on the sample is almost flat and the amplitude of theexit wave field well-confined. The field distribution at the detector in the far field showsno resemblance with the exit wave. There is no geometrical magnification of the exit wavefield behind the sample and the width D1 of the exit wave’s FOV in the reconstruction planeis 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 sampleis nearly spherical and less well-confined in amplitude. This leads to a diffracted ampli-tude which can be interpreted as magnified near-field propagated object function (with thespherical part of the illumination removed). The extension D′

1 = D2/M of the FOV in thesample plane is now proportional to the FOV width in the detector plane. (C) Vertical slicethrough the normalized reconstructed probe amplitude |P| for the Siemens star dataset. Redvertical lines mark the positions of the first side minima, within which 98 % of the inten-sity is located. (D) Vertical slice through the unwrapped reconstructed phase ϕ(P) of theprobe 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 z1.

squared modulus of the Fourier transform of the object transmission function convolved withthe Airy pattern of the pinhole. As a consequence, the diffraction pattern has no resemblance tothe object.

On the other hand there is the limiting case of large distances z1 between pinhole and sam-ple, i.e. the limit of very small Fresnel numbers F � 1, when the sample is illuminated by apinhole beam already propagated into the far field. The illuminating wave field at the sample isthen given as a product of a spherical phase term (with radius of curvature z1) and the Fouriertransform of the aperture function. Within the small-angle approximation the propagation ofthe exit wave over the distance z2 to the detector can then be described in an equivalent plane-wave geometry (with the spherical part of the exit wave removed) over an effective distancez2/M with geometrical magnification M = (z1 + z2)/z1 [8]. The FOV for a single exit wave inthe sample plane is then given by D′

1 = D2/M with the FOV width D2 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 signalshows significant resemblance to the original object.

Ptychographic CDI is usually performed with the sample illuminated by a relatively flat andsharply confined wavefront [15, 18–21], in the imaging regime represented by the setup shownin Fig. 6A. In fact, this can best be achieved by placing the sample into the focal plane of astrongly 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 ofboth limiting cases illustrated in Fig. 6A and B is adequate to completely describe the processof diffraction pattern formation: Consider the vertical slices through the amplitude and phaseof the probe function as reconstructed from 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 withinthe first side minima, a region with a relatively flat phase (variations up to 1.5π rad). This is thepart of the beam that leads to the strongest diffraction effects which can be roughly interpretedas 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 radiusz1 (see Fig. 6D) and leads to a weak, Fresnel-propagated image of a comparatively large part ofthe sample on the detector. Note that the increased amplitude in the probe reconstruction on theright in Fig. 6C is most probably an artifact due to the previously described product ambiguitybetween probe and object amplitude. These amplified outer components of the probe amplitudelead to the high-frequency artifacts appearing at the probe wave field back-propagated to thepinhole plane, as visible in Fig. 5B. They are much weaker in the probe reconstruction of thediatom dataset.

It has been shown previously that high-curvature beams can be used advantageously for CDIexperiments [32] and also ptychography [33] in a mode that is called Fresnel CDI. Here theillumination is actually reconstructed independently from a diffraction pattern of the emptybeam alone [34].

We now turn from the probe reconstruction towards a qualitative evaluation of typical ob-served diffraction patterns of the diatom and Siemens star (see Fig. 5D and E). They areboth characterized by a Fresnel-propagated direct image of the sample as well as a far-fielddiffracted reciprocal-space signal (see Fig. 5), similar to diffraction patterns encountered inFresnel-CDI [32]. In contrast to the the diatom dataset the near-field propagated (direct) imageof the Siemens star always covers the full detector area, as the sample laterally extends in alldirections over a very large area (dozens of microns in diameter) which is still partly illumi-nated by the outer parts of the far-field propagated pinhole beam. Note that this leads to anextraordinary broad mix of length scales: The central flattest and strongest part of the beamwith a diameter on the order of 2 μm hits the center of the Siemens star with smallest lengthscales down to 50 nm leading to diffracted signals at the edges of the detector. At the same timethe weaker and highly curved outer parts of the probe illuminate a sample area with a maxi-mum lateral extension of D′

1 � 72 μm (assuming an ideal paraxial spherical wave emanatingfrom the pinhole), leading to a direct image of very large structures towards the edges of thedetector. Although the reciprocal signal from the center is stronger than the direct signal fromthe edges they are both in the same area of the detector. Thus, an additional source of artifactsin the reconstruction could be a possible miss-interpretation of direct low-frequency signal asreciprocal high-frequency signal in the reconstruction process. For the Siemens star such a fail-ure is even more probable as there is a resemblance of both signals. The reconstruction fromthe diatom dataset, where the detected signal is dominated by the pinhole diffraction, showed amuch better convergence compared to the Siemens star dataset. This can be mostly attributed toto the smaller overall extend of the sample, making the holographic direct image less dominantand extended in the diffraction pattern.


We note that binning of detector pixels can also lead to increased artifacts as the compu-tationally relevant FOV in the sample area is given by D1 rather than D′

1 with the data beingnot translated into the effective plane-wave geometry for ptychographic reconstruction. For abinning factor B = 2, as used here, D1 � 43 μm, so that some weak signal at the outer regionsof the detector will be either not reconstructed or become a possible source of artifacts due tomiss-interpretation as high-frequency signal. However, in tests with B= 1 (scaling up computa-tion time roughly by a factor of 4) on the Siemens star dataset it was found that the direct signalfrom the sample at the outer parts of the detector is generally too weak to be reconstructedas a direct near-field propagated image. Therefore, the effect of shorter computation time wasfavored.

5. Summary, conclusion and outlook

In conclusion, we have demonstrated the successful application of ptychographic coherentdiffractive imaging to a fossil diatom with a contrast largely comparable to that of unstainedbiological cells at the used photon energy in the water window (here E = 517 eV). As a con-sequence, also for soft x-ray energies the usual field-of-view restrictions of conventional CDIcan be overcome. Imaging a heavy-element lithographic test pattern, a nearly full absorptionobject, 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 ob-tained which is in good agreement with estimated phase and amplitude modifications due tosuch an object. Here the small depth of focus of diffraction microscopy at small x-ray wave-lengths becomes apparent in the reconstruction. As a consequence, for large biological objectswhich are several microns thick along the direction of the beam, the projection approximation islikely to be violated at these wavelengths making it difficult to find a global focal plane for thewhole object. In such cases, one has to be aware of the complications for future 3D tomographicreconstructions based on ptychographic CDI in the water window.

For both datasets consistent reconstructions of the complex illumination function were ob-tained, allowing for a complete characterization of the wave field exiting the pinhole used forillumination. While the two probe reconstructions show a reasonable agreement in view of thesubstantially different nature of the samples, we noticed that in this case the weaker scatteringobject, the diatom, led to the physically more meaningful probe and object reconstruction. Thisindicates that for probing a wave field using ptychographic CDI not necessarily the strongest-scattering sample is to be preferred.

The observed challenges of pinhole-based ptychographic CDI at low wavelengths have beendiscussed, which are partly due to the fact that one easily enters imaging regimes where thepropagated pinhole beam exhibits significant phase curvature and is not sharply confined inthe sample plane. In such situations the observed diffraction patterns are characterized by anoverlay of a direct ’holographic’ image of a large fraction of the extended sample and a far-fielddiffraction 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 thesoft x-ray regime. A good way to achieve this experimentally can be the use of a zone platefocus as the probe. A current technical limitation to PCDI in the soft x-ray regime is the lack ofdetectors free of dark and readout noise and the relatively long readout times of present CCDsystems. As a consequence, for standard fully quantitative PCDI also in the soft x-ray regimeimprovements on the detector side are very desirable.

Notwithstanding the technical challenge in the proper choice of experimental parameters andinstrumentation as discussed above, we believe that the simple experimental concept of ptycho-graphic coherent diffraction imaging will be a very valuable tool in particular in the soft-x-rayrange, 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 illu-mination wave front is particularly important here, since short propagation distances betweenoptics and sample already lead to strong propagation effects and simple assumptions on theprobe function need to be considered carefully. In addition, wavefront reconstruction with res-olutions in the nanometer range represents a truly unique tool for instrument characterizationand development.

Acknowledgements

The work was funded by the German Ministry of Education and Research under grant numbers05KS7VH1 and 05K10VH4 and by the German Research Foundation within the collaborativeresearch center SFB 755 ”Nanoscale photonic imaging”. K.G. acknowledges financial supportfrom the Helmholtz society within the framework of the VI-203 of the Impuls- and Vernet-zungsfonds. We acknowledge P. Thibault for fruitful interaction on a related project, and forsharing helpful insights concerning ptychographic reconstruction, including the possibility touse a mask constraint for the probe function. We thank I. Thome for providing us with SEMimages of the pinhole.


ptychographic coherent x-ray diffractive imaging in the water

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