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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 1037–1050
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Ptychographic coherent x-ray diffractive imaging in the water window

K. Giewekemeyer, M. Beckers, T. Gorniak, M. Grunze, T. Salditt, and A. Rosenhahn  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 1037-1050 (2011)
http://dx.doi.org/10.1364/OE.19.001037


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Abstract

Coherent x-ray diffractive microscopy enables full reconstruction of the complex transmission function of an isolated object to diffraction-limited resolution without relying on any optical elements between the sample and detector. In combination with ptychography, also specimens of unlimited lateral extension can be imaged. Here we report on an application of ptychographic coherent diffractive imaging (PCDI) in the soft x-ray regime, more precisely in the so-called water window of photon energies where the high scattering contrast between carbon and oxygen is well-suited to image biological samples. In particular, we have reconstructed the complex sample transmission function of a fossil diatom at a photon energy of 517 eV. In imaging a lithographically fabricated test sample a resolution on the order of 50 nm (half-period length) has been achieved. Along with this proof-of-principle for PCDI at soft x-ray wavelengths, we discuss the experimental and technical challenges which can occur especially for soft x-ray PCDI.

© 2011 Optical Society of America

1. Introduction

With the advent of third-generation synchrotron radiation sources and free electron lasers coherent x-ray diffractive imaging (CDI or CXDI) has emerged as a new tool for structure analysis on the nanoscale [1

1. K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 1–99 (2010). [CrossRef]

,2

2. P. Thibault and V. Elser, “X-ray diffraction microscopy,” Annu. Rev. Condens. Matter Phys. 1, 237–255 (2010). [CrossRef]

]. In the classical CDI experiment [3

3. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999). [CrossRef]

] a coherent plane wave illuminates a sample of several microns in diameter, and the resulting diffracted intensity is recorded in the far field. In an iterative process the object transmission function is then recovered numerically from the measured intensity. This conceptually rather simple experimental scheme has been applied very successfully (see [1

1. K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 1–99 (2010). [CrossRef]

, 2

2. P. Thibault and V. Elser, “X-ray diffraction microscopy,” Annu. Rev. Condens. Matter Phys. 1, 237–255 (2010). [CrossRef]

] and references therein) and has been proven to be extremely powerful in terms of resolution [4

4. C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schoder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101, 090801 (2008). [CrossRef] [PubMed]

, 5

5. 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. U.S.A. 107, 7235–7239 (2010). [CrossRef] [PubMed]

]. Reconstruction is possible, if the diffraction pattern is band-limited and recorded on a fine enough grid to sample its smallest features [6

6. J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003). [CrossRef]

]. This restricts the application of the method to isolated specimens of a lateral extent much smaller than the beam diameter. In addition to slow convergence and uniqueness issues, this limitation has been a motivation for alternative approaches such as holography, based on deterministic single-step reconstruction [7

7. S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004). [CrossRef] [PubMed]

10

10. A. P. Mancuso, T. Gorniak, F. Staier, O. M. Yefanov, R. Barth, C. Christophis, B. Reime, J. Gulden, A. Singer, M. E. Pettit, T. Nisius, T. Wilhein, C. Gutt, G. Grübel, N. Guerassimova, R. Treusch, J. Feldhaus, S. Eisebitt, E. Weckert, M. Grunze, A. Rosenhahn, and I. A. Vartanyants, “Coherent imaging of biological samples with femtosecond pulses at the free-electron laser flash,” N. J. Phys. 12, 035003 (2010). [CrossRef]

].

Beyond an early non-iterative approach based on “Wigner-deconvolution” [11

11. H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996). [CrossRef]

] Ptychographic Coherent (X-ray) Diffractive Imaging (PCDI) has emerged [12

12. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004). [CrossRef]

, 13

13. 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]

] as a generalization of conventional CDI suitable also for samples of unlimited lateral extent. Here a finite sample area is illuminated by a coherent beam, a diffraction pattern is recorded, and subsequently the sample is translated laterally before recording a new diffraction pattern and repeating the process until a desired field of view (FOV) has been scanned through the beam. The sampling condition is now obeyed by the finite size of the illuminated area. A certain degree of overlap [14

14. 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]

] between neighboring illuminated areas allows for a high redundancy in the recorded data which strongly facilitates the reconstruction process. Importantly, there is no need for a planar illumination function any more, as in recent variants of PCDI [15

15. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008). [CrossRef] [PubMed]

17

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

] the complex illuminating wave field can be determined independently from the sample transmission function using the same experimental dataset. This allows for the routine application of many different (possibly distorted) illumination functions, such as the unfocused, Fresnel-diffracted beam of a circular pinhole [18

18. 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. U.S.A. 107, 529–534 (2010). [CrossRef]

, 19

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

] or highly-confined wave fields, either focused by Fresnel zone plates [15

15. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008). [CrossRef] [PubMed]

], compound refractive lenses [20

20. 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]

] or focusing mirrors [21

21. 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]

], or confined by X-ray waveguides [22

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

].

Here we report on an application of the method using soft x-rays in the so-called water window energy range where the refractive index ratio of carbon and oxygen yields an especially high contrast of biological specimens against their natural aqueous environment [23

23. D. Weiss, G. Schneider, B. Niemann, P. Guttmann, D. Rudolph, and G. Schmahl, “Computed tomography of cryogenic biological specimens based on x-ray microscopic images,” Ultramicroscopy 84, 185–197 (2000). [CrossRef] [PubMed]

,24

24. C. A. Larabell and M. A. Le Gros, “X-ray tomography generates 3-d reconstructions of the yeast, saccharomyces cerevisiae, at 60-nm resolution,” Mol. Biol. Cell 15, 957–962 (2004). [CrossRef]

]. More specifically, we have applied ptychographic CDI at a photon energy of 517 eV to reconstruct the complex transmission function of a moderately absorbing fossil diatom. A pinhole was used to define the illumination on the sample. To assess the possible spatial resolution we have imaged a lithographically fabricated tantalum test pattern with essentially binary contrast (transmission values 1 and 0) at the given photon energy. In both experiments the complex illuminating wave field at the sample was reconstructed, allowing for back-propagation to the plane of the pinhole which was used to illuminate the specimens.

2. Experiments

2.1. Setup

Experiments were carried out at the undulator beamline UE52 – SGM of the Berlin electron storage ring BESSY II, using the dedicated vacuum chamber HORST (holographic x-ray scattering chamber) developed at the University of Heidelberg for coherent imaging experiments with soft x rays [10

10. A. P. Mancuso, T. Gorniak, F. Staier, O. M. Yefanov, R. Barth, C. Christophis, B. Reime, J. Gulden, A. Singer, M. E. Pettit, T. Nisius, T. Wilhein, C. Gutt, G. Grübel, N. Guerassimova, R. Treusch, J. Feldhaus, S. Eisebitt, E. Weckert, M. Grunze, A. Rosenhahn, and I. A. Vartanyants, “Coherent imaging of biological samples with femtosecond pulses at the free-electron laser flash,” N. J. Phys. 12, 035003 (2010). [CrossRef]

]. The incident beam was focused by mirrors and/or confined by slits to a size of about 17.4(h) × 100(v) μm2 at a photon energy of 517 eV. After passing a pinhole (stainless steel, thickness ca. 13 μm, diameter on the order of 2 μm [25

25. The pinhole actually had a slightly elliptical shape with largest (smallest) diameter of about 2 μm (1.5 μm).

], 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 the beam on a rectangular grid with 800 nm step size in horizontal and vertical direction to allow for sufficient overlap between illuminated areas of adjacent scan points. A schematic of the experiment is depicted in Fig. 1. To assure high positioning accuracy, closed-loop piezo-electric positioning stages (Physik Instrumente, Germany) were used for scanning the sample through the beam. The resulting diffraction patterns were recorded at a distance z2 = 0.49 m away from the sample on a back-illuminated, peltier-cooled CCD detector (DX436, Andor Technology, UK) with a pixel width of 13.5 μm in horizontal and vertical direction and a total number of 2048 × 2048 pixels.

Fig. 1 Experimental setup for ptychographic coherent x-ray diffractive imaging of a fossil diatom: The illuminating wave field is confined by a small pinhole with a diameter on the order of 2 μm, before it impinges onto the sample after propagating over a distance z1 ≃ 1 mm. The sample, a diatom on a silicon nitride membrane, is scanned laterally through the beam on a rectangular grid while at each scan point a diffraction pattern is collected on a two-dimensional CCD detector placed at a distance z2 ≃ 0.49 m away from the sample. The same setup and measurement principle was then used in a second experiment to image a test pattern structured by nano-lithography.

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 translated through the beam at a distance of z1 ≃ 1 mm from the pinhole and diffraction patterns obtained at 14 × 24 scan points on a rectangular grid with a spacing of 800 nm in lateral and vertical direction were used for reconstruction. Each diffraction pattern was collected during an illumination time of 0.18 s, making use of the full dynamic range of the detector without using a beamstop to block the direct beam. The total exposure time was thus 60.48 s for a scanned area of ca. 191 μm2. A dark image with the same illumination time was used for background correction. For reconstruction a region of 1920 × 1920 pixels was used on the detector, leading to a real-space pixel width of 45 nm in the sample plane. To reduce computational complexity the diffraction data was binned down by a factor of 2 along each dimension, yielding an effective detector pixel width of 27 μm.

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

For the second imaging experiment, a Siemens star test pattern (model ATN/XRESO-50HC, NTT-AT, Japan) consisting of a 500-nm-thick nanostructured tantalum layer on a transparent membrane (Ru(20 nm)/SiC(200 nm)/SiN(50 nm)) was translated at a distance of z1 ≃ 1.4 mm from the pinhole on a rectangular grid with the same spacing as used before. To minimize the effect of drift in the positioning stages only a subregion of the total scanned area was selected for reconstruction, consisting of 9 × 7 scan points. 10 exposures with a duration of 0.22 s each were collected at every scan point, accumulated, and corrected by subtraction of an equivalent sum of dark images. The total exposure time here was thus 138.6 s for a scanned area of ca. 31 μm2. The combination of several exposures lead to an increased dynamic range of the diffraction patterns used for reconstruction. As for the first dataset, data from a detector region of 1920 × 1920 pixels was selected and binned by a factor of 2 along each dimension to make numerical calculations feasible within a duration of several hours.

An important step in the preparation of the data before reconstruction was the subtraction of a dark field from all CCD images, with identical total illumination time and exposure characteristics. A high dark current on the order of 795 counts/pixel/frame (with a standard deviation around 3 to 4 counts/pixel/frame) was inevitable and mostly generated by readout noise (independent of illumination time). The fast readout mode of the CCD had to be used to reduce thermal drift effects on the positioning stages in vacuum. The corrected intensity Icorr in each pixel was calculated here as Icorr = max{Imeas – (1 + 2σ)Idark, 0} with Imeas denoting the measured signal, Idark the dark signal and σ ≲ 0.01 denoting the standard deviation of the dark image, relative to its mean. Using this subtraction rule, remaining noise in Icorr due to camera readout could be strongly suppressed.

For reconstruction, the algorithm first introduced in [15

15. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008). [CrossRef] [PubMed]

] was applied. The reconstruction process yields independently the complex illumination or probe function P(r) and the complex object transmission function O(r) in the exit plane directly behind the sample. r denotes the two-dimensional spacial coordinate in the sample plane. At each out of NP scan points rj the exit wave field
ψj(r)=P(r)O(rrj)
(1)
is modelled as a product of the constant probe function and the laterally translated object transmission function. With the detector placed into the far field of the exit wave, propagation to the detector plane corresponds to a two-dimensional Fourier transform ℱ [ψj(r)] of the exit wave field with the measured intensity distribution Ij given as
Ij(q)=|[ψj(r)]|2,
(2)
where q denotes the two-dimensional reciprocal space coordinate. Starting with an initial guess {ψj(0)} of exit waves the algorithm [15

15. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008). [CrossRef] [PubMed]

, 26

26. 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]

] then iteratively finds a solution {ψj = P(r)O(rrj)}, consisting of NP exit waves ψj that each obey Eqs. (1) and (2). Most importantly, all these exit waves are formed by a product of the same probe P(r) and translated object function O(rrj), the two quantities one is usually interested in.

One important step during each iteration is the enforcement of consistency with the measured data. The simplest approach to enforce this consistency is to replace the Fourier modulus |[ψj(j)(r)]| of the current iterate ψj(j)(r) (here for every j = 1,...,NP) by the measured amplitude Ij and retain its phase part ψj(j)(r)/|ψj(j)(r)|. A mere replacement operation of the Fourier amplitude by the measured amplitude does not take into account the experimental noise present in Ij. To circumvent this problem we used the same modified projection operator as in [18

18. 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. U.S.A. 107, 529–534 (2010). [CrossRef]

], which allows a certain distance between the updated Fourier amplitude and the measured amplitude in the space of all possible exit waves. The parameter controlling the allowed distance was optimized towards best reconstruction results.

For reconstruction of the diatom dataset the algorithm was iterated for 200 iterations, starting with a nearly flat initial guess with small added random phase and amplitude distortions. To average out small fluctuations from one iteration to another the final object and probe reconstructions were obtained as a complex mean over the last 50 iterations.

For reconstruction of the Siemens star dataset the algorithm was iterated for 400 iterations, starting with an equally generated random complex field as before. Analogously, the final object transmission function was formed then as a complex mean over the last 50 iterations. Note that with ψj = P · Oj being a reconstructed solution at scan position rj also αP· α−1 Oj with α ∈ ℝ+ is a solution [15

15. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008). [CrossRef] [PubMed]

]. For this reason the object amplitude transmission |O| was forced to stay within Tmin < |O| < Tmax = 1.0 with Tmin = 0.0114 and Tmax = 1.0 being the minimum and maximum expected amplitude transmissions of the test sample. To suppress artifacts in the reconstruction due to complications discussed in more detail in section 4 an additional real space constraint had to be added which is more restrictive than the preceding one: Starting at iteration 152, at every fourth iteration the amplitude of the object function was set to the theoretical transmission value of Tmin = 0.0114 for 500 nm Ta at 517 eV photon energy, if |O| < 0.2.

3. Results

3.1. Diatom sample

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

A magnified inset of the object reconstruction within the area that has been covered by the central part of the probe during the scan is shown in Fig. 3A. Details of the ornamental perforations (holes several 100 nanometers in size) in the frustule (diatom shell) can be seen in subfigures 3B (phase) and 3D (amplitude). Note that in Fig. 2B and 3A phase values are shown modulo 2π, “wrapped” into an interval IP of length 2π. This wrapping of phase values is due to the multi-valued nature of the arg(z)-function of a complex number z and can lead to unphysical phase discontinuities. Unwrapping a discrete two-dimensional phase distribution, i.e. mapping of phase values from the interval IP into the space ℝ of physical phase values, is a well-known mathematical problem which can be very difficult to solve due to phase aliasing, noise and physical discontinuities [28

28. R. Gens, “Two-dimensional phase unwrapping for radar interferometry: developments and new challenges,” Int. J. Remote Sens. 24, 703–710 (2003). [CrossRef]

]. For the small subregion shown in subfigure Fig. 3B unphysical phase discontinuities in horizontal direction have been removed using the Matlab [29

29. Matlab is a registered trademark of The Mathworks Inc.

] built-in one-dimensional phase unwrapping routine unwrap.m, leading to the true ‘physical’ phase. To a good approximation, the fossil diatom can be considered to be composed of silicon-dioxide with a uniform density. Assuming a silicon dioxide mass density of 2.2 g/cm3 [30

30. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at e = 50-30,000 ev, z = 1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993). [CrossRef]

] one arrives at a phase shift of around 1π rad and amplitude transmission of T = 0.58 per 1 μm projected thickness [30

30. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at e = 50-30,000 ev, z = 1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993). [CrossRef]

], leading to a maximum thickness on the order of 2 – 3 μm.

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

An exact determination of the obtained resolution in direct space is difficult for biological objects which generally do not exhibit edges with a known sharpness. A rough estimate on the resolution can be given here based on the fit of an error-function to the sharp boundary of the diatom. The fit here yields an edge smoothness of 129 nm (FWHM).

Note that diffraction fringes are visible on the edges of the diatom in the reconstructed transmission function (see Fig. 3A), indicating a possible breakdown of the projection approximation. A coarse criterion for the neglect of diffraction effects during propagation through the sample, i.e. for the validity of the projection approximation, has been given based on the angle of total external reflection due to a lateral refractive index gradient [31

31. P. Cloetens, “Contribution to phase contrast imaging, reconstruction and tomography with hard synchrotron radiation,” Ph. D. thesis, Vrije Universiteit Brussel (1999).

]: The approximation is valid 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 sample. In addition, the resolution has to fulfill r>a2=λΔt in order to avoid Fresnel diffraction effects within the sample [31

31. P. Cloetens, “Contribution to phase contrast imaging, reconstruction and tomography with hard synchrotron radiation,” Ph. D. thesis, Vrije Universiteit Brussel (1999).

]. Assuming δ = 0.0012 and a thickness of 2 μm in the present example one arrives at a1 ≃ 100 nm and a2 ≃ 69 nm. With a resolution in the object reconstruction close to this value it is clear that the experimental configuration is at the validity limit of the projection approximation and the sample might extend the depth of focus at certain points.

3.2. Siemens star test object

The reconstructed amplitude of the Siemens star object transmission function is shown in Fig. 4A. The reconstruction shows details down to a half-period resolution on the order of 50 nm, the central angular width of the void stripes of the innermost ring in the test pattern.

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

As visible in Fig. 4B the phase is only reconstructed uniformly in the void segments, exhibiting random fluctuations in the filled regions. Qualitatively this can be understood in view of the low minimum intensity transmission of Tmin2=1.3104 leading to insufficient transmission for a non-random phase reconstruction. With an average background-corrected accumulated count number of 2 · 109 analog-to-digital units (“counts”) on the detector for each scan point the average “fluence” at the sample is around 1.2 · 108 counts/μm2 or 2.6 · 105 counts per pixel. This allows for a relative error in intensity transmission of 1/2.6105=2103 which is however insufficient to reliably detect a transmission of T2 = 1.3 · 10−4 as expected for the tantalum material. Instead, the average relative amplitude transmission between void and filled regions is around 22. Further reasons for the non-quantitative object reconstruction are discussed in section 4.

3.3. Probe reconstructions

The complex reconstructed probe functions, obtained from the diatom and Siemens star datasets are depicted in Fig. 5A and B, respectively, both back-propagated over the respective distance z1 to the plane of the aperture. A comparison to the scanning electron micrograph of the pinhole exit surface indicates a considerable degree of similarity between the overall shapes of the reconstructed wave fields and the pinhole structure. Notably, the probe reconstruction obtained from the scan of the diatom sample, which scatters much less than the Siemens star, exhibits a flat central phase and amplitude within the elliptical pinhole area. On the other hand, the probe reconstruction from the Siemens star dataset is characterized by high-frequency distortions in amplitude and phase which are most likely artifacts of the reconstruction (see section 4). Note here that a typical diffraction pattern from the diatom dataset (see Fig. 5D) is dominated by the diffracted signal from the pinhole while the Siemens star diffraction pattern is very strongly dominated by the sample diffraction which totally suppresses the signal from the pinhole (see Fig. 5E).

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

4. Discussion

In the reconstruction of both, the diatom and especially of the Siemens star dataset certain artifacts remain, even though the reconstruction of the Siemens star dataset was restricted with rather strong additional real-space constraints. For a possible explanation we briefly discuss the geometry of diffraction pattern formation in both experiments. Conceptually, there are two extreme imaging regimes in which one could work using the present imaging setup (see Fig. 6A and B). The first situation is characterized by a very small distance z1 between pinhole and sample and thus a large Fresnel number F = a2/(λz1) ≫ 1 (with pinhole diameter a), leading to an illumination that is sharply confined in amplitude and nearly flat in phase. In this configuration — which is closest to that of classical CDI where a plane wave is used to illuminate the isolated sample — the diffraction pattern at the detector is nearly given as the squared modulus of the Fourier transform of the object transmission function convolved with the Airy pattern of the pinhole. As a consequence, the diffraction pattern has no resemblance to the object.

Fig. 6 (A) Far-field geometry: For Fresnel numbers F = a2/(λ z1) ≫ 1 (with pinhole diameter a) the wavefront impinging on the sample is almost flat and the amplitude of the exit wave field well-confined. The field distribution at the detector in the far field shows no resemblance with the exit wave. There is no geometrical magnification of the exit wave field behind the sample and the width D1 of the exit wave’s FOV in the reconstruction plane is not related to the lateral size of the detector, but inversely proportional to its pixel width. (B) Effective near-field geometry: For Fresnel numbers F ≪ 1 the probe at the sample is nearly spherical and less well-confined in amplitude. This leads to a diffracted amplitude which can be interpreted as magnified near-field propagated object function (with the spherical part of the illumination removed). The extension D1=D2/M of the FOV in the sample plane is now proportional to the FOV width in the detector plane. (C) Vertical slice through the normalized reconstructed probe amplitude |P| for the Siemens star dataset. Red vertical lines mark the positions of the first side minima, within which 98 % of the intensity is located. (D) Vertical slice through the unwrapped reconstructed phase φ (P) of the probe function and the phase φ (S) of an ideal spherical wave S in paraxial approximation, emanating from the center of the pinhole and propagated over a distance z1.

On the other hand there is the limiting case of large distances z1 between pinhole and sample, i.e. the limit of very small Fresnel numbers F ≪ 1, when the sample is illuminated by a pinhole beam already propagated into the far field. The illuminating wave field at the sample is then given as a product of a spherical phase term (with radius of curvature z1) and the Fourier transform of the aperture function. Within the small-angle approximation the propagation of the 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 distance z2/M with geometrical magnification M = (z1 + z2)/z1 [8

8. C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006). [CrossRef]

]. The FOV for a single exit wave in the sample plane is then given by D1=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 signal shows significant resemblance to the original object.

Ptychographic CDI is usually performed with the sample illuminated by a relatively flat and sharply confined wavefront [15

15. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008). [CrossRef] [PubMed]

, 18

18. 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. U.S.A. 107, 529–534 (2010). [CrossRef]

21

21. 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]

], in the imaging regime represented by the setup shown in Fig. 6A. In fact, this can best be achieved by placing the sample into the focal plane of a strongly focused wave field [15

15. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008). [CrossRef] [PubMed]

,20

20. 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]

,21

21. 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]

] or very close to an opaque mask (e.g. a pinhole) [18

18. 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. U.S.A. 107, 529–534 (2010). [CrossRef]

,19

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

]. However, the geometry of the present experiment (F ≃ 1) leads to a situation where neither of both limiting cases illustrated in Fig. 6A and B is adequate to completely describe the process of diffraction pattern formation: Consider the vertical slices through the amplitude and phase of the probe function as reconstructed 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 within the first side minima, a region with a relatively flat phase (variations up to 1.5π rad). This is the part of the beam that leads to the strongest diffraction effects which can be roughly interpreted as far-field patterns of the illuminated sample area. The remaining part of the beam, however, is subject to a phase curvature which is almost as high as that of a spherical wave with radius z1 (see Fig. 6D) and leads to a weak, Fresnel-propagated image of a comparatively large part of the sample on the detector. Note that the increased amplitude in the probe reconstruction on the right in Fig. 6C is most probably an artifact due to the previously described product ambiguity between probe and object amplitude. These amplified outer components of the probe amplitude lead to the high-frequency artifacts appearing at the probe wave field back-propagated to the pinhole plane, as visible in Fig. 5B. They are much weaker in the probe reconstruction of the diatom dataset.

It has been shown previously that high-curvature beams can be used advantageously for CDI experiments [32

32. 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]

] and also ptychography [33

33. 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]

] in a mode that is called Fresnel CDI. Here the illumination is actually reconstructed independently from a diffraction pattern of the empty beam alone [34

34. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101–104 (2006). [CrossRef]

].

We now turn from the probe reconstruction towards a qualitative evaluation of typical observed diffraction patterns of the diatom and Siemens star (see Fig. 5D and E). They are both characterized by a Fresnel-propagated direct image of the sample as well as a far-field diffracted reciprocal-space signal (see Fig. 5), similar to diffraction patterns encountered in Fresnel-CDI [32

32. 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]

]. In contrast to the the diatom dataset the near-field propagated (direct) image of the Siemens star always covers the full detector area, as the sample laterally extends in all directions over a very large area (dozens of microns in diameter) which is still partly illuminated by the outer parts of the far-field propagated pinhole beam. Note that this leads to an extraordinary broad mix of length scales: The central flattest and strongest part of the beam with a diameter on the order of 2 μm hits the center of the Siemens star with smallest length scales down to 50 nm leading to diffracted signals at the edges of the detector. At the same time the weaker and highly curved outer parts of the probe illuminate a sample area with a maximum lateral extension of D172μm (assuming an ideal paraxial spherical wave emanating from the pinhole), leading to a direct image of very large structures towards the edges of the detector. Although the reciprocal signal from the center is stronger than the direct signal from the edges they are both in the same area of the detector. Thus, an additional source of artifacts in the reconstruction could be a possible miss-interpretation of direct low-frequency signal as reciprocal high-frequency signal in the reconstruction process. For the Siemens star such a failure is even more probable as there is a resemblance of both signals. The reconstruction from the diatom dataset, where the detected signal is dominated by the pinhole diffraction, showed a much better convergence compared to the Siemens star dataset. This can be mostly attributed to to the smaller overall extend of the sample, making the holographic direct image less dominant and extended in the diffraction pattern.

We note that binning of detector pixels can also lead to increased artifacts as the computationally relevant FOV in the sample area is given by D1 rather than D1 with the data being not translated into the effective plane-wave geometry for ptychographic reconstruction. For a binning factor B = 2, as used here, D1 ≃ 43 μm, so that some weak signal at the outer regions of the detector will be either not reconstructed or become a possible source of artifacts due to miss-interpretation as high-frequency signal. However, in tests with B = 1 (scaling up computation time roughly by a factor of 4) on the Siemens star dataset it was found that the direct signal from the sample at the outer parts of the detector is generally too weak to be reconstructed as a direct near-field propagated image. Therefore, the effect of shorter computation time was favored.

5. Summary, conclusion and outlook

In conclusion, we have demonstrated the successful application of ptychographic coherent diffractive imaging to a fossil diatom with a contrast largely comparable to that of unstained biological cells at the used photon energy in the water window (here E = 517 eV). As a consequence, also for soft x-ray energies the usual field-of-view restrictions of conventional CDI can be overcome. Imaging a heavy-element lithographic test pattern, a nearly full absorption object, allowed for reconstructions on the order of 50 nm (half-period) resolution.

For the semi-transparent diatom sample a full complex object wave reconstruction was obtained which is in good agreement with estimated phase and amplitude modifications due to such an object. Here the small depth of focus of diffraction microscopy at small x-ray wavelengths becomes apparent in the reconstruction. As a consequence, for large biological objects which are several microns thick along the direction of the beam, the projection approximation is likely to be violated at these wavelengths making it difficult to find a global focal plane for the whole object. In such cases, one has to be aware of the complications for future 3D tomographic reconstructions based on ptychographic CDI in the water window.

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

The observed challenges of pinhole-based ptychographic CDI at low wavelengths have been discussed, which are partly due to the fact that one easily enters imaging regimes where the propagated pinhole beam exhibits significant phase curvature and is not sharply confined in the sample plane. In such situations the observed diffraction patterns are characterized by an overlay of a direct ‘holographic’ image of a large fraction of the extended sample and a far-field diffraction pattern due to a very small sample region illuminated by the central part of the beam. Instead, a flat, sharply confined illumination is very desirable for ptychographic CDI also in the soft x-ray regime. A good way to achieve this experimentally can be the use of a zone plate focus as the probe. A current technical limitation to PCDI in the soft x-ray regime is the lack of detectors free of dark and readout noise and the relatively long readout times of present CCD systems. As a consequence, for standard fully quantitative PCDI also in the soft x-ray regime improvements on the detector side are very desirable.

Notwithstanding the technical challenge in the proper choice of experimental parameters and instrumentation as discussed above, we believe that the simple experimental concept of ptychographic coherent diffraction imaging will be a very valuable tool in particular in the soft-x-ray range, where the interaction with matter is comparably strong and the diffractive signal hence is high, up to high momentum transfer. The fact that PCDI allows for reconstruction of the illumination wave front is particularly important here, since short propagation distances between optics and sample already lead to strong propagation effects and simple assumptions on the probe function need to be considered carefully. In addition, wavefront reconstruction with resolutions in the nanometer range represents a truly unique tool for instrument characterization and development.

Acknowledgments

References and links

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2.

P. Thibault and V. Elser, “X-ray diffraction microscopy,” Annu. Rev. Condens. Matter Phys. 1, 237–255 (2010). [CrossRef]

3.

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C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schoder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101, 090801 (2008). [CrossRef] [PubMed]

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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. U.S.A. 107, 7235–7239 (2010). [CrossRef] [PubMed]

6.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003). [CrossRef]

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S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004). [CrossRef] [PubMed]

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C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006). [CrossRef]

9.

A. Rosenhahn, R. Barth, F. Staier, T. Simpson, S. Mittler, S. Eisebitt, and M. Grunze, “Digital in-line soft x-ray holography with element contrast,” J. Opt. Soc. Am. A 25, 416–422 (2008). [CrossRef]

10.

A. P. Mancuso, T. Gorniak, F. Staier, O. M. Yefanov, R. Barth, C. Christophis, B. Reime, J. Gulden, A. Singer, M. E. Pettit, T. Nisius, T. Wilhein, C. Gutt, G. Grübel, N. Guerassimova, R. Treusch, J. Feldhaus, S. Eisebitt, E. Weckert, M. Grunze, A. Rosenhahn, and I. A. Vartanyants, “Coherent imaging of biological samples with femtosecond pulses at the free-electron laser flash,” N. J. Phys. 12, 035003 (2010). [CrossRef]

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H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996). [CrossRef]

12.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004). [CrossRef]

13.

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]

14.

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]

15.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008). [CrossRef] [PubMed]

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M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinearoptimization approach,” Opt. Express 16, 7264–7278 (2008). [CrossRef] [PubMed]

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

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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. U.S.A. 107, 529–534 (2010). [CrossRef]

19.

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

20.

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]

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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]

33.

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]

34.

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OCIS Codes
(180.7460) Microscopy : X-ray microscopy
(340.7440) X-ray optics : X-ray imaging
(260.6048) Physical optics : Soft x-rays

ToC Category:
X-ray Optics

History
Original Manuscript: August 9, 2010
Revised Manuscript: October 29, 2010
Manuscript Accepted: December 2, 2010
Published: January 10, 2011

Virtual Issues
Vol. 6, Iss. 2 Virtual Journal for Biomedical Optics

Citation
K. Giewekemeyer, M. Beckers, T. Gorniak, M. Grunze, T. Salditt, and A. Rosenhahn, "Ptychographic coherent x-ray diffractive imaging in the water window," Opt. Express 19, 1037-1050 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-1037


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References

  1. K. A. Nugent, "Coherent methods in the x-ray sciences," Adv. Phys. 59, 1-99 (2010). [CrossRef]
  2. P. Thibault, and V. Elser, "X-ray diffraction microscopy," Annu. Rev. Condens. Matter Phys. 1, 237-255 (2010). [CrossRef]
  3. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens," Nature 400, 342-344 (1999). [CrossRef]
  4. C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schoder, and C. Riekel, "Coherent x-ray diffraction imaging with nanofocused illumination," Phys. Rev. Lett. 101, 090801 (2008). [CrossRef] [PubMed]
  5. 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. U.S.A. 107, 7235-7239 (2010). [CrossRef] [PubMed]
  6. J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, "Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method," Phys. Rev. B 67, 174104 (2003). [CrossRef]
  7. S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, "Lensless imaging of magnetic nanostructures by x-ray spectro-holography," Nature 432, 885-888 (2004). [CrossRef] [PubMed]
  8. C. Fuhse, C. Ollinger, and T. Salditt, "Waveguide-based off-axis holography with hard x rays," Phys. Rev. Lett. 97, 254801 (2006). [CrossRef]
  9. A. Rosenhahn, R. Barth, F. Staier, T. Simpson, S. Mittler, S. Eisebitt, and M. Grunze, "Digital in-line soft x-ray holography with element contrast," J. Opt. Soc. Am. A 25, 416-422 (2008). [CrossRef]
  10. A. P. Mancuso, T. Gorniak, F. Staier, O. M. Yefanov, R. Barth, C. Christophis, B. Reime, J. Gulden, A. Singer, M. E. Pettit, T. Nisius, T. Wilhein, C. Gutt, G. Grübel, N. Guerassimova, R. Treusch, J. Feldhaus, S. Eisebitt, E. Weckert, M. Grunze, A. Rosenhahn, and I. A. Vartanyants, "Coherent imaging of biological samples with femtosecond pulses at the free-electron laser flash," N. J. Phys. 12, 035003 (2010). [CrossRef]
  11. H. N. Chapman, "Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution," Ultramicroscopy 66, 153-172 (1996). [CrossRef]
  12. J. M. Rodenburg, and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004). [CrossRef]
  13. 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]
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