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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 17 — Aug. 13, 2012
  • pp: 19232–19254
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Hard X-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt  »View Author Affiliations


Optics Express, Vol. 20, Issue 17, pp. 19232-19254 (2012)
http://dx.doi.org/10.1364/OE.20.019232


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Abstract

Ptychographic coherent X-ray diffractive imaging (PCDI) has been combined with nano-focus X-ray diffraction to study the structure and density distribution of unstained and unsliced bacterial cells, using a hard X-ray beam of 6.2keV photon energy, focused to about 90nm by a Fresnel zone plate lens. While PCDI provides images of the bacteria with quantitative contrast in real space with a resolution well below the beam size at the sample, spatially resolved small angle X-ray scattering using the same Fresnel zone plate (cellular nano-diffraction) provides structural information at highest resolution in reciprocal space up to 2nm−1. We show how the real and reciprocal space approach can be used synergistically on the same sample and with the same setup. In addition, we present 3D hard X-ray imaging of unstained bacterial cells by a combination of ptychography and tomography.

© 2012 OSA

1. Introduction

To understand the important biophysical problem of DNA compactification in bacterial cells, ideally the three-dimensional density distribution at the sub-cellular level needs to be imaged with quantitative contrast values. Indeed, a simple unknown quantity relevant to the understanding of DNA compactification in bacterial nucleoids is the local mass density. This important question cannot be solved by electron microscopy or fluorescence microscopy studies, not for reason of resolution but of contrast. More generally, the problem of mapping out the native three-dimensional density distribution in biological cells, without staining and slicing, is a formidable task. Density contrast can provide useful insight into many biophysical and biological problems by bridging previously disjunct pieces of information: (a) averaged densities derived from centrifuging separated and purified organelles, which is one of the most common tools in biochemistry and molecular biology, and (b) the three-dimensional (3D) structure derived from imaging techniques such as confocal fluorescence microscopy or electron tomography.

To this end, we want to advocate the use of coherent X-ray imaging and tomography to shed light on this unsolved issue, since these techniques offer a unique potential for quantitative three-dimensional determination of density in unstained and unsliced biological cells and tissues. In particular, as discussed in [1

1. K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 529–534 (2010). [CrossRef]

], electron density can be derived from hard X-ray imaging independently of local stoichiometry for nearly all biologically relevant elements, if the photon energy is higher than the corresponding absorption edges.

We would like to complement ptychography with a higher resolution structural analysis in reciprocal space, i.e. by nano-beam diffraction. Note that notwithstanding impressive progress, coherent imaging of cellular samples [17

17. D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005). [CrossRef] [PubMed]

19

19. J. Nelson, X. Huang, J. Steinbrener, D. Shapiro, J. Kirz, S. Marchesini, A. M. Neiman, J. J. Turner, and C. Jacobsen, “High-resolution x-ray diffraction microscopy of specifically labeled yeast cells,” Proc. Natl. Acad. Sci. USA 107, 7235–7239 (2010). [CrossRef] [PubMed]

] has always been limited to moderate resolution, not reaching resolution of typical diffraction experiments on soft matter samples. Here we combine coherent imaging with nano-beam diffraction for structure analysis of biological cells. By simple experimental parameters (slit opening, detector distance, beamstop, defocus position), we can switch between a setting optimised for high resolution diffraction, and a setting optimised for coherent imaging, on the same cell (see schematic in Fig. 1). Compared to a standard nano-diffraction experiment the following advantages apply for a combined approach: (i) the illuminating beam (probe function) and corresponding interaction volume can be deduced quantitatively from ptychographic inversion instead of simple knife edge scans, (ii) correlative microscopy images of projected density and diffraction observables are obtained, (iii) one can easily and quickly determine where exactly the beam is positioned on the cell with a full field (defocus) image, before more lengthy scanning diffraction maps are recorded. Compared on the other hand to a pure coherent imaging study, analysis in reciprocal space can yield higher resolution by averaging over the illuminated spot, and by circumventing the stringent requirements (oversampling, signal-to-noise, coherence) of full phase reconstruction. Apart from density as a contrast value, many more structural parameters (inter-molecular spacings, short range correlations, lattice constants, form factors) can reasonably be interpreted by a local average on length scales of 10 – 1000nm. Fourier components corresponding to much smaller structures can thus be measured, very much like in a standard small angle X-ray diffraction (SAXS) experiment [20

20. O. Glatter and O. Kratky, Small Angle X-Ray Scattering (Acad. Pr., 1982).

, 21

21. A. Guinier and G. Fournet, Small-Angle Scattering of X-Rays (Wiley, 1955).

], but now scaled down to a sample volume corresponding to a specific part of a cell. Nano-diffraction of biological cells will thus result in a 2D image with a resolution determined by the focal size, with each pixel encoding different observables, in particular diffraction observables. This concept of scanning SAXS has been introduced first on the micron scale, with seminal work in the field of biological materials by Fratzl and co-workers [22

22. P. Fratzl, H. F. Jakob, S. Rinnerthaler, P. Roschger, and K. Klaushofer, “Position-resolved small-angle x-ray scattering of complex biological materials,” J. Appl. Crystallogr. 30, 765–769 (1997). [CrossRef]

25

25. A. Gourrier, W. Wagermaier, M. Burghammer, D. Lammie, H. S. Gupta, P. Fratzl, C. Riekel, T. J. Wess, and O. Paris, “Scanning x-ray imaging with small-angle scattering contrast,” J. Appl. Crystallogr. 40, 78–82 (2007). [CrossRef]

] (see also [26

26. O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009). [CrossRef]

,27

27. T. H. Jensen, M. Bech, O. Bunk, M. Thomsen, A. Menzel, A. Bouchet, G. L. Duc, R. Feidenhans’l, and F. Pfeiffer, “Brain tumor imaging using small-angle x-ray scattering tomography,” Phys. Med. Biol. 56, 1717–1726 (2011). [CrossRef] [PubMed]

]), and can now with further progress in X-ray focusing be easily extended to the nano-scale. The pixel of the real space image can encode: phase shift (i.e. projected density), ion concentration from fluorescence yield, small-angle scattering intensity extracted from the far-field pattern (darkfield), or other structural parameters derived from the full 2D diffraction pattern, e.g. fibre bundle orientation and spacing. Beyond a simple contrast value, for each pixel a fully quantitative diffraction pattern is available at much higher signal-to-noise than typically in CDI, since oversampling and coherence constraints can be relaxed. In contrast to electron microscopy, no sectioning of the cells is necessary for X-ray imaging.

Fig. 1 Schematic of the experimental setup used for X-ray cellular nano-diffraction and PCDI accentuating the important parameters: 1. gap of the source slit S0, which controls the flux and coherence properties of the beam, 2. the distance Z0 between the zone plate optics (FZP) and the sample, which defines the size of the beam at the cellular specimen, and 3. the detector distance Z12 with respect to the sample, which determines sampling of the recorded diffraction pattern and attainable diffraction angle. For different choices of these three parameters the biological specimen can then be translated in the XY – directions and additionally be rotated with respect to ϕ. In detail, the beamline setup consists of the X-ray undulator (period 19mm) source (S), horizontal and vertical slits (S1–S4), Si(111) monochromator (LN2-cooled), fast shutter (FS), filter (Fi), central stop (CS), order sorting aperture (OSA) and focus (F) with bacterial specimen. Flight tube and beamstops are not shown.

After this introduction, the experimental details are presented, including a brief account of sample preparation, the used instrumentation at the beamline, as well as the experimental parameters of ptychographic imaging, tomography, including details on dose estimation, and cellular nano-diffraction. Next, results are presented, first for ptychography obtained from a test pattern to retrieve the probe function followed by the cellular samples, then the cellular tomography, and finally the nano-diffraction data. The paper closes with some conclusions and a brief outlook.

2. Experiments

2.1. Sample preparation

Cells of the Deinococcus radiodurans wild-type strain were cultivated from freeze-dried cultures (DSM No. 20539 by the German Collection of Microorganisms and Cell Cultures) for one day at 30°C on petri dishes covered with nutrient medium (corynebacterium agar: 10g/l casein peptone, 5g/l yeast extract, 5g/l glucose, 5g/l NaCl, 15g/l agar). Prior to preparation the actively growing cells were washed off the culturing medium with ca. 1.5ml buffer solution (2g/l KH2PO4, 0.36g/l Na2HPO4 · 2H2O, pH 7.2). After placing a droplet of cell suspension onto the substrate, a polyimide foil of 12.5μm thickness (diffraction experiment) and Si3N4 membranes of 0.5μm thickness (PCDI experiment), the cells were allowed to adhere for 60 seconds. The remaining buffer was blotted, the foil mounted on top of a solid stainless steal rod cryo-plunged into liquid ethane to prevent crystallization [28

28. J. Dubochet, M. Adrian, J.-J. Chang, J.-C. Homo, J. Lepault, A.W. McDowall, and P. Schultz, “Cryo-Electron microscopy of vitrified specimens,” Q. Rev. Biophys. 21, 129–228 (1988). [CrossRef] [PubMed]

] and afterwards lyophilized in a home-built freeze-drier.

2.2. Beamline and instrumentation

The experiment has been carried out at the coherent Small Angle X-ray Scattering beamline (cSAXS) of the Swiss Light Source (SLS) at the Paul Scherrer Institut in Villigen, Switzerland. Starting from the undulator source consisting of Nu = 96 magnet periods with undulator period 19mm, the beam was defined by the following optical elements, as indicated schematically on Fig. 1: (i) horizontal slit S0 at 12.1m distance from source, (ii) horizontal and vertical slits S1 at 26m, (iii) a Si(111) double crystal monochromator at 28.6m (2nd crystal). The first crystal was LN2-cooled. The monochromator was set to a photon energy of 6.2keV. Following (iv) further slit systems S2, S3, and S4 as well as the fast shutter system (FS) for detector acquistion, the beam impinges on a Fresnel zone plate (FZP) with preceding central stop (CS). We used a a highly efficient FZP fabricated of Au on a Si3N4 membrane using 100keV e-beam lithography and electroplating [29

29. S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salome, and C. David, “High- efficiency fresnel zone plates for hard x-rays by 100 kev e-beam lithography and electroplating,” J. Synchrotron Radiat. 18, 442–446 (2011). [CrossRef] [PubMed]

, 30

30. S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology 21, 295303 (2010). [CrossRef] [PubMed]

] with a diameter of ∅︀ = 200μm, zone height of 1μm and outermost zone width dr = 100nm. The FZP was located at ∼ 34m from the source and at a distance of ∼ 0.1m to the sample. An order sorting aperture (OSA) with a diameter of ∅︀ = 20μm was placed between FZP and sample. The sample was positioned by a high precision piezo stage (Physik Instrumente, Germany) mounted on an air bearing rotation stage (Micos UPR160F) on top of a hexapod (PI M-850).

The first part of the experiment was optimised for ptychography: The coherent slit setting of S0 (gap width 10μm) was chosen. In addition, a long distance between sample and detection plane is favoured for sufficient sampling of the recorded diffraction pattern (fulfilment of the oversampling condition). Therefore, the zero read-out noise photon counting detector PILA-TUS 2M (SLS detector group, [31

31. P. Kraft, A. Bergamaschi, C. Broennimann, R. Dinapoli, E. F. Eikenberry, B. Henrich, I. Johnson, A. Mozzanica, C. M. Schleputz, P. R. Willmott, and B. Schmitt, “Performance of single-photon- counting pilatus detector modules,” J. Synchrotron Radiat. 16, 368–375 (2009). [CrossRef] [PubMed]

]) was placed at the long working distance of z12 ≃ 7.22m (ptychographic imaging of resolution chart) behind the sample and a helium-filled flight tube of length 7m was used. Ptychographic imaging results of D. radiodurans have been obtained during a second experiment at cSAXS using the same experimental setup. However, the detector was placed at z12 ≃ 7.65m (see [32

32. K. Nygård, O. Bunk, E. Perret, C. David, and J.F. van der Veen, “Diffraction gratings as small-angle X-ray scattering calibration standards,” J. Appl. Cryst. 43, 350–351 (2010). [CrossRef]

] for distance callibration) with an air path of ≃ 0.4m between the flight tube and the PILATUS. A cryojet, beamstops and filters were not used for ptychographic imaging.

The second part of the experiment was optimised for diffraction: a cryojet was used to cool the sample and the detector was placed z12 ≃ 2.28m behind the sample (according to a callibration using silver behenate), with 1475 × 1679 pixels of size 172μm × 172μm. A flight tube of length 2.063m was used between sample and detector to reduce air-scattering of the diffracted X-ray beam. Beam damage of the detector was suppressed by using multiple beamstops between sample and detection plane and/ or by using silicon filters of appropriate thickness. The coherence properties of the beam were altered by changing the gap of the horizontal slits S0. Importantly, increasing the gap yields a beam with lower spatial coherence, but more flux.

2.3. Ptychography

Before going into the experimental details of the ptychographic data, we will give a very brief formulation of the underlying theory (see also [33

33. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009). [CrossRef] [PubMed]

, 34

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

]). Assuming that the exit surface wave Ψj(x⃗) (ESW) directly behind the sample can be formulated as a product of the illuminating wave field P(x⃗x⃗j) and the complex transmission of the specimen S(x⃗), a Fourier transform relation holds between the ESW and the intensity Ij at a Fraunhofer or farfield detection plane
Ij=|{Ψj(x)}|2=|{P(xxj)S(x)}|2,
(1)
where x⃗, x⃗j ∈ ℝ2 are vectors in direct space of the plane of the sample. The index j indicates a lateral shift of the probe function P. In the process of measuring the diffraction signal, only the intensity of the diffracted beam can be recorded, thus preventing a direct inversion of Eq. (1) under general conditions. However, the lost phases can be recovered through iterative reconstruction if the problem fulfils the oversampling condition [35

35. 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]

] and is constrained by the experimental conditions. In this experiment, a set of diffraction patterns was collected by scanning the sample along concentric circles in a XY –plane (cf. [36

36. M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von Knig, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010). [CrossRef]

], Fig. 2(a)) allowing adjacent illuminated spots to overlap. The set of diffraction patterns is thus constrained through (i) a locally finite support of the ESW as introduced by the finite lateral extend of the probe of each diffraction pattern, (ii) the complete set of measured intensities and most importantly (iii) redundancy between the exit surface waves by overlapping illuminated regions. The introduced overlap constraint (influence of the overlap parameter see [37

37. O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008). [CrossRef]

]) is the basis for the ptychographic phasing method, which iteratively searches for an estimate of both the complex transmission S(x⃗) and the probe P(x⃗). Usually, further issues have to be considered in the reconstruction scheme for Eq. (1) to hold: These are experimental uncertainties such as positioning errors due to the limited experimental accuracy, drifts and vibrations, noise in the recorded data, unusable areas of the detector, a limited stationarity of the probe and partial spatial coherence. In particular, phasing of weakly scattering specimens such as bacterial cells is a challenging task [1

1. K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 529–534 (2010). [CrossRef]

, 36

36. M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von Knig, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010). [CrossRef]

].

Fig. 2 (a) Phase from ‘ePIE’ reconstruction of the tantalum test structure. Scan points are indicated as magenta points. A fit of the error function to one of the edges of the line scans yields a line-spread function with FWHM-values of 16nm in the vertical direction and 17nm in the horizontal direction. The inner part of (a) is shown in (c). (b) Complex representation of reconstructed probe at the plane of the sample (top) and horizontal and vertical line scans through the centre of the back-propagated probe (bottom) (cf. Media 1). The white dashed lines highlight the position of the different focal positions. (d) (top) Gaussian fits (solid red lines) to the peaks of the line scans of the probe intensity are depicted below together with the data from the line scans (black dots). We obtain FWHM-values of 87nm in the vertical direction, and 93nm in the horizontal direction. (d) A typical diffraction pattern of the PCDI dataset (log-scale) is shown in the lower part.

We made use of two different ptychographic algorithms as described in [33

33. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009). [CrossRef] [PubMed]

] (and references therein) and [34

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

], respectively. We refer to the former algorithm as ‘DM’ (difference-map) and to the latter as ’ePIE’ throughout the paper.

Following the strategy of [36

36. M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von Knig, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010). [CrossRef]

], we used a Siemens star resolution test chart (model ATN/XRESO-50HC, NTT-AT, Japan) in order to get a good estimate of the probe, which could be used as an initial guess for the phasing of the bacterial specimen. The sample was placed into a defocus position of ∼ 1mm. 323 diffraction patterns have been measured using a scan pattern with scan points on concentric circles (cf. Fig. 2(a)), a dwell time of 0.2s or equivalently, an exposure time of ≃ 1min in total. The distance between adjacent concentric circles was 0.5μm in comparison to ∼ 0.75μm × 1.5μm FWHM values of probe intensity, h × v respectively. A subset of 800 × 800 pixels of the recorded whole diffraction patterns (cf. Fig. 1) was used for phasing (∼ 10nm real space pixel size). Dead or hot pixels of the detector were masked and their intensity values were defined by the current iterate in the reconstruction scheme. Moreover, the ptychographic reconstruction algorithm (‘ePIE’ algorithm) was initiated using the complex field of a 1mm forward propagated, circular aperture with uniform amplitude and constant phase as probe function and uniform amplitude with constant phase as object guess. Additionally, the amplitude of the object was clipped onto the interval [0.7, 1] during each iteration. Note, that the interval has been chosen close to the theoretical expectation. The final reconstruction (cf. Fig. 2(a)) was obtained after 320 iterations by averaging over the last 200 iterations with an average interval of 2. Here, we found superior results using the ‘ePIE’ algorithm with parameters α = β = 1/2 (cf. Eq. (6), (7) in [34

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

]).

For the interpretation of the results we made use of the ‘phase retrieval transfer function’ (PRTF), which indicates the goodness of the reconstruction in comparison to the measured intensities (cf. [1

1. K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 529–534 (2010). [CrossRef]

, 17

17. D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005). [CrossRef] [PubMed]

, 38

38. H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006). [CrossRef]

], Fig. 3). Best results were obtained through the averaging procedure over many iterations and thereby cancelling randomly fluctuating phase contributions. Therefore, the closest estimate for the measured intensities will be that corresponding to the exit surface waves 〈Ψj(x⃗)〉it of the average over iterates. An overall estimate of the full ptychographic reconstruction can be achieved by averaging over all probe positions (corresponding to measured diffraction patterns j). In addition, we performed an azimuthal average. We thus use the PRTF as defined by:
PRTF(|q|)=IjPCDIjφIjmeasjφ=|{Ψjit}|jφIjmeasjφ.
(2)

Fig. 3 The PRTFs of the ptychographic reconstruction of the resolution test chart and of a single projection of cells (Fig. 4(b)) are depicted in (a), green curve and blue curve respectively. In addition, the PRTF of the test sample according to an average over a single scan point is shown as red curve (cf. Eq. (2)). The high frequency ranges of the azimuthally averaged power spectral densities (PSD) of the reconstructed phase distributions are displayed in (b). Real space half-periods of 50nm, 25nm and 15nm are indicated as dashed, solid and dashed-dotted black lines, respectively. Legend of (b) is legend in (a).

The phase reconstructions of D. radiodurans (cf. Fig. 4) were obtained using the ’DM’ algorithm for the simple reason that it can be parallelized. Here, both algorithms yield reconstructions of comparable quality, as judged from single projections. Two different regions with bacteria have been studied, Fig. 4(a) and Figs. 4(b)–4(d) respectively. The samples were measured in the focal region (probe size:∼ 0.5μm × 0.4μm full width at half maximum (FWHM) values of probe intensity, h × v respectively; see Table 1 for scan details).

Fig. 4 (a)–(c) Phase reconstructions of D. radiodurans (‘DM’ - algorithm) showing dark areas, which we attribute to DNA rich regions. (d) 3D visualisation of the tomographic reconstruction of the set of projection images such as seen in (b) (cf. Media 2). The tomographic reconstruction is shown at three different angles. A magenta label in the tomogram (front-view) depicts voxel-regions which have been used to estimate the mass density ρc in the high-density regions of the bacterial specimen (red). We obtained <ρc> ≈ 1.6g/cm3. (a), (b) same colorbar as in (c).

Table 1. Overview of PCDI data sets according to dwell time ΔT, grid spacing between adjacent rings Δr of a scan of one projection, fluence F of a single projection and dose estimations D of the whole reconstruction.

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We recorded 77 ptychographic datasets for a tomographic reconstruction on the cellular sample seen in Fig. 4(b). The angular increment between successive projections is ΔΦ = 2°, leaving a missing-wedge of ±13°. In each projection a dwell time of 0.2s was used. The grid spacing between adjacent concentric circles was 0.35μm. Three projections could not be reconstructed and have been omitted for the tomographic reconstruction. To quantify the errors resulting from the missing data, we have simulated the reconstruction based on a test phantom with parameters comparable to our experimental situation. An upper limit of 10% error in the reconstructed density values was found. These errors were dominated by the missing-wedge, while the three missing projections were found to have a negligible effect.

A subset of 192×192 pixels of each PILATUS image corresponding to a real space pixel size of 46nm has been chosen for all reconstructions. Importantly, the ptychographic reconstructions were initiated with the reconstructed illuminating function from the Siemens star as a probe guess and a uniform amplitude, constant phase object guess. In addition, we introduced a further constraint onto the data. We defined a rectangular support χ of the visible cell cluster and during each iteration the phase outside the support was set close to zero using the following phase constraint:
𝒫(O(x)):={|O(x)|exp(imin(Arg(O(x)),0))xχ|O(x)|exp(iγArg(O(x)))xχ,
(3)
where γ ∈ ℝ was chosen close to zero. It should be noted that 𝒫(O(x⃗)) enforces a negative phase shift on the object transmission. However, this represents no restriction since one usually is interested in the relative phase shift between different materials. Moreover, the object transmission can be easily adapted to cover the full phase range by multiplying 𝒫(O(x⃗)) by exp(). The final results were obtained by averaging over 500 iterations during the so-called steady state of the ‘DM’ algorithm [33

33. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009). [CrossRef] [PubMed]

].

2.4. Tomography

At first, the ptychographically reconstructed projections (PCDI parameters for 2D reconstructions as above) have been aligned with respect to the axis of rotation using a method discribed in [39

39. M. Guizar-Sicairos, S.T. Thurman, and J.R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008). [CrossRef] [PubMed]

]. Afterwards, the 3D tomographic reconstruction was done by applying the filtered backprojection algorithm using the standard ‘Ram-Lak’ or ramp filter [40

40. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

]. Visualisation was done with ’Avizo’-Software.

2.5. Dose estimations

We used the reconstructed Probe P from the ptychographic reconstruction to investigate the applied surface dose D. In a first step we calculated the 2D-photon density ργ(x⃗) by scaling the normalized intensity of the probe at each scan position j to the actual photon number Nj: ργ(x⃗) = ∑j Nj|P(x⃗x⃗j)|2 / ∫2 dS|P(x⃗)|2. Given the characteristic function χ(x⃗) of a certain part of the cell area of the scan, the photon energy E, the mass density of the cell material ρc and the absorption coefficient μ, one obtains the applied surface dose D [41

41. M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J. Electron Spectrosc. 170, 4–12 (2009). [CrossRef]

] for this region:
D=μEρc2dSχ(x)ργ(x)2dSχ(x).
(4)

All dose calculations in this work are based on the assumption that the main consistency of the cell material is protein of the empirical formula H50C30N9O10S1 of density ρc = 1.35g/cm3 [41

41. M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J. Electron Spectrosc. 170, 4–12 (2009). [CrossRef]

].

2.6. Cellular diffraction

In a first step diffraction on D. radiodurans data was taken by translating the object in the XY-plane of the focal region using a coherent slit setting of S0 on the sample region A of Fig. 5 (see Table 2 for a summary on scan details). The performed mesh scans differ in dwell time ΔT, step size Δxy between adjacent scan points and field of view (FOV). The data set A1 was recorded without a beamstop but with a Si filter of 50μm thickness. A2, A3, A4 were carried out with a beamstop and no filters. The two diffraction patterns of A4 were taken on a cell region and on an empty region of the polyimide foil, respectively.

Fig. 5 Sample region A - coherent setting: differential phase contrast (a), (c) and darkfield (b), (d) of cells are shown for the mesh scans A1 (top) and A2 (bottom) - (cf. Table 2). The colorbar in (c) denotes the movement of the first moment whereas in (d) the grey scale denotes the ratio of scattered to unscattered photons, ns and n0 respectively. The intensity of the beam from the ptychographic reconstruction of the Siemens star (cf. Fig. 2) after back propagation of 1mm into its focus is visualized on top of the darkfield (d). Note, that the propagation of 1mm corresponds to the lateral displacement of the sample between the two different measurements (±100μm due to an uncertainty of the position on the sample holder with respect to the curvature of the polyimide foil). The centre of the reconstructed beam is chosen to be at the cell-position of A4 (cf. Fig. 6). Sample region B - incoherent setting: a section of the scan B1 is shown as darkfield (f) and differential phase contrast (e). The position of the cell diffraction of B3 is indicated on top of the incoherent darkfield as red circle (f). Sample region B - coherent setting: darkfield (h) and phase contrast (g) of a corresponding coherent mesh scan B2 are shown. Numerical simulations C: synthetic cell sample (top) and calculated differential phase contrast images (below) according to Eq. (1) using the back propagated probe (cf. Fig. 2) at different positions on the optical axis. The distance of propagation is displayed and the corresponding intensity of the probe is visualised as an overlay using the same colorbar as in (d).

Table 2. Overview of diffraction data sets according to dwell time ΔT, step size Δxy and number of scan points Nx × Ny within the rectangular mesh. In addition, Si filter thickness (BS denotes usage of beamstops instead of Si filter) and nominal gap of slits S0 are shown as well as the results from diffraction analysis for the exponent of the diffraction curves ν, highest scattering vector q and dose estimations D.

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Secondly, more diffraction data was taken with the same settings on the sample region B (cf. Fig. 5, Table 2). However, a rather incoherent beam was used by opening the slits S0 to a width of Δ = 0.42mm. Two beamstops were used during recording of the datasets B1 and B3. Afterwards, the scan B2 was performed in the coherent setting without any beamstop but with a Si filter of 50μm thickness.

For each mesh scan four contrast values can be calculated from every single diffraction pattern: the integral signal (transmission), the ratio of the integral signals from a region defining the central beam and from its counter set (darkfield) and the relative centre of mass movement in X- and Y-direction (differential phase contrast, DPC-x and DPC-y respectively). The dark-field contrast is a measure for the scattering strength of the sample. In the case that attenuation effects in the sample are negligible and all photons behind the sample can be detected, it reflects the ratio of scattered photons to incident photons. As explained in [42

42. G. Morrison and B. Niemann, “Differential phase contrast x-ray microscopy,” in X-Ray Microscopy and Spectromicroscopy, Würzburg, 1996, J. Thieme, ed. (Springer, 1998), 85–94.

44

44. P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80, 043813 (2009). [CrossRef]

], in addition to transmission and darkfield the differential phase contrast DPC-x and DPC-y reflects the gradient in averaged phase shift over the illuminated sample spot. Note that by blocking a part of the primary beam using a beamstop, none of these contrasts reflect quantitative measurements anymore. However, the tails of the primary beam can still be used for the analysis to obtain sufficient contrast. Here, only a subset of 194 × 194 pixels around the centre of the beam of each diffraction pattern has been used to obtain the different contrasts.

Next, we segmented the corresponding diffraction patterns according to the contrast (dark-field or DPC) into ‘cell and sample holder’ Σ1 and ‘sample holder’ Σ2 (cf. e.g. subgraphs in Fig. 6, 7). This was done either by thresholding the darkfield image or by thresholding a contrast image followed by further image processing. Diffraction curves I(|q⃗|) have been calculated for both sets of diffraction patterns by averaging over the set 〈·〉Σi=1,2 and azimuthally averaging 〈·〉ϕ with respect to constant radii. The difference signal
ΔI(|q|)=Ij(q)ϕjΣ1Ij(q)ϕjΣ2
(5)
is the basis for our analysis. Here, the momentum transfer |q⃗| = k sinα/2 is used with α being the angle between optical axis and scattering vector, and k denoting the wavenumber. Note, that this diffraction data significantly exceeds the q-range of the ptychographic analysis. Due to spurious OSA scattering, some sectors of the diffraction patterns had to be excluded. Dead and hot pixels have been masked. The absence of correlations between the scattering signal from the sample holder and the scattering of the cellular specimen justifies background subtraction similar to conventional (incoherent) diffraction. The fact that we have used both coherent and incoherent illumination settings with consistent results can be regarded as a validation of this assessment. We stress that background subtraction based on a pixel by pixel analysis of the sample (sample pixels versus background pixels) was found to be significantly more robust than background subtraction based on a second empty sample.

Fig. 6 Sample region A - coherent setting: (top) Diffraction data of A2 and A3 (cf. Table 2). The sets of diffraction patterns were segmented into areas with Σ1 and without cell signal Σ2 using the differential phase contrast image of A2 (see inset). The difference between the two classes of diffraction sets averaged according to their set size is depicted next to the legend in the top right corner for A2. Diffraction signals were obtained by azimuthally averaging each diffraction pattern of a scan and thereafter averaging according to their segmentation. The segmented diffraction curves of A2 are shown and the difference signals are presented for both scans in combination with their power-law fits. Exponents of ν = −3.2 and ν = −3.7 were obtained for A2 and A3, respectively. (Bottom) presents the results of A4 (cf. Fig. 5(d)). The diffraction signals obtained after azimuthally averaging are shown in the graph. The difference between the diffraction data of the cell material and of the sample holder is depicted in the lower left corner. A slightly modified power-law yields an exponent of ν = −3.6. In both graphs the points q = q ≡ (c/a)1/ν are marked.
Fig. 7 Sample region B - incoherent setting: incoherent diffraction curves were obtained from data B1 and B3 (cf. Table 2). The image segmentation of B1 according to areas with cell signal Σ1 and without cell signal Σ2 is shown as an inset. Power-law fits yield exponents of ν = −3.5 and ν = −3.8 for B1 and B3, respectively. The points q = q ≡ (c/a)1/ν are marked for both curves.

One may be concerned about the effect which a highly convergent beam has on small-angle X-ray scattering, since SAXS measurements usually require collimated beams. By placing a sample in the planar wave-front of the focal plane, the far-field pattern of the sample structure is measured. More precisely, the measured far-field pattern corresponds to the Fourier transform of the sample convolved with the Fourier transform of the probe, which is a plane wave with for example a Gaussian envelope in the focal plane. Since for cellular nano-diffraction we are by definition interested in diffraction from structures, which are smaller than the beam size, one realizes that the convolution of the far-field pattern with the Fourier transform of the probe has only a negligible effect, if the structure size is much smaller than the beam size, which is the case in the present work. Furthermore, one may even argue that it will be easier to control the wave-front flatness of a coherent nano-beam by ptychography than the collimated (stochastic) wave-fronts of macroscopic SAXS measurements.

3. Results

3.1. Ptychographic imaging of the resolution chart and probe retrieval

The ptychographic reconstruction of the 500nm thick layer of tantalum suitably matches the corresponding theoretical phase shift of ΔΦ(E = 6.2keV) = −0.34π and transmission of I/I0 = 0.77 (not shown). Its innermost part consisting of 50nm structures is clearly resolved (cf. Fig. 2). The ‘overall’-PRTF (cf. Eq. (2)) in Fig. 3(a) (green curve) shows signal at higher frequencies beyond the black dash-dotted line of ν = 34μm−1 or equivalently of dx = 15nm half-period length. In addition, we calculated the PRTF according to an average over a single scan point (red curve). Note that the contribution of the probe to the diffraction patterns is confined to a small circular area up to νmax ≈ 5μm−1. The disc-like image of the FZP and OSA can be clearly distinguished from the contribution of the Siemens star at high frequencies. We thus attribute the resolution to the sample and not the probe. Let us briefly consider the imaging regime of the diffraction patterns. Using a Gaussian beam model, the radius of curvature of the probe R(z) = z(1+(z/z0)2), z0 being the Rayleigh range, can be approximated by R(z) ≈ z at a distance of ca. 1mm to the focus. Estimating the probe diameter to be 1–3μm one thus obtains Fresnel numbers F in the range of 4 ≤ F ≤ 45. Note that in this regime one may rather speak of ptychographic Fresnel coherent diffractive imaging (cf. [10

10. 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] [PubMed]

, 45

45. D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic fresnel coherent diffractive imaging,” Phys. Rev. A 80, 063823 (2009). [CrossRef]

]). The relationship between the resolution and the highest angle at which signal is recorded may thus not be applicable due to the phase curvature of the illuminating probe (cf. Fig. 2(b)). As an alternative, we make use of the power spectral density (PSD) and line scans at the sharp edges to determine the resolution (see appendix for details of calculations of the PSD). A fit of the vertical and horizontal edges to an error function yields line-spread functions with FWHM-values of 16nm (v) and 17nm (h), respectively. In addition, the PSD of the reconstructed phase (cf. Fig. 3(b)) supports the conclusion that features in the range corresponding to 15 – 10nm half-period length are resolved.

The reconstruction of the probe can be used to analyse the complex field of the focus [46

46. A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010). [CrossRef]

48

48. Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011). [CrossRef]

]. Consider the following operation on the complex field of the probe P (’sharpness’, cf. [49

49. M. Guizar-Sicairos, “Methods for coherent lensless imaging and x-ray wavefront measurement,” Ph.D. Thesis, University of Rochester (2010).

])
ϒ(Δz):=2dS|𝔉{P}(x,Δz)|4,
(6)
where 𝔉{·}(·, Δz) denotes the free space back propagation by Δz. We have determined the points of highest focusing by locating two maxima of ϒ(Δz). The numerical back propagation of the complex wave field thus reveals an astigmatism in the focal region (cf. Media 1). The difference of propagation distances yields a distance of ΔFhv ≃ 0.7mm between the two orthogonal directions of highest focusing. The field along the optical axis is displayed as planes of constant x and y through the centre (cf. Fig. 2(b)). A Gaussian fit to the regions of highest focusing yields FWHM-values of 89nm in the vertical direction and 93nm in the horizontal direction.

3.2. Ptychographic imaging of Deinococcus radiodurans

There are features in the reconstructions visible within sizes of one to two pixels indicating a half period length resolution in the range of the pixel dimension of dx = 46nm. The PRTF reflects a good reconstruction up to the highest frequencies. In comparison to the PRTF of the Siemens star (cf. Fig. 3(a)), we observe a lower trend in the high frequency region of the reconstruction of Fig. 4(b). The PSD (blue curve, Fig. 3(b)) of the reconstruction in Fig. 4(b) clearly shows structures up to dx = 50nm (half period length). The cutoff is very small and can be better seen in the 2D PSD (see appendix) indicating a resolution in the range of dx = 50 – 55nm.

The tomographic reconstruction Fig. 4(d) reveals the 3D structure of bacterial cells seen in (Fig. 4(b)). A video is provided in the supplementary material (cf. Media 2). We can identify the 3D distribution of the high-density regions (red). Assuming the ratio of the mass number and the atomic number on average over a voxel to be nearly constant, the 3D phase distribution ϕ(r⃗) can be used to estimate the effective mass density of the cell [1

1. K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 529–534 (2010). [CrossRef]

]
ρc(r)2uλr0ϕ(r),
(7)
where u denotes atomic mass unit, r0 the classical electron radius and λ the wavelength. For cell material of H50C30N9O10S1 the error is less than 10% [1

1. K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 529–534 (2010). [CrossRef]

]. We used the tomographic reconstruction to estimate the mass density of the high-density regions. With the help of software tools for analysing tomographic data, we defined a sub-volume which can be seen as a magenta label in the lower part of the tomogram (front-view, Fig. 4(d)). The average over the sub-volume yields <ρc> ≈ 1.6g/cm3.

The results show that ptychographic tomography is a suitable approach not only for strongly scattering samples (e.g. mineralized tissue [14

14. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010). [CrossRef] [PubMed]

] and stained bacteria [15

15. M. Guizar-Sicairos, A. Diaz, M. Holler, M. S. Lucas, A. Menzel, R. A. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19, 21345–21357 (2011). [CrossRef] [PubMed]

]) but also for weakly scattering specimens such as freeze-dried cells. Note that ptychography from unmineralized tissues and in particular cells is limited by the signal to noise ratio. Thus weakly scattering samples are much more difficult to reconstruct than for example mineralized tissues such as bone or more generally solid state samples.

In comparison to the reconstruction of the single projection in Fig. 4(b) which has been taken before the tomographic dataset, the quality of the tomographic reconstruction is affected by the overall consistence of all ptychographic reconstructions. The inner structure slightly deviates from the first phase map indicating a moderate radiation induced change of the inner structure. The applied dose for the tomographic dataset was estimated to be ∼ 2.4 · 108 Gy. Another tomographic dataset has been taken after the first one in order to achieve an angular increment of ΔΦ = 1°. However, the combined tomogram (not shown) shows less contrast in the high-density regions and was thus discarded. The image in Fig. 4(c) has been recorded after an integral dose of > 5 · 108 Gy and clearly shows radiation induced structure changes in comparison to its initial state in Fig. 4(b). Note, that the structural changes may also have an effect on <ρc>. Our dose estimations are summarized in Table 1 (see also Fig. 8). We thus conclude that in contrast to single projections, medium and high resolution tomography will necessitate cryogenic sample conditions.

Fig. 8 Photon surface densities ργ of diffraction datasets A2 (a), A3 (b), B1 (c) and ptychographic dataset (Fig. 4(a)) (d). The location of bacterial cells is indicated by black lines. (a)–(c) same colorbar as in (d).

3.3. Diffraction data of Deinococcus radiodurans

The upper graph in Fig. 6 shows the background corrected cellular diffraction signal as measured for ΔT = 1s and ΔT = 10s, leading to q = 0.75nm−1 and q = 0.95nm−1, respectively. This is in agreement to experimental expectation. The ΔT = 100s data set recorded at a single spot, displayed in the lower graph of Fig. 6 shows an even stronger signal, extending to q = 1.88nm−1. This demonstrates that cellular structure can be analysed by diffraction at resolutions (in rec. space) which are currently not accessible by X-ray microscopy, either in the conventional form or the more recent Coherent Diffractive Imaging (CDI). The data shown in Fig. 6 also illustrates the difficulties associated with spurious scattering by the OSA, which is difficult to suppress due to thermal effects induced by the cryojet.

Finally we note, that at small q the A4 data set shows a transition of the power law decay to a plateau (not fully reached due to the BS). This behaviour was captured by adapting the fitting function to I = a · qν (1 − exp(−(d · q/2π)2)) + c, with d denoting the crossover. The resulting value of d ≈ 60nm gives a typical length scale of the largest structure.

Next, we would like to know which effect opening the slit S0 and the associated relaxation of coherence has on the resolution. Note that the high flux does not necessarily have to result in higher q due to a possible decrease in signal to noise. However, the results shown in Fig. 7 of the B1 data set, show q = 2.3nm−1 which is indeed the highest value obtained. Furthermore, it seems that dose fractionation over the entire aggregate of cells presents an advantage over a single point long measurement where we see a moderate increase of resolution (in q-space), at least for detectors which are free of read-out noise.

We present the results of the dose estimations in Fig. 8, as described in the method section (cf. Eq. (4)). Note that our dose estimations are spatially more precise than a simpler estimation based on an average fluence, which can be best seen in Fig. 8(b). After all, a comparison of the attainable resolution in diffraction settings always should be accompanied by a corresponding comparison in dose, in view of the important issue of radiation damage in biological specimens. In case of the sample region A (coherent setting) the average photon flux was measured with the PILATUS to be ∼ 7 · 108 photons/s. Assuming the same photon flux at each scan position we obtain applied doses of 3.6·107 Gy, 2.2·107 Gy and 1.3·108 Gy for A2, A3 and A4, respectively. In case of the incoherent setting of sample region B we measured the increase in flux to be in the range of a factor of 10–17 due to the increased gap of the slit S0. Therefore, we assumed an average photon flux of ∼ 1010 photons/s yielding applied doses of 5.5·107 Gy and 2·109 Gy in case of the incoherent datasets B1 and B3, respectively. The results are summarized in Table 2.

4. Conclusion, discussion, and outlook

In summary, we have carried out an X-ray structural analysis on bacterial cells of D. radiodurans, combining coherent imaging and cellular nano-diffraction. For this study, the bacterial cells were vitrified by plunge freezing followed by freeze-drying. However, in order to minimize radiation damage, samples were at least for experiments with highest flux (incoherent setting) kept at cryogenic temperatures by use of a cryogenic N2 jet. This shows that the experimental scheme is compatible with future extension to frozen hydrated samples with near optimum structure preservation.

Secondly, we have shown how the cellular diffraction signal can be extracted and analysed. Furthermore, we have compared the cellular diffraction curves as a function of momentum transfer after azimuthal averaging, for different parameters of the illuminating beam, notably flux and coherence. The observed curves could all be described by a power law decay with an exponent ν in the range −4 ≤ ν ≤ −3. Importantly, the cellular diffraction experiments open the analysis to maximum scattering vectors, up to a momentum transfer of q ≈ 2nm−1, which is to date unachievable by CDI. No indications for ordered structures associated with DNA compactification were observed in this range. Note, however, that the absence of peaks or modulations in the diffraction curves does not preclude ordered structures with signals below the dominating power-law decay.

In addition, we have shown how the ptychographically reconstructed probe can be directly used for enhanced control of the nano-diffraction experiment. Most importantly, the illuminated region of the sample can be well characterised. In particular, the size and shape of the beam does not have to be measured by knife-edge scans with the associated inaccuracies. Moreover, the exact complex-valued wave front is accessible, from which all beam parameters desired to better control the nano-diffraction experiment can be derived, such as wavefront curvature for example. Furthermore, the scanning X-ray microscopy images in phase contrast and darkfield contrast can be analysed with the full probe information at hand. Finally, we have shown how dose estimations become more precise by taking into account the exact beam profile, as well as the overlap between serial exposures during scanning.

However, we are also aware of the limitations in the presented approach. While the resolution can be extended by analysis in reciprocal space, this resolution applies to the averaged structure. The real space resolution of this averaging process as given by the focal spot is clearly lower in cellular nano-diffraction than in ptychographic imaging or more generally in CDI. At the same time, the overhead for detector read-out during scanning of three degrees of freedom needed for tomography limits the reasonable field of view, and hence in most cases the size of the object. Thus, for multicellular organisms, adaptable focal sizes and scanning steps would be desirable.

5. Appendix

5.1. Calculation of the PSD

The usage of a window function for the calculation of the PSD is common in digital signal processing. A very useful window is the so-called Kaiser-window w(x) (cf. [51

51. K. Giewekemeyer, “A study on new approaches in coherent x-ray microscopy of biological specimens,” Ph.D. Thesis, Universität Göttingen (2011).

]):
w(x)={I0(β1(2x/L)2)/I0(β),L/2xL/20,|x|>L/2
(9)
where I0 is the modified Bessel-function of the first kind, β is a free parameter and L is the window length. Note that one recovers the rectangular window for β = 0 which is intrinsically used for the calculation of the discrete Fourier transform of any discretised signal of finite length. The Kaiser-window with β > 0 helps to reveal the high frequency content of a signal that would be otherwise suppressed by noise or a huge amount of low frequencies in the data. We used the Kaiser-window for the calculation of the PSDs (Fig. 9).

Fig. 9 (a) 2D PSD of ‘ePIE’ reconstruction of Siemens star test pattern. Note that the Siemens star structure can be clearly seen in the radial direction. (b) 2D PSD of ptychographic (‘DM’) reconstruction of D. radiodurans in Fig. 4(b). Scale bars correspond to spatial frequencies.

5.2. Discussion knife-edge scanning

At first, let us consider a simulated ’dumbbell-shaped’ probe intensity (cf. Fig. 10(a)), which violates the assumption of factorization. Certainly, this model-probe may have pronounced and somewhat exaggerated side-lobes. However, the reader may note that the probe reconstructed experimentally in this work also exhibits side-lobes at least in the plane shown in Fig. 10(d). For this choice of probe, the fit of the error function to the calculated knife-edge scan along the horizontal direction (cf. Fig. 10(b)) yields FWHM = 0.635μm in comparison to the theoretical value of FWHM = 0.1μm at the centre (cf. Fig. 10(c)).

Fig. 10 (a) Intensity of a model-probe exhibiting a dumbbell-shape. The calculated knife-edge scan along the horizontal direction is shown in (b). The white, dashed line in (a) indicates the region of highest focusing which is shown in (c) (solid, black line). The resulting Gaussian shape as estimated from the knife-edge scan is depicted as solid, red curve. (d) shows the intensity of the reconstructed probe at Δz = −0.686mm behind the plane of the sample. A white, dashed line indicates the part which has been used for estimating the beam width. The corresponding result of the calculated knife-edge scan is shown in (e). The FWHM values from calculated knife-edge scans are shown as function of distance to the sample plane in (f). The red and the blue curve denote horizontal and vertical beam-width, respectively. In addition, ϒ(Δz) is shown as solid, black curve which has been scaled arbitrarily for reasons of visualisation.

Next, let us consider the experimental case of the reconstructed probe field intensity (ptychographic reconstruction of the Siemens star resolution chart, cf. Fig. 2). After a calculation of the probe at several planes behind the sample, we performed ’numerical’ knife-edge scans along the horizontal and vertical direction in a few hundred planes to obtain the beam-width along the optical axis. The horizontal and vertical beam-widths as a function of the distance to the sample plane are shown in Fig. 10(f) as red and blue curves, respectively. Note that the minimal beam-widths do not exactly coincide with the maxima of ϒ(Δz) (Eq. (6)), indicating that the horizontal and vertical axes do not coincide with the minor and major axes of the beam. As is known from knife-edge scanning of astigmatic Gaussian beams [53

53. Hans R. Bilger and Taufiq Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. 24, 686–690 (1985). [CrossRef] [PubMed]

], it is not generally sufficient to perform knife-edge scans along the horizontal and vertical axes. Contrarily, one needs to scan at several angles to determine the major and minor axes in one plane. Furthermore, it becomes clear that features such as the double-peak and donut-shape (at certain positions of the optical axis, cf. Fig. 5(C)), will be extremely difficult to infer from a set of knife-edge scans along two orthogonal axes.

Finally, let us consider the intensity of the probe at Δz = −0.686mm (cf. Fig. 10(d)) behind the plane of the sample. In order to get a FWHM estimate that becomes close to the theoretical value of FWHM = 0.093μm (cf. Fig. 2) the fit of the error function to the knife-edge scan needs to be restricted to values close to the centre yielding a FWHM = 0.104μm (cf. Fig. 10(e)). The outcome of the FWHM value of the fit could also be improved by manually weighting the data points near the centre. However, without a two-dimensional representation of the probe intensity, justification for both means would not be granted. We conclude that even if the information on the phase front of the beam is not needed, reconstruction of the intensity distribution from selected knife-edge scans is problematic and does not stand up to the full probe reconstruction by ptychography. Therefore, we recommend that ptychography should replace knife-edge scans whenever possible, i.e. if the beam is sufficiently coherent.

Acknowledgments

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M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010). [CrossRef] [PubMed]

15.

M. Guizar-Sicairos, A. Diaz, M. Holler, M. S. Lucas, A. Menzel, R. A. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19, 21345–21357 (2011). [CrossRef] [PubMed]

16.

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012). [CrossRef]

17.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005). [CrossRef] [PubMed]

18.

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” New J. Phys. 12, 035008 (2010). [CrossRef]

19.

J. Nelson, X. Huang, J. Steinbrener, D. Shapiro, J. Kirz, S. Marchesini, A. M. Neiman, J. J. Turner, and C. Jacobsen, “High-resolution x-ray diffraction microscopy of specifically labeled yeast cells,” Proc. Natl. Acad. Sci. USA 107, 7235–7239 (2010). [CrossRef] [PubMed]

20.

O. Glatter and O. Kratky, Small Angle X-Ray Scattering (Acad. Pr., 1982).

21.

A. Guinier and G. Fournet, Small-Angle Scattering of X-Rays (Wiley, 1955).

22.

P. Fratzl, H. F. Jakob, S. Rinnerthaler, P. Roschger, and K. Klaushofer, “Position-resolved small-angle x-ray scattering of complex biological materials,” J. Appl. Crystallogr. 30, 765–769 (1997). [CrossRef]

23.

S. Rinnerthaler, P. Roschger, H. F. Jakob, A. Nader, K. Klaushofer, and P. Fratzl, “Scanning small angle x-ray scattering analysis of human bone sections,” Calcified Tissue Int. 64, 422–429 (1999). [CrossRef]

24.

O. Paris, D. Loidl, H. Peterlik, M. Muller, H. Lichtenegger, and P. Fratzl, “The internal structure of single carbon fibers determined by simultaneous small- and wide-angle scattering,” J. Appl. Crystallogr. 33, 695–699 (2000). [CrossRef]

25.

A. Gourrier, W. Wagermaier, M. Burghammer, D. Lammie, H. S. Gupta, P. Fratzl, C. Riekel, T. J. Wess, and O. Paris, “Scanning x-ray imaging with small-angle scattering contrast,” J. Appl. Crystallogr. 40, 78–82 (2007). [CrossRef]

26.

O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009). [CrossRef]

27.

T. H. Jensen, M. Bech, O. Bunk, M. Thomsen, A. Menzel, A. Bouchet, G. L. Duc, R. Feidenhans’l, and F. Pfeiffer, “Brain tumor imaging using small-angle x-ray scattering tomography,” Phys. Med. Biol. 56, 1717–1726 (2011). [CrossRef] [PubMed]

28.

J. Dubochet, M. Adrian, J.-J. Chang, J.-C. Homo, J. Lepault, A.W. McDowall, and P. Schultz, “Cryo-Electron microscopy of vitrified specimens,” Q. Rev. Biophys. 21, 129–228 (1988). [CrossRef] [PubMed]

29.

S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salome, and C. David, “High- efficiency fresnel zone plates for hard x-rays by 100 kev e-beam lithography and electroplating,” J. Synchrotron Radiat. 18, 442–446 (2011). [CrossRef] [PubMed]

30.

S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology 21, 295303 (2010). [CrossRef] [PubMed]

31.

P. Kraft, A. Bergamaschi, C. Broennimann, R. Dinapoli, E. F. Eikenberry, B. Henrich, I. Johnson, A. Mozzanica, C. M. Schleputz, P. R. Willmott, and B. Schmitt, “Performance of single-photon- counting pilatus detector modules,” J. Synchrotron Radiat. 16, 368–375 (2009). [CrossRef] [PubMed]

32.

K. Nygård, O. Bunk, E. Perret, C. David, and J.F. van der Veen, “Diffraction gratings as small-angle X-ray scattering calibration standards,” J. Appl. Cryst. 43, 350–351 (2010). [CrossRef]

33.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009). [CrossRef] [PubMed]

34.

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

35.

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]

36.

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von Knig, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010). [CrossRef]

37.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008). [CrossRef]

38.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006). [CrossRef]

39.

M. Guizar-Sicairos, S.T. Thurman, and J.R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008). [CrossRef] [PubMed]

40.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

41.

M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J. Electron Spectrosc. 170, 4–12 (2009). [CrossRef]

42.

G. Morrison and B. Niemann, “Differential phase contrast x-ray microscopy,” in X-Ray Microscopy and Spectromicroscopy, Würzburg, 1996, J. Thieme, ed. (Springer, 1998), 85–94.

43.

G. Morrison, W. Eaton, R. Barrett, and P. Charalambous, “STXM imaging with a configured detector,” J. Phys. IV France 104, 547–550 (2003). [CrossRef]

44.

P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80, 043813 (2009). [CrossRef]

45.

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic fresnel coherent diffractive imaging,” Phys. Rev. A 80, 063823 (2009). [CrossRef]

46.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010). [CrossRef]

47.

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010). [CrossRef] [PubMed]

48.

Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011). [CrossRef]

49.

M. Guizar-Sicairos, “Methods for coherent lensless imaging and x-ray wavefront measurement,” Ph.D. Thesis, University of Rochester (2010).

50.

J. Zimmerman and J. Battista, “A ring-like nucleoid is not necessary for radioresistance in the deinococcaceae,” BMC Microbiol. 5, 17 (2005). [CrossRef] [PubMed]

51.

K. Giewekemeyer, “A study on new approaches in coherent x-ray microscopy of biological specimens,” Ph.D. Thesis, Universität Göttingen (2011).

52.

T. Weitkamp, “Imaging and tomography with high resolution using coherent hard synchroton radiation,” Ph.D. Thesis, University of Hamburg (2002).

53.

Hans R. Bilger and Taufiq Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. 24, 686–690 (1985). [CrossRef] [PubMed]

OCIS Codes
(100.5070) Image processing : Phase retrieval
(100.6950) Image processing : Tomographic image processing
(170.1530) Medical optics and biotechnology : Cell analysis
(340.7440) X-ray optics : X-ray imaging

ToC Category:
Image Processing

History
Original Manuscript: May 10, 2012
Revised Manuscript: June 8, 2012
Manuscript Accepted: June 13, 2012
Published: August 8, 2012

Virtual Issues
Vol. 7, Iss. 10 Virtual Journal for Biomedical Optics

Citation
R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, "Hard X-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction," Opt. Express 20, 19232-19254 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-17-19232


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References

  1. K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA107, 529–534 (2010). [CrossRef]
  2. S. Levin-Zaidman, J. Englander, E. Shimoni, A. K. Sharma, K. W. Minton, and A. Minsky, “Ringlike structure of the deinococcus radiodurans genome: a key to radioresistance?,” Science299, 254–256 (2003). [CrossRef] [PubMed]
  3. M. Eltsov and J. Dubochet, “Fine structure of the deinococcus radiodurans nucleoid revealed by cryoelectron microscopy of vitreous sections,” J. Bacteriol.187, 8047–8054 (2005). [CrossRef] [PubMed]
  4. M. Eltsov and J. Dubochet, “Rebuttal: Ring-like nucleoids and dna repair in deinococcus radio-durans,” J. Bacteriol.188, 6052 (2006). [CrossRef]
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  6. A. Minsky, E. Shimoni, and J. Englander, “Rebuttal: study of the deinococcus radiodurans nucleoid,” J. Bacteriol.188, 6059 (2006). [CrossRef]
  7. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett.98, 034801 (2007). [CrossRef] [PubMed]
  8. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett.85, 4795–4797 (2004). [CrossRef]
  9. 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,” Nature400, 342–344 (1999). [CrossRef]
  10. 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] [PubMed]
  11. 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]
  12. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21, 2758–2769 (1982). [CrossRef] [PubMed]
  13. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science321, 379–382 (2008). [CrossRef] [PubMed]
  14. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010). [CrossRef] [PubMed]
  15. M. Guizar-Sicairos, A. Diaz, M. Holler, M. S. Lucas, A. Menzel, R. A. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express19, 21345–21357 (2011). [CrossRef] [PubMed]
  16. A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B85, 020104 (2012). [CrossRef]
  17. D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA102, 15343–15346 (2005). [CrossRef] [PubMed]
  18. K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” New J. Phys.12, 035008 (2010). [CrossRef]
  19. J. Nelson, X. Huang, J. Steinbrener, D. Shapiro, J. Kirz, S. Marchesini, A. M. Neiman, J. J. Turner, and C. Jacobsen, “High-resolution x-ray diffraction microscopy of specifically labeled yeast cells,” Proc. Natl. Acad. Sci. USA107, 7235–7239 (2010). [CrossRef] [PubMed]
  20. O. Glatter and O. Kratky, Small Angle X-Ray Scattering (Acad. Pr., 1982).
  21. A. Guinier and G. Fournet, Small-Angle Scattering of X-Rays (Wiley, 1955).
  22. P. Fratzl, H. F. Jakob, S. Rinnerthaler, P. Roschger, and K. Klaushofer, “Position-resolved small-angle x-ray scattering of complex biological materials,” J. Appl. Crystallogr.30, 765–769 (1997). [CrossRef]
  23. S. Rinnerthaler, P. Roschger, H. F. Jakob, A. Nader, K. Klaushofer, and P. Fratzl, “Scanning small angle x-ray scattering analysis of human bone sections,” Calcified Tissue Int.64, 422–429 (1999). [CrossRef]
  24. O. Paris, D. Loidl, H. Peterlik, M. Muller, H. Lichtenegger, and P. Fratzl, “The internal structure of single carbon fibers determined by simultaneous small- and wide-angle scattering,” J. Appl. Crystallogr.33, 695–699 (2000). [CrossRef]
  25. A. Gourrier, W. Wagermaier, M. Burghammer, D. Lammie, H. S. Gupta, P. Fratzl, C. Riekel, T. J. Wess, and O. Paris, “Scanning x-ray imaging with small-angle scattering contrast,” J. Appl. Crystallogr.40, 78–82 (2007). [CrossRef]
  26. O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys.11, 123016 (2009). [CrossRef]
  27. T. H. Jensen, M. Bech, O. Bunk, M. Thomsen, A. Menzel, A. Bouchet, G. L. Duc, R. Feidenhans’l, and F. Pfeiffer, “Brain tumor imaging using small-angle x-ray scattering tomography,” Phys. Med. Biol.56, 1717–1726 (2011). [CrossRef] [PubMed]
  28. J. Dubochet, M. Adrian, J.-J. Chang, J.-C. Homo, J. Lepault, A.W. McDowall, and P. Schultz, “Cryo-Electron microscopy of vitrified specimens,” Q. Rev. Biophys.21, 129–228 (1988). [CrossRef] [PubMed]
  29. S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salome, and C. David, “High- efficiency fresnel zone plates for hard x-rays by 100 kev e-beam lithography and electroplating,” J. Synchrotron Radiat.18, 442–446 (2011). [CrossRef] [PubMed]
  30. S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology21, 295303 (2010). [CrossRef] [PubMed]
  31. P. Kraft, A. Bergamaschi, C. Broennimann, R. Dinapoli, E. F. Eikenberry, B. Henrich, I. Johnson, A. Mozzanica, C. M. Schleputz, P. R. Willmott, and B. Schmitt, “Performance of single-photon- counting pilatus detector modules,” J. Synchrotron Radiat.16, 368–375 (2009). [CrossRef] [PubMed]
  32. K. Nygård, O. Bunk, E. Perret, C. David, and J.F. van der Veen, “Diffraction gratings as small-angle X-ray scattering calibration standards,” J. Appl. Cryst.43, 350–351 (2010). [CrossRef]
  33. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009). [CrossRef] [PubMed]
  34. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109, 1256–1262 (2009). [CrossRef] [PubMed]
  35. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A15, 1662–1669 (1998). [CrossRef]
  36. M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von Knig, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys.12, 035017 (2010). [CrossRef]
  37. O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy108, 481–487 (2008). [CrossRef]
  38. H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A23, 1179–1200 (2006). [CrossRef]
  39. M. Guizar-Sicairos, S.T. Thurman, and J.R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett.33, 156–158 (2008). [CrossRef] [PubMed]
  40. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).
  41. M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J. Electron Spectrosc.170, 4–12 (2009). [CrossRef]
  42. G. Morrison and B. Niemann, “Differential phase contrast x-ray microscopy,” in X-Ray Microscopy and Spectromicroscopy, Würzburg, 1996, J. Thieme, ed. (Springer, 1998), 85–94.
  43. G. Morrison, W. Eaton, R. Barrett, and P. Charalambous, “STXM imaging with a configured detector,” J. Phys. IV France104, 547–550 (2003). [CrossRef]
  44. P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A80, 043813 (2009). [CrossRef]
  45. D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic fresnel coherent diffractive imaging,” Phys. Rev. A80, 063823 (2009). [CrossRef]
  46. A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett.96, 091102 (2010). [CrossRef]
  47. C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy110, 325–329 (2010). [CrossRef] [PubMed]
  48. Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B83, 214109 (2011). [CrossRef]
  49. M. Guizar-Sicairos, “Methods for coherent lensless imaging and x-ray wavefront measurement,” Ph.D. Thesis, University of Rochester (2010).
  50. J. Zimmerman and J. Battista, “A ring-like nucleoid is not necessary for radioresistance in the deinococcaceae,” BMC Microbiol.5, 17 (2005). [CrossRef] [PubMed]
  51. K. Giewekemeyer, “A study on new approaches in coherent x-ray microscopy of biological specimens,” Ph.D. Thesis, Universität Göttingen (2011).
  52. T. Weitkamp, “Imaging and tomography with high resolution using coherent hard synchroton radiation,” Ph.D. Thesis, University of Hamburg (2002).
  53. Hans R. Bilger and Taufiq Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt.24, 686–690 (1985). [CrossRef] [PubMed]

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