1. Introduction

Hard x-ray scanning microscopy is used in a variety of scientific fields such as physics and chemistry, as well as biomedical, earth, environmental, materials, and nanoscience [1]. Due to their large penetration depth x rays are ideally suited for non-destructive microscopy of the bulk of an object [2, 3] or to image in-situ a specimen inside a special sample environment [4]. Different x-ray analytical techniques, such as x-ray diffraction, x-ray fluorescence analysis, and x-ray absorption spectroscopy, yield structural, elemental, and chemical contrast, respectively [5–8].

In scanning microscopy, the x-ray beam from a highly brilliant synchrotron radiation source is focused by an x-ray optic onto the sample in a strongly reducing geometry. Ideally, the beam size at the sample is limited by diffraction at the aperture of the optic. In that case, the smallest beam sizes and highest spatial resolutions are achieved. While the short wavelength allows in principle to reach diffraction limited beams in the sub-nanometer range [9–11], todays x-ray optics are mainly technology limited, reaching focal spot sizes well below 100 nm [12–14].

Generally, the spatial resolution of a scanning microscope is limited by the size of the probe, i. e., the structural variations inside the sample are convolved with the nanobeam. The exact knowledge of the optical field around the focus is crucial to properly interpret and deconvolve the scanning micrographs, and to understand and correct for aberrations of x-ray optics. This complex wave field can be determined using ptychographic scanning diffraction microscopy [15–20]. In this coherent imaging technique [15, 21], a sample is scanned through the nanobeam, recording at each scanning position a far-field diffraction pattern [Fig. 1(a)]. From these data, the complex transmission function [Fig. 1(b)] of the object and the complex wave field of the illuminating beam [Fig. 1(c)] can be reconstructed. With the complex wave field reconstructed in the object plane, in principle, the full caustic of the nanobeam can be reconstructed by numerical propagation, giving a full optical characterization of the nanobeam and the aberrations of the instrument [15–20, 22].

 

Fig. 1 (Color) (a) Ptychography: the sample is scanned trough the focused x-ray beam. Diffraction patterns are recorded in the far field at each scanning position. Ptychographic reconstruction: (b) phase of a test object (microchip [2]) and (c) reconstructed complex wave field both at position −500 μm upstream of the focal plane of the microscope. The rectangle in (b) delineates the area covered by the ptychographic scan.

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In this article we address the question of how reliable the reconstruction of the complex optical wave field is, considering that the reconstructed object function often suffers from artifacts due to positioning errors of the scanner and mechanical instabilities during the exposure [19]. While the object function can be verified by other microscopy techniques, there is no independent other method that yields the same wealth of information for the wave field. Therefore, here, we propose a verification by self-consistency: ptychograms of a test object are recorded at different positions along the beam, reaching well into the far field of the nanobeam. The resulting complex wave fields are then used to predict the wave fields at the other positions along the beam by numerical (Fresnel-Kirchhoff) propagation. A comparison with the measured wave fields shows how consistent these results describe a common nanobeam.

2. Ptychographic Experiments

The ptychography experiments were carried out at the hard x-ray scanning microscope at beam-line ID13 of the European Synchrotron Radiation Facility (ESRF). The hard x rays from an undulator source were monochromatized (energy E = 15.25 keV, wave length λ = 0.813 Å) by a channel-cut monochromator (Si 111) located at 31 m from the source and focused in the x-ray scanning microscope (location: L _s = 96 m from the source) by nanofocusing refractive x-ray lenses made of silicon [12]. Two such cylinder lenses are aligned in a crossed geometry [cf. Fig. 1(a)] to generate a nearly Gaussian point focus about 13 mm behind the last lens with full width at half maximum (FWHM) lateral extensions of 74 × 81 nm² in the horizontal and vertical directions (H × V), respectively. Due to the generally small numerical aperture (NA ≈ 10⁻³) of hard x-ray optics, the depth of focus is slightly larger than 100 μm. This also justifies the use of the paraxial approximation in numerical propagation.

The effective FWHM source size d _s is 150 μm horizontally and 50 μm vertically as measured by fringe visibility experiments [23], resulting in a lateral coherence length at the instrument of 53 × 160 μm² (H × V), here, defined by $l_t\hspace{0.17em}=\hspace{0.17em}\frac{\lambda L_{\text{s}}}{d_{\text{s}}}$. As this lateral coherence length at the instrument is much larger than the aperture of the optic [40 × 36 μm² (H × V)], the focus is diffraction limited and the focused beam has a high degree of lateral coherence. The coherence in the focus is determined by propagating the mutual intensity from the source, through the focusing optic to the focus as described in [24]. At the focus, the lateral coherence length is 325 × 990 nm² (H × V), well exceeding the size of the central maximum of the focus.

Ptychograms were recorded at different positions of the object along the optical axis, e. g., −1000, −500, 0, 500, and 1000 μm from the focal plane. As a test object a front-end processed microchip (512 Mb DDR 2 RAM by Qimonda in 80 nm technology) was used [2]. For each ptychogram an area of 2 × 2 μm² was scanned in 50 × 50 steps (step size 40 nm), recording at each position of the scan a far-field diffraction pattern with a single photon counting MAXIPIX detector placed at a distance L = 1.9 m downstream of the focus (exposure time 0.3 s per point). The overall acquisition time including overheads was 43 min. The MAXIPIX detector has N ² = 256 × 256 pixels (pixel size d ² = 55 × 55 μm²). As the translation along the optical axis was not perfectly aligned with the beam, the scanned area on the sample varied slightly for the different ptychograms.

3. Reconstruction and Consistency of Optical Wave Field

All ptychograms were reconstructed using the algorithm by Maiden and Rodenburg [17] (100 iterations). Figure 1(b) exemplarily shows the reconstructed test object 500 μm upstream of the focus (position −500 μm) together with the reconstructed x-ray wave field at that position along the optical axis [Fig. 1(c)]. The pixel size in both reconstructions is λL/(Nd) = 11.0 nm. Due to the extended illumination, the object is also reconstructed beyond the scanned area [cf. Fig. 1(b)]. The strongly phase shifting plug vias in the microchip (cf. [2] for a detailed description of the sample) are visible as dark vertically elongated dots in Fig. 1(b). Due to mechanical instabilities of the x-ray microscope, they show slight artifacts in the form of horizontally displaced fringes. Therefore, it is important to verify that similar artifacts do not appear in the reconstructed wave field. This is done here, assuming that artifacts in the reconstructed wave field would yield inconsistencies when propagated to another position along the optical axis.

The first row of Fig. 2 shows the complex wave fields reconstructed from the different ptychograms along the optical axis. For comparison, the second row of Fig. 2 shows the wave fields calculated by propagating that at the focal position to the different positions along the optical axis. The propagation is done using Fresnel-Kirchhoff integration (scalar field) in paraxial approximation. This is very well justified in the hard x-ray range at the 10 nm resolution level. As the overall phase in each reconstructed optical field is arbitrary, it was fixed to match that in the propagated optical fields. In addition, the optical axis was centered to the nearest pixel in each of the reconstructed and propagated wave fields.

 

Fig. 2 (Color) First row: ptychographic reconstructions of the x-ray wave field measured independently at different positions from the focus along the optical axis. Second row: given the wave field in the focus (0 μm), the wave fields at the other positions are obtained by Fresnel-Kirchhoff propagation. Complex amplitudes are coded according to the color wheel. Comparing the propagated and reconstructed wave fields shows excellent agreement. The reconstructed projected vertical beam profile (intensity) is shown at the bottom.

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Figure 2 shows a striking correspondence of the measured and propagated wave fields along the caustic. Exemplarily, the difference of two corresponding wave fields is shown in Fig. 3. The measured and propagated wave fields [Fig. 3(a) and 3(b)] can hardly be distinguished visually. The difference [Fig. 3(c) and 3(d)] amounts to about 10% in relative L ² norm and is mainly a result of sub-pixel misalignment. This is shown in Fig. 3(d): each of the Fresnel zones in the difference image changes phase by about 180° between upper right and lower left [cf. color wheel in Fig. 3(c)]. This means that the propagated field is slightly shifted towards the upper right with respect to the measured field, making it smaller in amplitude at the lower left and higher at the upper right.

 

Fig. 3 (Color) (a) illumination reconstructed (measured) at position −1000 μm upstream of the focus and (b) that obtained by propagation from the wave field in the focus (same as in Fig. 2). (c) Difference between reconstructed and propagated illumination at that position on the same color scale as in (a) and (b). (d) difference with 10× enhanced contrast. The 180° phase shift of the Fresnel zones (cf. color wheel) from upper right to lower left in the difference map indicates a (sub-pixel) misalignment along the lower-left-upper-right diagonal.

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Similarly consistent results are obtained by choosing any reconstructed wave field and propagating it to the other positions along the beam. In the case shown here, the reconstruction is faithful over a range larger than 20 times the depth of focus. Ptychographic reconstruction of the correct illumination fails when a significant part of the illumination falls out of the field of view (FOV) that is limited by the angle d/L subtended by a pixel of the detector (FOV = λL/d), i. e., when the sampling requirements for the far-field diffraction patterns are no longer met (cf. also [25]). In the given example, this gradually happens around 1500 μm from the focus. When the optical wave field falls fully into the field of view — this is very well fulfilled in focus — the caustic can be calculated over much longer distances by appropriately zero padding the FOV before numerical propagation. This allows one to propagate the wave field to the exit plane of the nanofocusing optic.

4. Conclusion

In conclusion, ptychography is very well suited to reliably reconstruct the complex wave field in a confined beam, even when mechanical instabilities and unwanted sample motion introduce artifacts in the reconstruction of the object. No prior knowledge of the test sample is required, except that it should be optically thin and have a sufficiently high structural diversity on the length scales of the beam. In addition, the test object does not have to be in focus: reliable beam reconstructions can be made from a distance. This is important, for example, when the beam is to be assessed at a position that is not accessible by a test sample, e. g., at the exit of a wave guide [26] or a pinhole [18]. The only restriction is the size of the beam at the sample position that must be compatible with the sampling of the far-field diffraction patterns. The method is not limited to the hard x-ray range, but can be applied in other (longer) wavelength regimes as well.