Figure 1 shows a schematic of the experiment performed with the *G* *öttingen* *I* *nstrument for* *N* *ano* *I* *maging with* *X* *-rays* (GINIX) installed at the coherence beamline P10 of the new storage ring PETRA III at Hasylab, DESY (Kalbfleisch et al. 2011; Salditt et al. 2011). The undulator beam was monochromatized by a *Si*(111) double crystal to 13.8 keV and then focused by two Kirkpatrick-Baez (KB) mirrors polished to fixed elliptical shape, yielding a focus size of *D*
_{horz} = 370 nm and *D*
_{vert} = 200 nm full width at half maximum (FWHM), as measured by scanning x-ray waveguides horizontally and vertically through the beam. The total flux was 2.4 × 10^{11} counts per second at 70 mA ring current, as measured by a pixel detector (Pilatus, Dectris) positioned at 5.29 m in the widened far-field of the KB beam. The waveguide (WG) system was positioned in the focal plane of the KB mirror, using a miniaturized fully motorized goniometer with optical encoders (Attocube). Alignment of the waveguide as well as the sample mounted on a dedicated tomography stage was facilitated by use of two on-axis optical microscopes, one directed downstream and one upstream with the beam, see Additional file 1 for details. Two different single photon counting pixel detectors, with 172 μm (Pilatus, Dectris) and 55 μm (Maxipix) pixel size were used. The waveguide system consisted of two crossed planar waveguide slices, each with a transmission optimized sputtered thin film sequence *Ge/Mo/C/Mo/Ge*, with 35 nm amorphous *C* as the guiding layer (Krüger et al. 2010
2012; Salditt et al. 2011). Note that due to grazing angles, the guiding layer width of 35 nm corresponds to only two propagating modes in the waveguide. Absorption of radiative modes in the cladding then results in a strong damping of all radiation except of the two fundamental modes. Finally, the interference of the modes can lead to a an effective beam width which is substantially smaller than the guiding layer thickness (Krüger et al. 2012). The small exit beam width of the waveguide beam is evidenced by the large divergence angle of its far-field intensity distribution, measured with a pixel detector (Pilatus) at a distance of 5.29 m behind the WG (data shown in Figure 1(b), recorded at 15 keV photon energy). The relatively homogenous part of the waveguide far-field intensity distribution (dashed rectangle in Figure 1(b)) is well suited for imaging and has a divergence angle of 5 mrad. The near-field intensity distribution in the effective focal plane is obtained by inverting the diffraction pattern using the error reduction algorithm (Krüger et al. 2010
2012 (see Figure 1(c)). Gaussian fits of the central peak along the horizontal and vertical direction give a width of 10 nm × 9.8 nm (FWHM), respectively (see Figure 1(d)). 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), suspended on Si_{3}N_{4}-foils (Silson) with 1 μm thickness and 5 × 5 mm^{2} lateral dimensions, shock frozen by cryogenic fixation in ethane (cryo-plunging), and subsequently freeze dried, see Additional file 1 for details. To record projection propagation images in full field mode the sample (for an optical micrograph of the cells, see Figure 1(a)) was placed at a (defocus) distance *z*
_{1} = 8 mm from the WG exit plane, where the divergent WG beam has broadened to a field of view (FOV) of 40 × 40 μm^{2}, as calculated from the measured far-field divergence angle of 5 mrad. As shown previously, the imaging experiment can then be described in a well-known equivalent parallel-beam geometry (Mayo et al. 2002; Fuhse et al. 2006) with a demagnified (effective) detector pixel size of *Δ*
_{
D
} / *M* and a (de)magnification factor of *M* = (*z*
_{1} + *z*
_{2}) / *z*
_{1} as well as an effective sample-detector distance *z*
_{eff} = *z*
_{1}
*z*
_{2} / (*z*
_{1} + *z*
_{2}) = *z*
_{2} / *M*. Note that here *z*
_{2} ≫ *z*
_{1}, so that *z*
_{eff} ≃ *z*
_{1} = 8 mm and *M* ≃ *z*
_{2} / *z*
_{1} = 660, resulting in an effective detector pixel size of ≃ 83 nm. For 3D imaging, 83 projection images *I*
_{
ϕ
} were collected over 162 degrees with a total exposure time of 10 minutes for each angle *ϕ*, distributed over *N* = 15 detector accumulations, which were subsequently corrected for lateral drift by cross-correlation methods with sub-pixel accuracy (Guizar-Sicairos et al. 2008). The total fluence for each projection was ≃4.38·10^{6} photons / μm^{2}, corresponding to a dose of ≃1.9·10^{3} Gy, based on calculations presented in (Howells et al. 2009). Next, the normalized intensity distribution was calculated as \stackrel{\u0304}{{I}_{\varphi}}(x,y)={I}_{\varphi}/{I}_{0} by division with the empty beam intensity distribution *I*
_{0}(*x* *y*), followed by a filtering of residual low frequency variations, see Additional file 1 for details on image processing and filtering. Based on the projection approximation, the measured signal at the detector normalized by the empty beam intensity distribution, described in the equivalent parallel-beam geometry is then given as

\begin{array}{c}\stackrel{\u0304}{{I}_{\varphi}}(x,y)\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}\left|{D}_{{z}_{\text{eff}}}\right[\chi (x,y)]{|}^{2}\phantom{\rule{1em}{0ex}},\end{array}

(1)

where {D}_{{z}_{\text{eff}}} denotes the two-dimensional Fresnel propagator over the distance *z*
_{eff} along the optical axis and *χ*(*x* *y*) the sample transmission function (Giewekemeyer et al. 2011). Backpropagation of Eq. (1) leads to a single-step holographic reconstruction of the normalized intensity \stackrel{\u0304}{{I}_{\varphi}}(x,y), which is however intrinsically spoiled by the so-called twin-image. As presented previously in Giewekemeyer et al. (2011), a significant improvement of the reconstruction can be achieved by application of a modified version of the Hybrid-Input-Output (HIO) algorithm (Paganin (2006)) well suited for pure phase objects. Note that the HIO algorithm is a classical iterative algorithm, which cycles between reciprocal space and real space. In reciprocal space, the measured information is fed in, and in real space, additional ‘real space’ constrains are applied, which in many cases leads to convergence towards the solution which satisfied both constraints. Most common a priori knowledge in real space is based on support information, i.e. in which region of the illuminated area the sample is positioned and/or amplitude constraint, i.e. knowledge about the amplitude of the object, for example when absorption can be neglected. According to the HIO version published in Giewekemeyer et al. (2011), the real space upgrade consists of two constraints, one for the amplitude, and one for the phase of the wave, as specified by the following two equations. The update of the current amplitude |*χ*
_{
n
}| is given by

\left|{\chi}_{n+1}\right|=\left|{\chi}_{n}\right|-\beta \xb7\left(\left|{\chi}^{\prime}n\right|-1\right),

(2)

as proposed in Gurevey (2003). Secondly, a phase constraint is included in the iteration, namely,

\begin{array}{l}\phi \left({\chi}_{n+1}\right(x,y\left)\right)\\ =\left\{\begin{array}{c}\phi \left({\chi}_{n}\right(x,y\left)\right)-\gamma \xb7\phi \left({\chi}^{\prime}n\right(x,y\left)\right)\phantom{\rule{0.3em}{0ex}}\forall (x,y)\notin S\\ \phantom{\rule{4.0em}{0ex}}min\left\{\phi \right({\chi}^{\prime}n(x,y)),0\}\phantom{\rule{0.3em}{0ex}}\forall (x,y)\in S.\end{array}\right.\end{array}

(3)

\left|{\chi}^{\prime}n\right| denotes the amplitude of the *n*-th iterate after application of the detection plane constraint, i.e. {\chi}^{\prime}n:={P}_{M}\left({\chi}_{n}\right) with {P}_{M}\left({\chi}_{n}\right)={D}_{-{z}_{\text{eff}}}[\sqrt{\overline{I}}\xb7\text{exp}(i\phi \left({\stackrel{~}{\chi}}_{n}\right)\left)\right] denoting the modulus replacement operation in the detection plane and *φ*(*z*): = arg(*z*) for any complex number z\in \mathbb{C}. The amplitude constraint (2) slowly pushes |*χ*
_{
n
}| towards 1 and the phase constraint (3) causes a gentle decrease of the phase to a constant *C* (*C* = 0 was chosen here) in the area, where no object is located. The phase inside the support area S\subset {\mathbb{R}}^{2}, however, is left untouched, as long as it is not larger than *C*, allowing for phase changes *Δφ*(*x*, *y*) in one direction only, as expected for objects with |*Δφ*(*x*, *y*)| < *Π*. The speed of convergence is determined by the feedback parameters *γ* ∈[0, 1] and *β* ∈[0, 1]. For tomographic datasets it is crucial to determine the accurate support area *S* automatically for each projection, which is achieved by a thresholding approach, as explained in Additional file 1.

Before application to a biological sample, suitable test structures can be used to control the contrast transfer of the setup at the relevant resolution range. Figure 2 shows the example of a lithographic pattern consisting of 50 nm lines and spaces in a 500 nm thick tantalum layer (NTT-AT, Japan, model # ATN/XRESO-50HC), see (a) for a SEM micrograph. The measured hologram (b), after correction by the empty beam, corresponds to a total field of view of 3.4 μm (h.) × 2.2 μm (v.). Analysis of line cuts as shown in (c) shows that sufficient contrast of *ΔI* / *I* ≃ 0.2 is achieved. For the data on the test structure, we have used a waveguide system with 60 nm amorphous *C* as the guiding layer resulting in a total flux of 5 × 10^{8} cps exiting the waveguide system at 80 mA ring current. The test structure was positioned at *z*
_{1} = 1.063 mm. Due to the sequential arrangement of the crossed waveguide system the distances from both individual planar waveguide slices to the sample differ by 450 μm, which is the thickness of the second waveguide slice. With the detector (Maxipix) placed at *z*
_{2} = 5.36 m the effective pixel side lengths are ≃ 16 nm in horizontal and ≃ 11 nm in vertical direction. Note that the resulting anisotropy of effective propagation distance and sampling along the horizontal and vertical direction can easily be implemented in the numerical Fresnel propagation. The total exposure time of the hologram was 1 second, distributed over 200 detector accumulations followed by an empty beam recording.