1. Introduction

X-ray propagation full field imaging and tomography have become powerful imaging techniques, based on successful development of phase retrieval algorithms [1–5]. In the detection plane, the contrast for both absorbing and phase shifting features is generated by the inherent near-field diffraction behind a sample which is illuminated (probed) by an extended x-ray wavefront P of sufficiently high spatial coherence. The exit wavefront and hence the complex-valued sample transmission function of the object O can be reconstructed from the intensity distribution in the detection plane at some (defocus) distance z behind the object plane, based, for example, on the transport-of-intensity equation (TIE) or various iterative algorithms [1, 6, 7]. A wide spectrum of applications ranging from material science [8], paleontology [9], histology [10], small arthropod biology [11] has emerged, taking advantage of the unprecedented 3D imaging capability, including fast tomography applications of dynamic processes [12]. While this development has been driven by the brilliance and coherence of synchrotron radiation, to some extent the method can be translated to compact laboratory-scale μ-CT instruments, preserving fully quantitative reconstruction [13]. Furthermore, in order to achieve higher spatial resolution, propagation imaging and tomography can be extended from the parallel beam case to quasi-spherical beams, allowing for adjustable geometric magnification [14–16]. With this approach, projection micrographs and tomograms of biological cells [16, 17], tissues and small organisms [18] can be recorded with quantitative density contrast and pixel size below 50 nm.

The necessary diverging spherical wavefronts for high magnification are created by nano-focus x-ray optics, such as Kirkpatrick-Baez (KB) mirror systems [19, 20]. For such optical systems, already small imperfections like mirror figure errors result in strong phase aberrations due to the small wavelength of x-rays [21]. The empty beam far-field patterns are therefore strongly affected and never correspond to the clean homogeneous flat field distribution ideally assumed. This aberrated empty beam, however, constitutes the probing field P for O. According to current wisdom, the problem is dealt with by simple division of the recorded image with object by that of the empty beam [17, 22]. In fact, the raw images are hardly ever presented in publications. Hence, significant errors can be expected in the subsequent phase retrieval process. While the simple flat field correction scheme by division is actually a crude approximation, it still works in some cases, whereas going terribly wrong in others. Figure 1 illustrates the basic problem investigated here. The inhomogeneities in P cause distortions which cannot be divided out. Strong artifacts of the mirror system are still observed in the ‘corrected’ hologram of a polystyrene sphere. The goal of the present work is to investigate the errors of the common flat field correction. In which cases are they strong, and in which cases are they negligible? In a second step, algorithmic approaches beyond the simple empty beam correction are presented, and tested numerically as well as experimentally.

 

Fig. 1 Illustration of the flat field problem, for a 10μm (diameter) polystyrene sphere positioned at a defocus distance of z₁ = 10 mm behind the focal plane of the KB mirror system at beamline ID22NI (ESRF, Grenoble), recorded at a detector distance of z₁ + z₂ = 525 mm and photon energy of E = 17 keV. (a) Empty beam recording showing the artifacts stemming from mirror imperfections. (b) Raw hologram of the sphere. (c) Corrected hologram as obtained by division of (b) by (a), i.e. the standard flat field or empty beam correction scheme. All data sets are normalized to the respective mean intensity. Scale bar of 2μm indicates the corresponding distance in the object plane.

Download Full Size | PDF

Simultaneous reconstruction of probing beam P and object O have been achieved by ptychographic algorithms [23, 24], based on the redundancy generated by lateral scanning with partial overlap. The method was first developed for far-field coherent diffraction data collected from a compact probe in the object plane [23, 24], but some recent generalizations to near-field (Fresnel) diffraction data have been demonstrated [25]. However, to this end P had to be further scrambled by a wavefront diffuser in order to generate enough diversity. Further, for the case of an illumination by a Fresnel zone plate, special reconstruction schemes were devised [26], based on constraints in several planes including the pupil of the zone plate to define the probe’s support. Unfortunately, both ptychography based on lateral scanning and the multi-plane Fresnel-CDI scheme [26, 27] turn out to be unsuited for the problem of the empty beam reconstruction in full field propagation imaging, since the necessary redundancy/diversity is generally not provided by lateral scanning. For a discussion of redundancy/diversity as well as of the sampling criteria relevant for ptychography, see also [28]. This is why we turn to an approach based on multiple detection planes in this work in order to reconstruct P. A multi-plane reconstruction scheme was first introduced in optics and electron optics in 1973 [29]. Since then this scheme and variations thereof have been used in microscopy to recover the complex valued point-spread-function (PSF) [30], and also in the field of x-ray imaging to reconstruct an exit wavefield P · O [31, 32]. However, in [31, 32] the multi-plane detection has not been used to recover P itself, simultaneously with the object O, which is provided here. We therefore extend near-field imaging in a similar manner as ptychography has previously done for far-field coherent imaging [23, 24].

The manuscript is organized as follows: Section 2 further defines the problem of the wavefront distortions and the associated correction schemes. The notation and the basic concepts of the work are introduced. Section 3 presents the experimental setup and parameters. Section 4 presents a single distance approach for the reconstruction of P, which we denote as divide&update. Finally, we turn to the reconstruction of P based on multiple detection planes in Section 5, before the paper closes with conclusions and outlook in Section 6.

2. How wrong is the standard empty beam correction?

Before we turn to reconstructions of P let us address the validity of the standard empty beam correction, i.e. the approximation

 

Fig. 2 (a) Sketch of the imaging scheme with the mirrors, the focus, the object and the detector. The probing beam P is generated by a pair of KB-mirrors at the distance f in front of the focal plane. The sample is placed at a distance z₁ behind the focal plane, and the sample to detector distance is z₂. (b) The equivalent parallel beam geometry corresponding to (a) with z_eff = z₁z₂/(z₁ + z₂) and a geometrically reduced detection plane with M = (z₁ + z₂)/z₁. (c) The phantom of P used in the numerical simulations. A mandrill monkey image serves as the input for the phases, Dürer’s Melencolia I as input for the amplitudes, both images are taken from MATLAB. The phases are in the range of [−0.4 0.4] rad and the amplitudes in [0.8 1.2]. The image size is 512 × 512 pixel. (d) The object O used for the numerical simulations consisting of a phase shifting 2D sine grating, here shown for a periodicity of 32 px and a phase shift of −0.3 rad. (e) The test object used for multiple plane reconstructions. The image of a stone sculpture serves as phases. The phases are in the range [−0.4 0] rad, the amplitudes are uniformly 1, thus we have a purely phase shifting object.

Download Full Size | PDF

In the simulation, we then evaluate δ̄ for the probe phantom shown in Fig. 2(c), denoted as the mandrill-Dürer-probe, where the amplitudes of P are given by the Dürer carving, and the phases by a mandrill monkey. The test object is chosen as a purely phase shifting object consisting of a sine grating with a grid constant (periodicity) Δx_O. Both probe and object are represented by a N × N matrices with N = 512. The error (of the conventional empty beam correction according to Eq. (4) is then evaluated for a constant Fresnel number F = Δx²/(λz_eff) = 0.0015, as a function of Δx_O in the range from 16 px to 128 px and as a function of the filter cutoff ξ, which was varied in the range from 2π/(NΔx) to π/Δx, with Δx the pixel size in the plane of the object. Note that this formulation in terms of the natural units of pixel units and Fresnel number has been chosen in view of the generality. The value of F corresponds in particular to the following (experimental) parameters: λ = 0.07 nm, Δx = 25 nm, z₁ = 6 mm, z₂ = 519 mm and z_eff = 5.93 mm. Next, we note that the varied range of the filter cutoff parameter can be transformed to a range of a from 0.34 px to 349.92 px, according to Eq. (5), evaluated for the above (experimental) parameters.

The expectation is that δ̄ increases when either the source size a is increased, or the feature size Δx_O is decreased. Figure 3 shows the simulation results for δ̄ for different Δx_O as function of ξ/ξ_max, with ξ_max = π/Δx. The alternative x-axis label on the top indicates the corresponding a. As expected δ̄ decreases if the bandwidth decreases, i.e. if P is filtered more strongly with a smaller cut-off frequency ξ. Furthermore, the error δ̄ is always larger for smaller grid parameters than for larger grid parameters. The smaller the structure, the more stringent is the requirement to keep the source size a small or the correlation length of the wavefront aberrations Δx_P large. From this we can infer that small structures in the object will not be transferred correctly after application of the empty beam division, while the contrast of larger objects may not suffer so much. Interestingly, the experimental example shown in Fig. 1 seems to be in contradiction to the simulation results. The hologram of a 10 μm sphere which is illuminated by a beam emanating from a KB-focus with 100 nm spot size should not be affected by artifacts due to empty beam division. The paradox is resolved by the fact that while the full width at half maximum (FWHM) of the focal spot is well below 100 nm, the KB focus is characterized by long tails which extend over several μm. It is these tails which render the effective source size too large for a ‘clean’ empty beam division. This conclusion has been validated experimentally by placing a pinhole with 1.4 μm diameter in the focal plane, which then results in a less disturbed, i.e. cleaner empty beam, see Fig. 4.

 

Fig. 3 The error δ̄ as function of the normalized cut-off frequency ξ/ξ_max in the filter applied to the probe P, for different periodicities Δx_O (grid spacings). The alternative x-label on the top indicates the effective source size a in units of pixels, corresponding to ξ. The simulation can be carried out in unit-less pixel variables and Fresnel numbers.

Download Full Size | PDF

 

Fig. 4 Examples of images recorded for a gold grid test pattern with 1.5 μm spacing at a defocus distance z₁ = 26 mm. (a) Empty beam measurement P. (b) Recorded (raw) hologram of P · O. (c) Standard empty beam correction obtained by division of (b) by (a). (d), (e), (f) same as (a), (b), (c), but with a pinhole of 1.4 μm (diameter) inserted in the focal plane, which cuts the tails of the KB-beam, leading to a filtering of P. All data sets are shown after normalization to the average intensity. Scale bar indicates 2 μm in the sample plane.

Download Full Size | PDF

3. Experimental setup

The experiment was performed at the nano-focusing beamline ID22NI at ESRF (Grenoble) at a photon energy of 17 keV (0.0729 nm). The source size (FWHM of the focus of the KB-mirror-system [21]) was determined to be 67 nm(vertical) and 73 nm(horizontal), as measured by a fluorescence knife edge scan. The sample and detector stage can be moved within a defocus range of 0 – 300 mm and 300 – 1500 mm, respectively. A FReLoN HD 4m detector (ESRF) equipped with a LSO:Tb (20 μm) scintillator and an optical magnification scheme was used, with an effective pixel size of 756 nm. In the first configuration the detector was placed at z₁ + z₂ = 525 mm. The variation of z_eff is achieved by placing the sample at different z₁. Figure 4 shows the measured intensity patterns recorded in this setup. The left column corresponds to the empty beam, the center to the raw hologram of a grid and the right column to the empty beam corrected hologram, for the data recorded without (top row) and with (bottom row) a ‘cleaning’ pinhole of 1.4 μm diameter positioned in the focal plane. P and P · O were recorded quickly after each other, so that the flat field correction is not affected by temporal drifts.

In the second configuration the detector is moved and the sample (gold grid) is kept at a constant z₁ (25 mm). This approach also yields different z_eff but the relative variation in z_eff is much smaller if only z₂ is changed. The detector was placed at z₂ = {500, 501, 510, 600} mm. For each distance 100 images with and without O were recorded over a total of 40 minutes. Figure 5 shows data recorded using this configuration. Here, the time span between the measurements of P and P · O was 20 minutes. This delay has visible influence on the quality of the flat field correction (c). Thus the primary prerequisite for the empty beam division is of course a stationary illumination field, or a correspondingly small sampling interval, in order to avoid temporal drift. While this trivial reason for possible artifacts is not our main concern, it does affect some of the experimental data. For this reason phase reconstructions run on data as Fig. 5(c) yield poor results as depicted in Fig. 5(d). The multiple detection plane reconstruction (Section 5) suffers also from this issue. For comparison, using data, collected over a considerably smaller time interval, as shown in Fig. 4(c) to run the phase reconstruction yields Fig. 7(c), as discussed further below. All relevant experimental parameters for the measurements shown are summarized in Tab. 1.

 

Fig. 5 Examples from a data set using the same object as in Fig. 4, but recorded in the detector scan (z₂ variation at fixed z₁). (a) Raw hologram of P · O for z₂ = 510 mm. (b) Corresponding empty beam P. (c) Empty beam correction: (a) divided by (b). (d) Phases of a reconstruction based on GS using (c) as measurement constraint and the projection on amplitude 1 and negative phases as sample constraint (cf. Section 4). (a), (b), (c) share the same colorbar shown below (c). (a) and (b) show the average of 100 measurements of the object and the empty beam, respectively, normalized by the mean intensity. Scale bar indicates 4 μm in the sample plane.

Download Full Size | PDF

Table 1. Main parameters for the datasets shown in Figs. 1, 4, 5.

View Table

4. Reconstructions from a single distance

In order to overcome the errors involved with the conventional empty beam division, as illustrated above, we now turn to reconstruction strategies of P and O. To this end, we use the common class of projection algorithms for the reconstruction [5], such as Gerchberg-Saxton (GS), Hybrid-Input-Output (HIO), Relaxed Averaged Alternate Reflection (RAAR) [33] etc. The algorithms were implemented using the SciPAL library [34] and are based on two alternating projections in two different planes: the projection P_M adapts the current iterate of the wave field Ψ to the recorded measurements in the plane of detection

 

Fig. 6 Algorithm sketch for divide&update (d&u) and simulated reconstructions. (a) Principal sketch of d&u. Further explanations in Section 4. (b) Phases of the reconstructed mandrill-Dürer-probe and (c) corresponding amplitudes. (d) Phases of the first guess of O as obtained by the GS algorithm applied to the empty beam corrected hologram. (e) Phases of O after 200 iterations of d&u, showing the improvement by the nested update. (f) and (g) show reconstructions from a pure phase probe, i.e. a beam with phases represented by the mandrill image, and amplitudes equal to one (denoted as mandrill-1-probe). Besides the negative phase constraint also the projection to uniform amplitude is used. (f) Phases of P, amplitudes of P not shown. (g) Phases of O after 200 iterations of d&u. Scale bar indicates 4 μm.

Download Full Size | PDF

The right side of the sketch shows $P_M({\mathrm{\Psi }}_n^{P\cdot O})$ and $P_M({\mathrm{\Psi }}_n^P)$ where n denotes the iteration index. The left side shows the update of P and O. In order to obtain a separation of the exit wave field P · O, we need an initial guess for O. A guess for O is easily obtained by running a phase reconstruction step on the empty beam corrected measurements. One iteration of d&u starts with applying $P_M({\mathrm{\Psi }}_n^{P\cdot O})$ and $P_M({\mathrm{\Psi }}_n^P)$ on P and O. P₀ and (P · O)₀ (the initializations) are chosen with uniform amplitude 1 and phase 0. After application of $P_M({\mathrm{\Psi }}_n^{P\cdot O})$ we can use the guess of O to obtain an update for P (blue arrow). We likewise use $P_M({\mathrm{\Psi }}_n^P)$ to update O (red arrow). The updated versions O_n+1 and P_n+1 are then used to compute the new exit wave (P · O)_n+1 (green arrows). This concludes one iteration of d&u. In practice, a few enhancements were added to the algorithm, and several additional constraints were tested. First, in the spirit of hybrid input-output, the update P_n+1 is calculated as an average of P_n and the update, same holds for O_n+1. Second, O_n+1 and P_n+1 were always projected on negative phases, see Eq. (8). We can consider this negativity (or positivity) constraint to be based on a physical relationship, such as positivity of the electron density (implying negativity of the object phase with respect to the vacuum wave). Therefore, one may be tempted to conclude that it would not hold for scattering length distributions which can change sign (as for neutrons), or for a sample embedded in a matrix. In the latter case the density of the object could be larger or lower than the matrix. Since a global phase offset is irrelevant, translating or reflecting all pixel values to negative or positive sign thus is not restrictive in our case, but is found to stabilize the algorithms by ‘aligning’ the phases. Finally, we projected O to unit amplitude 1, implying the constraint for a purely phase shifting O.

4.1. Simulation results

Figures 6 (b) and (c) show the result for the simulation of the d&u algorithm, after 200 iterations of d&u. The phantom used for P is the mandrill-Dürer probe introduced in Fig. 2(c). While the reconstructed phases shown in (b) resemble the mandrill image, the reconstructed amplitudes (c) are far off the Dürer input and exhibit significant contaminations of the phase (mandrill image). Hence, the phases of P are reconstructed to some extent, while the amplitudes are not. The object O, on the other hand shows a better quality, compare the result shown in (e) to the initial guess (d), which was computed from the standard empty beam corrected hologram by a GS-algorithm. The reconstruction in the d&u scheme is clearly better then the GS-solution which is spoiled by the wrongful empty beam division. However, the d&u result still shows residual cloudy artifacts, which show that the separation of O and P is not sufficient. If we use a purely phase shifting P in the simulation and imply the corresponding constraint, the reconstruction quality is excellent, see the reconstructions of P in (f), and of O in (g), respectively. The amplitudes of P are not shown but exhibit ±15% (max.) variations around 1. This variation is due to the fact, that the final reconstruction is generated after a last application of P_M, we use this procedure for every reconstruction.

4.2. Experimental data

Figure 7 shows the d&u reconstruction of the experimental data recorded for the gold grid. Close inspection of the reconstructed phases of P (a) show again regular structures which are artifacts stemming from the gold grid. The amplitudes of P (b) look reasonable, with no bigger imprints of the grid. Here, the reconstruction quality cannot be improved significantly by assuming a constraint of coupled real and imaginary component of the index of refraction (dipersion and absorption correction), such as proposed in [35]. The results also do not improve if we take a second z₁ position for the same object into account such that two positions in the same P are probed.

 

Fig. 7 Reconstructions of experimental data with the divide&update algorithm. (a) Phases and (b) amplitudes of the reconstructed P. (c) Phases of the first guess of O (gold grid) as obtained by the GS algorithm applied to the empty beam corrected hologram. (d) Phases of O after 200 iterations of d&u, clearly showing the improvement over (c). Scale bar indicates 4 μm in the sample plane.

Download Full Size | PDF

5. Reconstructions from multiple detector distances

The previous section has shown that images of the object and probe at a single distance do not fully constrain the reconstruction of P. We thus turn to an approach of using multiple measurements of the beam in parallel. By moving the detector coordinate z₂, the effective propagation distance z_eff is modified both for P · O and the empty beam P, which is measured at each detector distance. This experimental setup is sketched in Fig. 8. For the geometry of high magnification, z₂ ≫ z₁, z_eff changes only very little with the variation of z₂, since for z₂ → ∞, z_eff → z₁. We use the multi-distance (multi-magnitude) projection algorithm proposed by Allen et al. [36] to reconstruct the wave fields. The algorithm is denoted as mmp (multi-magnitude projection) and is rather straight forward. It uses only the measurements in the different detection planes as constraints, with no constraint in the sample plane. Using 4 detection distances, we have

 

Fig. 8 Detector distance variations. In this approach z₂ is varied and z₁ is kept constant.

Download Full Size | PDF

5.1. Simulation

The simulated reconstruction was carried out for the same four detector distances as in the experimental case discussed below, but in the equivalent (parallel beam) geometry with z_eff = {23.810, 23.812, 23.832, 24} mm. Poisson noise was added to each pixel of the simulated holograms of P and P · O for an expected number of 10⁹ detector photons. For P the mandrill-Dürer-probe was used as above. For the phase of O, the image of a stone sculpture was used with phases in the range of [−0.4 0] rad, see Fig. 2(e). Amplitudes of O were set to 1 (purely phase shifting object).

Figure 9 shows the simulation results for 1000 mmp iterations. The reconstructed phases and amplitudes of P, as shown in (a) and (b), respectively, are in very good agreement with the phantom, see Fig. 2(c). The amplitudes show only a faint imprint of the phases. The reconstruction of P · O (c) and (d) is also very good, and illustrates the fact that the exit wavefield is dominated by P, which corresponds to the case of a heavily aberrated probe. The information on O is visible only in form of contours of the stone sculpture in the phases. Dividing the recovered P · O by P finally yields O (e) and (f), again of very good quality, cf. Fig. 2(e). The phases of O shown in (e) also only expose a faint imprint of the mandrill’s whiskers. The amplitudes of O show an imprint of the phases of P and O but the distortions are only in the range of ±10% (max.). The error metric Δ on the reconstructions is defined by

 

Fig. 9 Reconstruction of the simulated detector scan experiment with 4 distances using the multiple magnitude projection algorithm (Eq. (9)). The left column shows phase and the right column shows amplitude. (a), (b) Reconstructed P. (c), (d) Reconstructed P · O. (e), (f) Reconstructed O, from division of P · O by P. The Δ in the title denotes the reconstruction error as defined in Eq. (10). Scale bar indicates 4 μm in the sample plane.

Download Full Size | PDF

5.2. Experimental data

Motivated by the successful numerical results, the next step was to apply the mmp algorithm to experimental data. The data was recorded in the second configuration as described in Section 3, exemplary data is depicted in Fig. 5. Due to the cone beam geometry and different magnifications the recorded images must be rescaled and aligned. The images were cropped to the smallest field of view. We have reconstructed the two data sets with and without O independently.

Figure 10 shows the results of the reconstruction of experimental data. The reconstructions of phase and amplitude of P are shown in (a) and (b), respectively. The result looks similar but more plausible than the reconstruction of P in Figs. 7(a) and 7(b). Comparing the reconstruction of P and P · O in (c) and (d) we can also identify the same structures in both, which indicates that the reconstructions are consistent. However, when extracting O from the reconstruction of P · O by dividing by P, certain artifacts remain, and the result looks worse than the result for O reconstructed from a single distance as before in the d&u approach, shown in Fig. 7(d). This is attributed to the fact that this data set is more affected by temporal drift of P, simply because the delay between measuring P and P · O was much too long, namely about 20 min. It is surprising that the reconstruction of P is still possible at this point. For an optimized scan, a time interval between subsequent images of less than a minute is realistic, in view of the necessary detector movement. This could sufficiently avoid drift of P and enable reconstructions of similar quality as those in the numerical simulation.

 

Fig. 10 Reconstruction of the experimental detector scan with the multiple magnitude projection algorithm (Eq. (9)). The left column shows phase and the right column shows amplitude of the reconstruction. (a), (b) Reconstructed P. (c), (d) Reconstructed P · O. (e), (f) Reconstructed O, from division of P · O by P. The pixel size is 36 nm. The colorbar below the last image in a column applies to all images of the column. Scale bar indicates 4 μm in the sample plane.

Download Full Size | PDF

6. Summary and Conclusion

The first aim of this work was to illustrate the problems associated with the standard flat field correction, both numerically and with experimental data. To this end, we have used an error metric δ Eq. (4). We have investigated, under which conditions the division of intensity images of P and P · O in the detection plane is a sufficient approximation, in terms of object and source size. From the simulation we can conclude that the reconstruction of the probe P is an important step towards higher resolution in propagation imaging, since otherwise good data quality can only be expected for feature sizes much larger than the source size.

As a first attempt, we have presented 2 algorithms to reconstruct the empty beam P in x-ray propagation imaging. The first algorithm d&u, devised for a single distance data set, uses a nested scheme to update P and O. This scheme works well for purely phase modulated P and O, but it breaks down for the general case of an amplitude and phase modulated P. We have also inferred from this approach that it is not sufficient to use one image of P and P · O each, even if O is known. The second algorithm mmp is already known in literature but has not been used to reconstruct P. The algorithm requires multiple distance measurements of P. mmp yields good reconstructions in simulations even for noisy data with small variation in z_eff. In order to probe P at different distances in the experiment, the sample-detector distance z₂ was changed, which is intrinsically different from changing the sample position z₁. Analyzing this data set showed that in order to get a good reconstruction of P the time between measurements should be short. This is of course also well known for the standard empty beam correction. However, in order to cover a sufficient diversity (range in Fresnel numbers), the detector has to be moved over a rather wide range, which makes this approach more prone to temporal drift. Flight tubes are another experimental difficulty in implementing this approach, unless the detector is contained in the flight tube. For the experimental strategy, the number of detector positions and the shift δz₂ must be optimized in view of sufficient diversity while keeping the total measurement time as small as possible in view of drift elimination. Interestingly, a recent generalization of ptychography to longitudinal scanning (OAP, on-axis ptychography) has also shown, that a defocus scan with constant detector position, i.e. moving only the sample, is not sufficient for simultaneous reconstruction of P and O [38]. However, numerical simulations have yielded good results already for two detector distances. This still awaits experimental verification, but it can be anticipated that more than one algorithm will be suitable to perform the simultaneous reconstruction P and O.

7. Appendix A

The error of the standard flat field correction can be analyzed based on a rigorous mathematical proof starting from the stationary wave equation and resulting in an upper bound for the error as a function of all relevant parameters. This result will be presented elsewhere [39], and used an alternative definition of the error metric

(11)$$\delta =\frac{{\Vert {𝒟}_{z_{\text{eff}}}(P\cdot O)-\text{exp}(-i2\pi /\lambda z_{\text{eff}}){𝒟}_{z_{\text{eff}}}(O){𝒟}_{z_{\text{eff}}}(P)\Vert }_{\infty }}{{\Vert {𝒟}_{z_{\text{eff}}}(P)\Vert }_{\infty }{\Vert {𝒟}_{z_{\text{eff}}}(O)\Vert }_{\infty }},$$

which differs from the heuristic expression given in Eq. (4). The numerator of δ measures the difference of 𝒟_zeff(P · O) and the product 𝒟_zeff(O)𝒟_zeff(P). This difference is then normalized by the modulus of the propagated P and O. It is shown in [39] that the supremum of Eq. (11) is bounded by

(12)$$\frac{\sqrt{2}\hspace{0.17em}{z}_{\text{eff}}\hspace{0.17em}\lambda }{8\hspace{0.17em}\pi \hspace{0.17em}\mathrm{\Delta }{x}_P\hspace{0.17em}\mathrm{\Delta }{x}_O}\frac{{\int }_{{ℝ}^2}\left|ℱ(P)\right|\text{d}k\hspace{0.17em}\hspace{0.17em}{\int }_{{ℝ}^2}\left|ℱ(O)\right|\text{d}k}{{\Vert {𝒟}_{z_{\text{eff}}}(P)\Vert }_{\infty }|{{𝒟}_{z_{\text{eff}}}(O)\Vert }_{\infty }}.$$

Figure 11 shows the results obtained with the metric Eq. (11). The simulation was carried out, following the same procedure as described in Section 2 also using the same set of parameters. Both errors Fig. 3 and Fig. 11 show the same behavior, regarding the strength of filtering (ξ/ξ_max varied) and the variation of object periodicity Δx_O. In fact it is possible to define several other metrics, showing the same general behavior as here, but without the mathematical foundation as Eq. (11).

 

Fig. 11 The mean error of Eq. (11) as a function of the normalized cut-off frequency ξ/ξ_max in the filter applied to the probe P, for different periodicities Δx_O (grid spacings). The alternative x-label on the top indicates the effective source size a in units of pixels, corresponding to ξ.

Download Full Size | PDF

Acknowledgments

We thank Matthias Bartels, Martin Krenkel and Markus Osterhoff for help at the beamtime, and also for numerous discussions and fruitful collaboration in related projects. Funding by SFB 755 Nanoscale Photonic Imaging is gratefully acknowledged.

References and links

1. P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999). [CrossRef]  

2. D. M. Paganin, Coherent x-ray optics (Oxford University Press,2006). [CrossRef]  

3. K. A. Nugent, “The measurement of phase through the propagation of intensity: An introduction,” Contemp. Phys. 52, 55–69 (2011). [CrossRef]  

4. H. M. Quiney, “Coherent diffractive imaging using short wavelength light sources,” J. Mod. Opt. 57, 1109–1149 (2010). [CrossRef]  

5. D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: Theory and numerical methods,” SIAM Rev. 44, 169–224 (2002). [CrossRef]  

6. T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997). [CrossRef]  

7. M. Krenkel, M. Bartels, and T. Salditt, “Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging,” Opt. Express 21, 2220–2235 (2013). [CrossRef]   [PubMed]  

8. E. Maire, J.-Y. Buffière, L. Salvo, J. J. Blandin, W. Ludwig, and J. M. Letang, “On the application of x-ray microtomography in the field of materials science,” Adv. Eng. Mater. 3, 539–546 (2001). [CrossRef]  

9. P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, and J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006). [CrossRef]  

10. B. D. Metscher, “Biological applications of x-ray microtomography: Imaging microanatomy, molecular expression and organismal diversity,” Microsc. Anal. (Am Ed) 27, 13–16 (2013).

11. T. van de Kamp, P. Vagovič, T. Baumbach, and A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011). [CrossRef]  

12. J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013). [CrossRef]  

13. M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, and T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013). [CrossRef]  

14. A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997). [CrossRef]  

15. P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14626–14630 (2006). [CrossRef]   [PubMed]  

16. M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy , 1,1–10 (2012).

17. K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011). [CrossRef]  

18. C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, and T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol. 57, 5309–5323 (2012). [CrossRef]   [PubMed]  

19. G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, and J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012). [CrossRef]  

20. S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011). [CrossRef]  

21. R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011). [CrossRef]  

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

23. 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]  

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

25. M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: Phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1–6 (2013). [CrossRef]  

26. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature (London) 2, 101–104 (2006).

27. F. Döring, A.-L. Robisch, C. Eberl, M. Osterhoff, A. Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M. Bartels, T. Salditt, and H. U. Krebs, “Sub-5 nm hard x-ray point focusing by a combined Kirkpatrick-Baez mirror and multilayer zone plate,” Opt. Express 21, 19311–19323 (2013). [CrossRef]   [PubMed]  

28. T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013). [CrossRef]  

29. D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D: Appl. Phys. 6, 2200–2216 (1973). [CrossRef]  

30. B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004). [CrossRef]   [PubMed]  

31. L. J. Allen, W. McBride, and M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004). [CrossRef]  

32. L. J. Allen, W. McBride, N. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004). [CrossRef]   [PubMed]  

33. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005). [CrossRef]  

34. S. C. Kramer and J. Hagemann, “SciPAL: Expression templates and composition closure objects for high performance computational physics with CUDA and OpenMP,” ACM Trans. on Parallel Comput. (submitted).

35. J. N. Clark, C. T. Putkunz, M. A. Pfeifer, A. G. Peele, G. J. Williams, B. Chen, K. A. Nugent, C. Hall, W. Fullagar, S. Kim, and I. McNulty, “Use of a complex constraint in coherent diffractive imaging,” Opt. Express 18, 1981–1993 (2010). [CrossRef]   [PubMed]  

36. L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001). [CrossRef]  

37. L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001). [CrossRef]  

38. A.-L. Robisch and T. Salditt, “Phase retrieval for object and probe using a series of defocus near-field images,” Opt. Express 21, 23345–23357 (2013). [CrossRef]   [PubMed]  

39. C. Homann, J. Hagemann, T. Salditt, and T. Hohage, “Remarks on the product approximation in the detector plane,” (in prepartion).