X-ray wavefront sensing techniques play an important role in both in situ metrology of X-ray optics and X-ray phase contrast imaging. In this letter, we report an approach to measure wavefront aberrations simply using abrasive paper. The wavefront phase change induced by the sample under test was extracted from the speckle displacement by applying a cross-correlation algorithm to two series of speckle images collected using two one-dimensional scans, whilst scanning the abrasive paper in a transverse direction to the incident X-ray beam. The angular sensitivity of the proposed method is shown to be around 2 nanoradians. The potential of the proposed technique for characterizing X-ray optics and the study of biomedical specimens is demonstrated by imaging representative samples.
© 2015 Optical Society of America
Wavefront sensing techniques can precisely measure the wavefront and associated aberrations and thus provide quantitative information about the phase change induced by inserting a sample into the photon beam. In the visible light regime, a Shack-Hartmann sensor is one of the most commonly used instruments for wavefront sensing, and plays an important role in the testing of adaptive optics and lenses . In the hard X-ray regime, advanced X-ray wavefront sensing techniques are increasingly used for in situ metrology of state-of-the-art X-ray optics, widely used at synchrotron radiation facilities and X-ray astronomy telescopes . Similarly, X-ray phase-contrast imaging is proven to greatly enhance the visualisation of samples with weak absorption compared to conventional, absorption contrast imaging. Generally, methods based on the measurement of the wavefront can be divided into three categories : direct measurement of the wavefront phase, such as interferometry using single crystals ; measurement of the first derivative (gradient) of the wavefront phase, such as Shack-Hartmann sensors , coded aperture methods  and grating-based interferometry ; and measurement of the secondary derivative (curvature) of the wavefront phase, such as propagation-based methods . These techniques have been successfully employed for characterizing X-ray wavefronts in many diverse fields [9–14]. However, such techniques generally require complex optical elements, sophisticated experimental setups, or stringent photon beam conditioning.
X-ray near-field speckle has been demonstrated for characterizing two dimensional (2D) wavefronts by tracking speckle displacements from two speckle images [15–18]. Nevertheless, both the spatial resolution and the angular sensitivity are limited by the pixel size of the detector. Attempts have been made to overcome these limitations by employing 2D raster scans , but acquiring the large number of images can be time consuming. Recently, the speckle based technique has been successfully employed to extract the second derivative of the wavefront by scanning a membrane in one dimension (1D) . In this letter, we propose an approach to directly measure the first derivative of the wavefront phase by using speckle scanning technique using two 1D scans only. We show that the pixel-wise wavefront phase can be found by applying a cross-correlation algorithm on two series of signals collected by 1D scanning of abrasive paper, thereby reducing data acquisition time and delivered dose. Importantly, an angular sensitivity of 2 nanoradians was achieved using the proposed technique. We demonstrate its potential for in situ metrology and X-ray phase contrast imaging by investigating suitable samples.
The principle of the proposed method is to perform a pixel-wise, tracking analysis of speckle displacements. As illustrated in Fig. 1, the speckle pattern is generated when abrasive paper, with randomized high-spatial frequency features, is placed into a partially coherent X-ray beam. A sequence of speckle images is collected by scanning the abrasive paper transverse to the incident X-ray beam with no sample present. This process is then repeated with a sample inserted into the X-ray beam, upstream of the abrasive paper. When the abrasive paper is scanned along the horizontal (or vertical) direction (), the intensity signal oscillates as a function of () at pixel . To aid clarity, we discuss only the vertical scan case. The variation in the intensity of the speckle pattern can be expanded as a Fourier series:
The vertical speckle displacement can be expressed as the relative phase change between the sample and reference beams21]. Once the displacement is calculated, it can be used to obtain the vertical wavefront gradient . Since the first derivative of the wavefront is directly proportional to the local gradient of the sample’s phase shift , it can be written as22].
The principle of the technique has been validated with experiments at Diamond Light Source’s (DLS) B16 Test beamline . The test sample was mounted on a motorized translation stage located L1 = 47m from the X-ray source. In order to improve the speckle tracking accuracy, the phase membrane was replaced with abrasive paper to increase the speckle visibility. Abrasive paper, with an average grain size of 5µm, was mounted on a two dimensional piezo stage, L2 = 225mm downstream from the sample. Images of the speckle pattern were collected using a high-resolution X-ray microscope placed L3 = 1065mm downstream from the abrasive paper. The X-ray camera is based on a PCO 4000 CCD detector and uses a Ce-doped YAG scintillator. As a demonstration of the capabilities of the proposed technique for in situ metrology, a 1D compound refractive lens (CRL) was characterized. X-rays with an energy of 15 keV were selected from a bending magnet source using a silicon, double-crystal monochromator. The X-ray detector is equipped with a 10 × microscope objective and has an effective pixel size of 3.6 μm × 3.6 μm after 4 × 4 binning. Acquisition time for each speckle image was 2s. The 1D beryllium CRL has a concave parabolic shape with an aperture of 3 mm horizontally and 0.9mm vertically. The radius of curvature R0 at the apex of the parabola, as specified by the manufacturer, is 200μm. For a single lens, the focal length f can be calculated from R0 by 
4. Results and discussion
As illustrated in Fig. 2, two sets of images were recorded: with (a) and without (b) the CRL in the X-ray path. A total of 60 images were acquired by scanning the abrasive paper transverse to the X-ray beam with a step size of μ = 0.5μm. Speckle intensity profiles for the two marked pixels are shown in Fig. 2c. It can be seen that the speckle (blue triangle) was shifted to the right for a marked pixel at the top of the CRL, whereas the speckle pattern was shifted to the left for a pixel at the bottom (red circle). A cross-correlation algorithm was employed to derive the corresponding correlation curves, as shown in Fig. 2d for the two pixels. The vertical displacement for each pixel was determined by locating the maxima in the correlation coefficient curves, and the wavefront gradient calculated using Eq. (3).
Figure 3a shows the vertical wavefront gradient αy, which is related to the y position and the focal length for a single refractive lens:Fig. 3(a) and Fig. 3(b) respectively. As shown in the line profile, the vertical wavefront gradient is linearly proportional to the position y. The focal length was calculated from a linear fit to the slope using Eq. (5). Surprisingly, a focal length f of 61.9 ± 0.1m was obtained, which is significantly smaller than the value of 66m expected from the manufacturer’s specification. To investigate the X-ray result, the curvature of the lens was independently analyzed in a “blind test” using a visible light micro-interferometer (MicroXAM) in the DLS Metrology lab. The radius of curvature R0 at the apex of the parabola, as measured by the micro-interferometer, varied between 182.5μm and 192.8μm along the length of the lens. The R0 value calculated from Eq. (4) from the X-ray data was 186.8μm, which is highly consistent with the micro-interferometer result. The agreement between the two independent measurements indicates that the fabrication error in the radius of the lens is ~6%. Such errors can significantly affect the focusing performance of an X-ray lens. It is worth mentioning that only an area of 0.2mm (H) × 0.15mm (V) could be measured in each micro-interferometer scan due to the strong radius of curvature of the CRL. In contrast, the whole cross-sectional area of 3mm (H) × 0.9mm (V) of the CRL was measurable using the X-ray speckle technique. The wavefront phase change induced by the 1D CRL was reconstructed from the horizontal and vertical wavefront phase gradients. As illustrated in Fig. 3b, the phase profile shows a concave parabolic shape. The corresponding aberrations in the parabolic form can be retrieved by subtracting the best-fit linear (quadratic) term from the wavefront gradient (phase). The wavefront gradient (phase) error for the region marked in Fig. 3a (Fig. 3b) is shown in Fig. 3c (Fig. 3d). The standard deviation of the wavefront gradient error for the 1D CRL is 54nrad, and the corresponding standard deviation of phase error is 0.04rad. These errors most likely result from a non-perfect fabrication process: either a variation in the thickness of Be in the lens; or asymmetry in the two opposite parabolic surfaces of the CRL. Such precise measurement of the refractive properties is essential for further improvement to the fabrication process, to achieve ever smaller X-ray focal spots. In order to verify the angular sensitivity, the usual procedure to calculate the standard deviation of phase gradient in the empty space was used [25, 26]. Following the same approach, we have tracked the speckle displacement from two repeated series of reference images to check the angular sensitivity of the proposed technique. As described in reference , three predominant parameters, namely, the tracking accuracy (η), the abrasive paper to detector distance (L3) and the scanning step size (μ), determine the angular sensitivity of the speckle scanning technique. In addition, the angular sensitivity can be further improved by averaging several series of speckle images to reduce random noise. Hence, the achievable angular resolution of the speckle scanning technique can be formulated asFig. 4, the standard deviation of the reference beam was reduced to 2.1nrad when averaging over 5 exposures, which agrees well with the reduction predicted by using Eq. (6). As expected, the angular sensitivity increased to 3.8nrad when changing the step size from 0.5μm to 1μm. We would like to highlight that this value is almost an order of magnitude better than the best values previously reported for a grating interferometer (14nrad)  and analyzer-based imaging (15nrad) . Even though the value of 1.9nrad has recently been reported with an edge illumination technique using a non-imaging germanium detector, a sensitivity of 21nrad was only demonstrated with an area detector and a large distance between the sample and the detector (∼15m) . From Eq. (6), the sensitivity of our technique has the potential for further improvement with larger L3 or smaller μ.
To demonstrate the potential of the presented technique for phase contrast imaging, we measured a biological sample. A 4 × microscope objective with a pixel size of 2.25 μm × 2.25 μm was employed to increase the field of view, and a higher X-ray energy of 21 keV was used to reduce the X-ray dose absorbed by the sample. A section of fin from a Mackerel fish was encapsulated in a plastic tube (10 mm diameter) and fixated in water. A series of scans were acquired in the horizontal and vertical directions, with a step size of 1μm and 60 images for each set, with and without the sample present in the X-ray beam. Figure 5a and Fig. 5b show the horizontal and vertical phase gradient images respectively. As indicated in the region marked by an ellipse, the high visibility of the vertical features is noticeable in the horizontal phase gradient map, whilst such information is lost in the vertical gradient map. The opposite situation is highlighted in a rectangle region. Figure 5c shows the phase image reconstructed from the two transverse phase gradients. As shown in the region enclosed by the rectangle, the features at the boundary between fish fin and trunk are clearly observed in the phase image.
In summary, we have demonstrated that the speckle scanning technique can be efficiently used for sensing X-ray wavefronts with high angular sensitivity. The spatial resolution for the proposed technique depends on the detector pixel size rather than the subset size (over ten pixels) for the conventional speckle tracking technique Moreover, data acquisition times can be significantly reduced using 1D scans rather than 2D raster scans. For example, the number of images is only 60 for a single direction for the proposed technique, but it would have required 3600 images for the 2D raster scan. Importantly, the angular sensitivity of the presented method has been demonstrated to be around 2 nanoradians, which has the potential to be further reduced to sub-nanoradians levels by using large detector distances or smaller scanning steps. It should be mentioned that the number of images has to be increased with reduced scan steps in order to further improve the angular sensitivity. In such a case, it would be even impractical to perform 2D raster scan due to the huge amount of data that would be required to be collected. Such ultra-high precision wavefront sensing is highly desirable for the accurate characterization of X-ray optics for next-generation synchrotron radiation sources, X-ray Free Electron Lasers (XFELs) and Astronomy telescopes. In addition, the speckle scanning technique can be easily implemented on existing imaging beamlines to perform phase contrast imaging. It can also be extended to 3D tomography, especially on high brilliance light sources, by using high efficiency X-ray cameras.
This work was carried out with the support of Diamond Light Source Ltd. We thank Simon Alcock & Ioana Nistea for testing the CRL lens using a micro-interferometer, and for Simon’s critical editing of the manuscript.
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