Two crossed waveguides (2DWG) were measured at the new endstation GINIX for coherent nano-focus imaging (Kalbfleisch et al.
) installed at the P10 coherence beamline, Petra III (DESY, Hamburg, Germany). The endstation uses elliptically figured Pd-coated silicon and Pd-coated silica Kirkpatrick–Baez (KB) mirrors for vertical and horizontal focusing, respectively. The photon energy is defined by a fixed-exit double-crystal Si(111) monochromator, and can easily be changed without much re-alignment of the nano-focus. Some of the imaging experiments during the commissioning phase have been carried out at 7.9 keV, others at 13.8 keV, some at 15 keV. The parameters of the KB focus vary with energy, storage ring operation, and alignment status. During the 13.8 keV alignment, the focal spot size was measured to
= 370 nm in the horizontal and
= 120 nm in the vertical direction, as measured by scanning the planar WG through the KB beam. The maximum integrated intensity in the focal spot of the KB beam was
= 2.4 × 1011
, as measured by a pixel detector (Pilatus 300K, Dectris). The waveguide was positioned in the focal spot of the KB system using a goniometer mounted upside down on a vibration-reduced extension arm with three miniaturized translations and two miniaturized rotations (Attocube Systems), along two directions, orthogonal to the optical axis
. A more detailed description of the endstation can be found by Kalbfleisch et al.
). A noise-free single-photon-counting detector [Pilatus 300K, Dectris (Kraft et al.
)] with a pixel size of 172 µm and an active area of 487 × 619 pixels was used to measure the far-field pattern of the WG at a distance of
= 5.29 m.
) shows the measured far-field pattern of a crossed waveguide system (2DWG-1) where the individual WG slices, denoted WG1-1 and WG2-1, have a guiding layer thickness of 35 nm each. The length of WG1-1 and WG2-1 are
= 400 m and
= 207 m, respectively, leading to a combined thickness of
= 607 m. The incoming KB beam subsequently illuminated WG1 which was placed horizontally and WG2 which was placed vertically.
Figure 9 (a) Fraunhofer diffraction pattern of the 2DWG-1, pre-focused by KB mirrors, at 15.0 keV (logarithmic scale, scalebar 0.02 Å−1, 100 s dwell time). (b) WG near-field distribution in the effective confocal plane of (more ...)
As described by Krüger et al.
), the 2DWG near-field was reconstructed using the iterative error-reduction (ER) algorithm. Fig. 9(b
) shows the exit wavefield reconstruction after ten iterations of the ER algorithm. Note that the reconstructed near-field must be associated with an effective confocal plane of the 2DWG. Fig. 9(c
) shows the line profile of the reconstruction in the horizontal (top) and vertical (bottom) direction along with Gaussian fits. The FWHM obtained from the fits are 10.0 nm and 9.8 nm in the horizontal and vertical direction, respectively. The high beam confinement is in agreement with the autocorrelation, which yields a FWHM of 18.3 × 17.8 nm. The respective FWHMs of the reconstruction are close to the values determined earlier for the same 2DWG, but with a different experimental set-up and at higher photon energy, reported by Krüger et al.
). The integrated photon flux exiting the 2DWG-1 was maximum at 2.0 × 107
Higher photon flux exiting a 2DWG can be reached by choosing a shorter waveguide length (adapted to the photon energy). We have performed the same measurements with a second crossed waveguide system, denoted as 2DWG-2, having a combined thickness of only
= 490 µm (WG1-2 vertically placed:
= 270 µm; WG2-2 horizontally placed:
= 220 µm) at
= 13.8 keV. Fig. 9(d
) shows the measured far-field pattern of the 2DWG-2. The far-field pattern indicates similar characteristics as 2DWG-1. A maximum photon flux of 1.0 × 108
exiting the 2DWG-2 was measured. In analogy to the reconstruction presented in Fig. 9(b
), the near-field reconstruction shown in Fig. 9(e
) exhibits a high beam confinement in the effective confocal plane of the 2DWG-2. Line scans with corresponding Gaussian fits yield a FWHM of 10.7 nm and 11.4 nm in the horizontal and vertical direction, respectively.
Waveguides can be used as illumination source for propagation imaging in projection geometry, as demonstrated here for a test sample placed at a distance
= 2.0 mm from the 2DWG-1 (
= 15 keV). The hologram is recorded at a distance
= 5.29 m from the sample using a single-photon-counting pixel detector (Pilatus, Dectris). Fig. 10(a
) shows schematically the experimental set-up used at the P10 beamline (nano-focus endstation operated by University of Göttingen) for imaging of weakly scattering samples. The sample stage is equipped with a group of
piezos (Physik Instrumente) on top of an air-bearing rotation (Micos). Additional
stages (Micos) below the rotation allow for distance variations of the sample to the WG. The distance between WG and sample is further controlled by two on-axis optical microscopes, one in front of the WG and one behind the sample.
Figure 10 (a) Experimental set-up for coherent nano-focus imaging at the P10 beamline, Petra III. (b) Schematic of the experimental set-up for waveguide-based imaging using KB mirror pre-focusing. The sample is placed at a distance from the waveguide and the hologram (more ...)
) shows the holographic phase reconstruction of a Siemens star test pattern (NTT-AT, Japan; model ATN/XRESCO-50HC), recorded at the P10 beamline using the Pilatus pixel detector (Dectris). A mesh of 7 × 6 scan points was recorded with the sample shifted in the
-plane (exposure time 10 s each), i.e.
a total number of 42 holograms.
Each hologram was reconstructed individually, and the resulting reconstructions were then stitched together. For holographic reconstruction the projection geometry used here was mapped onto parallel-beam propagation by a variable transformation based on the Fresnel scaling theorem. Given the distance
between the WG and the sample, parallel-beam reconstruction by Fresnel backpropagation of the recorded intensity can be applied using the effective defocus
= 2.0 mm. At the same time the hologram is magnified corresponding to the geometrical projection by a factor of M
= 2646. Accordingly, given the 172 µm pixel size, the effective (de-magnified) pixel size in the sample plane is 65 nm. Corresponding to this sampling, the sector ring down to 100 nm lines and spaces is represented, but not the innermost sector ring down to the 50 nm lines and spacings. Holographic reconstruction is a robust one-step reconstruction scheme and the reconstruction is unique. However, the reconstructed phase distribution is adulterated by the so-called twin-image leading to artifacts, i.e.
the reconstructed phase values are not quantitatively correct. For the present object and photon energy, a phase difference of 0.46 rad between the void areas and the Ta structure of the test pattern is expected. A pixel detector with smaller pixel size would thus improve the resolution at constant field of view, or allow for a larger field of view (as controlled by defocus distance) for constant resolution.
As another example, Fig. 10(d
) shows an image (reconstructed phases), recorded at the ID22-NI undulator beamline of ESRF, using the same waveguide and test pattern, but in this case a pixel detector with 55 µm pixel size (Maxipix). The experimental set-up is described in detail by Krüger et al.
). At a defocus distance of
= 7 mm, the effective pixel size in the sample plane is 124.6 nm. Unfortunately, smaller
values were prohibited at this set-up by bulky positioning stages and sample mounts. The total photon flux impinging onto the sample was 7.6 × 107
photons (17.5 keV, exposure time 1 s), providing a signal-to-noise ratio which is high enough for phase retrieval by an iterative algorithm. Compared with holographic reconstruction, iterative algorithms enable quantitative phase reconstruction without twin image artifacts. Here, we have used a modified Gerchberg–Saxton (GS) algorithm (Gerchberg & Saxton, 1972
), enhanced by an additional reconstruction tool proposed by Marchesini et al
) using a blurred version of the current estimate of the object under reconstruction. The blurring smoothes out noise and provides a form of regularization.
The blurring was carried out by convolving the reconstructed wavefield with a Gaussian of width σ at each iteration step. The width σ is set to 1 pixel (FWHM of 2.3548σ). The projection operator
in the sample plane acts on the amplitude of the convolved estimate of the object
where conv denotes the convolution operator and
is a Gaussian of width σ. We denote this scheme as GS-Gaussian. The additional convolution reduces the spatial resolution of the reconstructed object owing to blurring. However, resolution can be recovered by subsequent GS iterations. Fig. 10(c
) shows the phase reconstruction after
= 50 GS-Gaussian iteration steps followed by
= 14 GS iteration steps. The two maxima of the phase histogram yield a relative phase shift of 0.38 rad, close to the expected phase shift of 0.4 rad.
The examples shown above show that the waveguide-based illumination system yields full-field phase-contrast hard X-ray images with adjustable magnification, resolution and field of view, at relatively low dose. Importantly, the waveguide acts as a coherence filter enhancing the image quality with respect to propagation imaging based on partially coherent illumination, or illumination systems with wavefront distortions. Rather than reconstructing both the wavefield and object, which is necessary for distorted phase fronts, a simple division by the empty beam yields very clean holograms. The disadvantage is a compromise in flux, and, as is always the case for high-magnification projection microscopy, a considerable sensitivity to mechanical vibrations. The theoretical resolution of the present waveguide system is in the range of 10 nm, corresponding to the beam confinement. This resolution range could not be reached or even tested in the present example, since the resolution was limited by pixel size as dictated by the defocus distance and detector pixel size. With improved instrumentation, in particular with high-resolution detectors, which have in the meantime been installed at the P10 nano-focus endstation, higher-resolution images are now in reach.
Finally, we comment on the astigmatism which is an intrinsic feature of the crossed WG system, leading in the present case to 200 µm offset between the vertical and horizontal source plane. At small
= 2 mm, as in the example shown in Fig. 10(c
), this astigmatism leads to an optically visible ellipticity of about 10% in the reconstructed image. In future, we plan to remove this artifact by a simple generalization: the Fresnel propagators used in the reconstruction algorithms shall be adapted to the correct anisotropic propagation distance, each for the
plane, respectively. However, this is beyond the scope of the present work which concentrates on waveguide fabrication and characterization, instead of imaging.