Speckle based X-ray wavefront sensing with nanoradian angular sensitivity

Hongchang Wang; Yogesh Kashyap; Kawal Sawhney

doi:10.1364/OE.23.023310

1. Introduction

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 [1]. 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 [2]. 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 [3]: direct measurement of the wavefront phase, such as interferometry using single crystals [4]; measurement of the first derivative (gradient) of the wavefront phase, such as Shack-Hartmann sensors [5], coded aperture methods [6] and grating-based interferometry [7]; and measurement of the secondary derivative (curvature) of the wavefront phase, such as propagation-based methods [8]. 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 [19], 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) [20]. 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.

2. Principle

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 $x_{p}$ ( $y_{p}$ ), the intensity signal $I (x, y)$ oscillates as a function of $x_{p}$ ( $y_{p}$ ) at pixel $(l, m)$ . 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:

I^{s, r} (x, y, y_{p}) = \sum_{n = 0}^{N} {A_{n}^{s, r} (x, y) \exp [i B_{n}^{s, r} (x, y) (y_{p} + φ_{n}^{s, r} (x, y)]},

where

s

and

r

denote the sample and reference beams,

n

is the order of the Fourier series, and

A_{n}

,

B_{n}

and

φ_{n}

are Fourier coefficients. When a sample is inserted into the X-ray beam, the speckle pattern shifts due to inhomogeneous phase gradients in the sample.

Fig. 1 Schematic of the experimental setup. The test sample is placed upstream of the abrasive paper, which is mounted on a precision piezo translation stage. The speckle pattern is recorded using a high resolution, X-ray area detector.

Download Full Size | PDF

The vertical speckle displacement $v (x, y)$ can be expressed as the relative phase change between the sample and reference beams

v (x, y) = φ_{n}^{s} (x, y) - φ_{n}^{r} (x, y) .

Displacement is assumed to be equal for all orders, and is extracted with sub-pixel tracking accuracy (η) by applying a cross-correlation algorithm to the two signals [21]. Once the displacement is calculated, it can be used to obtain the vertical wavefront gradient

α_{y} (x, y)

[16]. Since the first derivative of the wavefront is directly proportional to the local gradient of the sample’s phase shift

Φ (x, y)

, it can be written as

\frac{\partial Φ (x, y)}{\partial y} = \frac{2 π}{λ} α_{y} (x, y) \approx \frac{2 π}{λ} \frac{(L_{1} + L_{2} + L_{3}) v (x, y) μ}{L_{1} L_{3}},

where μ is the piezo scanning step size, and L₁, L₂ and L₃ represent the distances between X-ray source, sample, abrasive paper and detector. Here, it should be mentioned that a few (typically 3-5) neighboring pixels for the reference speckle image are chosen to perform the cross-correlation so that the speckle displacement can still be tracked even though a strong speckle displacement exist perpendicular to the analyzed direction. Following the same procedure, a horizontal scan is performed to obtain the horizontal phase gradient. The phase shift induced by the sample is then reconstructed from the two, transverse phase gradients [22].

3. Experiment

The principle of the technique has been validated with experiments at Diamond Light Source’s (DLS) B16 Test beamline [23]. 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 [24]

f = \frac{R_{0}}{2 δ} .

For beryllium, with a refractive index decrement δ of 1.5 × 10⁻⁶ at 15keV, the expected focal length is 66m. The lens was oriented to focus in the vertical plane, with no other focusing optical elements upstream of the lens.

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).

Fig. 2 (Color online) A series of speckle images for the reference beam (a) and with a 1D CRL inserted into the X-ray beam (b). The speckle intensity scans (c) and the corresponding correlation coefficient (d) at the two marked positions.

Download Full Size | PDF

Figure 3a shows the vertical wavefront gradient α_y, which is related to the y position and the focal length for a single refractive lens:

α_{y} \approx \tan α_{y} = \frac{y}{f} .

The line profile along the centre of the CRL for the first derivative of the wavefront and the wavefront phase are shown on the left side of the corresponding plots in Fig. 3(a) and Fig. 3(b) respectively. As shown in the line profile, the vertical wavefront gradient

α_{y}

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

Δ θ

, 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 [19], 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 as

Δ θ \propto \frac{μ η}{\sqrt{N} L_{3}} .

where N is the number of series of speckle images to be averaged. The sensitivity from two series of reference images (single exposure) was 4.5nrad for the experimental setup of μ = 0.5μm, L₃ = 1.06m, and η = 0.01. As shown in Fig. 4, the standard deviation of the reference beam was reduced to 2.1nrad when averaging over 5 exposures, which agrees well with the

1 / \sqrt{5}

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) [25] and analyzer-based imaging (15nrad) [26]. 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) [27]. From Eq. (6), the sensitivity of our technique has the potential for further improvement with larger L₃ or smaller μ.

Fig. 3 Vertical wavefront gradient (a), and the reconstructed phase (b) for a 1D CRL from a series of speckle images with a single exposure. The vertical wavefront gradient error (c) and phase error (d) induced by the 1D CRL for the regions marked in (a) and (b). The standard deviation of the line profile of the vertical wavefront gradient error (c) induced by the 1D CRL is 54nrad, which is significantly larger than the sensitivity of the proposed technique. The scale bar is equal to 0.2mm.

Download Full Size | PDF

Fig. 4 Phase gradient from two series of reference images (averaged over 5 exposures) with step size μ = 0.5 and μ = 1.0μm along the marked line in Fig. 3 (a). The sensitivity for step size μ = 0.5 and μ = 1.0μm is 2.1nrad and 3.8nrad respectively.

Download Full Size | PDF

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.

Fig. 5 Horizontal wavefront gradient (a), vertical wavefront gradient (b) and phase image (c) of a section of the fin of a Mackerel fish from a series of speckle images with single exposure. The scale bar is equal to 0.5mm.

Download Full Size | PDF

5. Conclusions

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.

Acknowledgments

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.

References and links

1. “Program of the 1971 Spring Meeting of the Optical Society of America,” J. Opt. Soc. Am.61, 648–697 (1971).

2. K. Sawhney, H. Wang, J. Sutter, S. Alcock, and S. Berujon, “At-wavelength Metrology of X-ray Optics at Diamond Light Source,” Synchrotron Radiat. News 26(5), 17–22 (2013). [CrossRef]

3. S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Philos. Trans. R. Soc., A 372(2010), 20130021 (2014). [CrossRef] [PubMed]

4. U. Bonse and M. Hart, “An X-ray Interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965). [CrossRef]

5. P. Mercere, S. Bucourt, G. Cauchon, D. Douillet, G. Dovillaire, K. A. Goldberg, M. Idir, X. Levecq, T. Moreno, P. P. Naulleau, S. Rekawa, and P. Zeitoun, “X-ray beam metrology and x-ray optic alignment by Hartmann wavefront sensing,” Proc. SPIE 5921, 592109 (2005). [CrossRef]

6. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074106 (2007). [CrossRef]

7. C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002). [CrossRef]

8. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996). [CrossRef]

9. H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011). [CrossRef] [PubMed]

10. A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Noninterferometric phase-contrast images obtained with incoherent x-ray sources,” Appl. Opt. 50(12), 1765–1769 (2011). [CrossRef] [PubMed]

11. H. Wang, K. Sawhney, S. Berujon, J. Sutter, S. G. Alcock, U. Wagner, and C. Rau, “Fast optimization of a bimorph mirror using x-ray grating interferometry,” Opt. Lett. 39(8), 2518–2521 (2014). [CrossRef] [PubMed]

12. A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997). [CrossRef]

13. A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Methods Phys. Res. A 352(3), 622–628 (1995). [CrossRef]

14. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]

15. S. Bérujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-Dimensional X-Ray Beam Phase Sensing,” Phys. Rev. Lett. 108(15), 158102 (2012). [CrossRef] [PubMed]

16. S. Berujon, H. Wang, I. Pape, and K. Sawhney, “X-ray phase microscopy using the speckle tracking technique,” Appl. Phys. Lett. 102(15), 154105 (2013). [CrossRef]

17. K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012). [CrossRef]

18. H. Wang, S. Berujon, J. Herzen, R. Atwood, D. Laundy, A. Hipp, and K. Sawhney, “X-ray phase contrast tomography by tracking near field speckle,” Sci. Rep. 5, 8762 (2015). [CrossRef] [PubMed]

19. S. Berujon, H. Wang, and K. Sawhney, “X-ray multimodal imaging using a random-phase object,” Phys. Rev. A 86(6), 063813 (2012). [CrossRef]

20. S. Berujon, H. Wang, S. Alcock, and K. Sawhney, “At-wavelength metrology of hard X-ray mirror using near field speckle,” Opt. Express 22(6), 6438–6446 (2014). [CrossRef] [PubMed]

21. B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009). [CrossRef]

22. C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007). [CrossRef] [PubMed]

23. K. J. S. Sawhney, I. P. Dolbnya, M. K. Tiwari, L. Alianelli, S. M. Scott, G. M. Preece, U. K. Pedersen, R. D. Walton, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “A Test Beamline on Diamond Light Source,” AIP Conf. Proc. 1234, 387–390 (2010). [CrossRef]

24. B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999). [CrossRef]

25. F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52(23), 6923–6930 (2007). [CrossRef] [PubMed]

26. P. C. Diemoz, A. Bravin, M. Langer, and P. Coan, “Analytical and experimental determination of signal-to-noise ratio and figure of merit in three phase-contrast imaging techniques,” Opt. Express 20(25), 27670–27690 (2012). [CrossRef] [PubMed]

27. P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-Ray Phase-Contrast Imaging with Nanoradian Angular Resolution,” Phys. Rev. Lett. 110(13), 138105 (2013). [CrossRef] [PubMed]

Speckle based X-ray wavefront sensing with nanoradian angular sensitivity

Abstract

1. Introduction

2. Principle

3. Experiment

4. Results and discussion

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (6)

Optics Express