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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 6 — Mar. 24, 2014
  • pp: 6438–6446
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At-wavelength metrology of hard X-ray mirror using near field speckle

Sebastien Berujon, Hongchang Wang, Simon Alcock, and Kawal Sawhney  »View Author Affiliations


Optics Express, Vol. 22, Issue 6, pp. 6438-6446 (2014)
http://dx.doi.org/10.1364/OE.22.006438


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Abstract

We present a method to measure the surface profile of hard X-ray reflective optics with nanometer height accuracy and sub-millimetre lateral resolution. The technique uses X-ray near-field speckle, generated by a scattering membrane translated using a piezo motor, to infer the deflection of X-rays from the surface. The method provides a nano-radian order accuracy on the mirror slopes in both the tangential and sagittal directions. As a demonstration, a pair of focusing mirrors mounted in a Kirkpatrick-Baez (KB) configuration were characterized and the results were in good agreement with offline metrology data. It is hoped that the new technique will provide feedback to optic manufacturers to improve mirror fabrication and be useful for the online optimization of active, nano-focusing mirrors on modern synchrotron beamlines.

© 2014 Optical Society of America

1. Introduction

Reflective optics are traditionally employed to focus or collimate the X-ray beam on synchrotron beamlines. Figure errors, with spatial frequencies < 1 mm−1, on X-ray mirrors are known to generate optical aberrations, which degrade the focusing performance. Over the last few decades, continuous improvements to figure errors have tremendously enhanced the capabilities of modern synchrotron beamlines in term of spatial resolution and signal-to-noise ratio. However, as optical requirements continue to increase, the quest continues to find metrology techniques capable of characterizing next-generation, nano-focussing, X-ray mirrors with sub-nm figure errors [1

1. S. Berujon, At-Wavelength Metrology of Hard X-Ray Synchrotron Beams and Optics (University of Grenoble, Online, 2013).

].

The metrology challenges are even more demanding for long (> 1 m) or strongly curved mirrors since stitching or flat-field compensation techniques must often be employed for traditional offline techniques, such as Fizeau interferometry. Custom-built, slope measuring profiles, including the Diamond-NOM [2

2. S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: A non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res., Sect. A 616, 224–228 (2010). [CrossRef]

] and long trace profilers, are also typically operated for offline characterization of X-ray mirrors with nanometer height accuracy. Despite their undoubtable value, laboratory based instruments are unable to fully investigate mirrors under their final working conditions, including thermal bumps induced by the high power synchrotron X-ray beam.

To overcome limitations of offline metrology, increasing efforts are devoted to finding at-wavelength techniques [3

3. 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, 17–22 (2013). [CrossRef]

]. In addition, the short wavelength of X-rays pushes the theoretical limits imposed by diffraction of visible light.

In parallel, the proliferation of active optics on synchrotron beamlines encourages the development of quick, reliable and accurate at-wavelength X-ray metrology techniques for optimization. Active optics are attractive for two main reasons: they compensate for imperfections in the optical surface or in the incoming beam wavefront; and focus the reflected beam to a desired size over a range of sample positions. In both cases, it is vitally important that suitable metrology tools are available to fully control the optical surface [4

4. K. Sawhney, S. Alcock, J. Sutter, S. Berujon, H. Wang, and R. Signorato, “Characterisation of a novel super-polished bimorph mirror,” J. Phys: Conf. Ser. 425, 052026 (2013).

, 5

5. J. Sutter, S. Alcock, and K. Sawhney, “In situ beamline analysis and correction of active optics,” J. Synchrotron Radiat. 19, 960–968 (2012). [CrossRef] [PubMed]

].

Several at-wavelength metrology techniques are today available at synchrotrons for characterization and/or optimization of reflective optics. The most widely used is the ’pencil beam technique’ introduced in the late 20th century [6

6. O. Hignette, A. K. Freund, E. Chinchio, P. Z. Takacs, and T. W. Tonnessen, “Incoherent X-ray mirror surface metrology,” Proceedings of SPIE 3152, 188–199 (1997). [CrossRef]

]. A pair of narrow slits and an imaging detector are used to infer the trajectory of the reflected X-rays. Although this incoherent method is easy to use, the technique offers a reduced sampling resolution and a sensitivity limited by diffraction. More recent and sophisticated techniques, including Hartmann sensors [7

7. M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instrum. Methods Phys. Res., Sect. A 616, 162–171 (2010). [CrossRef]

], Fresnel propagation iterative algorithms [8

8. H. Yumoto, H. Mimura, S. Matsuyama, S. Handa, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “At-wavelength figure metrology of hard X-ray focusing mirrors,” Rev. Sci. Instrum. 77, 063712 (2006). [CrossRef]

], grating based instruments [9

9. S. Berujon, H. Wang, E. Ziegler, and K. Sawhney, “Shearing interferometer spatial resolution for at-wavelength hard X-ray metrology,” AIP Conference Proceedings 1466, 217–222 (2012). [CrossRef]

, 10

10. S. Rutishauser, A. Rack, T. Weitkamp, Y. Kayser, C. David, and A. T. Macrander, “Heat bump on a monochromator crystal measured with X-ray grating interferometry,” J. Synchrotron Radiat. 20, 300–305 (2013). [CrossRef] [PubMed]

] and ptychography [11

11. C. M. Kewish, M. Guizar-Sicairos, C. Liu, J. Qian, B. Shi, C. Benson, A. M. Khounsary, J. Vila-Comamala, O. Bunk, J. R. Fienup, A. T. Macrander, and L. Assoufid, “Reconstruction of an astigmatic hard X-ray beam and alignment of K-B mirrors from ptychographic coherent diffraction data,” Opt. Express 18, 23420–23427 (2010). [CrossRef] [PubMed]

] all provide valuable metrology data. However, all but one of these methods cannot yield a map of the full optical surface as metrology laboratory instruments do [12

12. S. Berujon and E. Ziegler, “Grating-based at-wavelength metrology of hard X-ray reflective optics,” Opt. Lett. 37, 4464–4466 (2012). [CrossRef] [PubMed]

].

We present an online technique with nanoradian sensitivity and sub-millimeters lateral resolution for the mapping of both the sagittal and tangential slope errors of hard X-ray mirrors. The technique is based on the speckle effect from a randomly scattering object. It does not require special optics or rely on the high coherence properties of the X-ray beam.

2. Background

Speckle is a well known phenomenon within the visible spectrum [13

13. J. W. Goodman, Speckle Phenomena in Optics; Theory and Applications, 1st edition (Roberts & Company Publishers, Greenwood Village, 2006).

], which has been extensively exploited since the advent of laser sources. Speckle arises from the mutual interference of light generated by imperfections and the rough structure of the objects located in the light path. Sometimes considered as unwanted noise within imaging techniques, speckle has proven highly useful in a range of scientific disciples including metrology [14

14. R. S. Sirohi, Speckle Metrology, Optical Science and Engineering (Marcel Dekker, In., 1993).

], astronomy, flow visualization, and colloidal fluid investigations. Until recently photon correlation spectroscopy was the only speckle based technique to be extended into the X-ray regime.

In recent years, the characterization of X-ray near-field speckle came to complement and enlarge the fields of application offered by X-ray speckle [15

15. R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008). [CrossRef]

]. One interesting property of X-ray near-field speckle is that propagation distortion is solely determined by the wavefront shape. This feature was quickly exploited in deterministic methods, where speckle was employed as a wavefront intensity modulator [16

16. S. Berujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional X-ray beam phase sensing,” Phys. Rev. Lett. 108, 158102 (2012). [CrossRef] [PubMed]

, 17

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

].

When placing a scattering diffuser, composed of small objects, into a fully or partially coherent beam, a speckle pattern can be recorded using a suitable detector with high resolution power. Within the near-field, the spectrum of the recorded X-ray pattern is determined by: the size of the scattering objects; modulated by the detector response functions; the beam coherence function; and finally by the Talbot effect [15

15. R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008). [CrossRef]

]. Despite not being a strong requirement, using a large bandwidth source permits access to the full spectrum of the diffuser image, thereby avoiding the total loss of certain frequencies due to the Talbot effect [18

18. J. Rizzi, T. Weitkamp, N. Gurineau, M. Idir, P. Mercre, G. Druart, G. Vincent, P. da Silva, and J. Primot, “Quadri-wave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett. 36, 1398–1400 (2011). [CrossRef] [PubMed]

]. The detector response function is also responsible for the degradation of the quality and sharpness of the observable speckle pattern. Indeed, it acts as a low-band pass filter blurring and smoothing the high frequencies. It is important to tune the size of the speckle grains to the detector resolution for a good contrast in the interference pattern. The transverse coherence of the beam is the largest impact factor on the observation of speckle, and sets limits for the largest scattering feature observable, and on the maximum distance at which the interference pattern remains correlated [19

19. R. Cerbino, “Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem,” Phys. Rev. A. 75, 053815 (2007). [CrossRef]

]. Nevertheless, a few microns of transverse coherence are sufficient to generate usable speckle patterns [16

16. S. Berujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional X-ray beam phase sensing,” Phys. Rev. Lett. 108, 158102 (2012). [CrossRef] [PubMed]

].

A common method of improving the contrast in speckle images is to correct for detector noise and spatial fluctuations of the incoming beam. Speckle patterns recorded by the detector Irec can be corrected Icor using:
Icor=IrecIdarkIflatIdark
(1)
where Idark is an average of several acquisitions with the beam shutter closed, and Iflat is the flat-field image acquired without the diffusor in the beam path.

3. Wavefront sensing and mirror shape calculation

The mirror measurement principle is to infer the deflection upon reflection of the wavefront modulation markers, i.e. the near-field speckle. As they match the trajectory of the rays, the speckle grains are linked to the wavefront W through the equation ndrds=gradW where r is a position vector of a ray point and n the optical index [20

20. E. Born and E. Wolf, Principle Of Optics, 7th edition (Cambridge University, 2008).

].

The geometrical setup and considerations for the calculations are shown in Fig. 1 where, for clarity, the problem is restricted to a plane defined by the (ey, ez) basis. The detector pixel rows are assumed to be parallel to the ex axis. We denote the pixel spacing by p, and the mirror tilt angle Θ is set to zero when the mirror is parallel to the direction of the beam, i.e. when its two extrema are aligned with the source point. Since the angles involved are small (typically Θ < 10 mrad), we can make the small angle approximation.

Fig. 1 (a) Geometry considerations. (b) Speckle patterns acquired in two consecutive rows of the detector. (c) Cross-correlation map of the two patterns shown in (b).

The diffusor is scanned along the ey axis at a constant speed, and its position defined by the variable τ. The geometrical wavefront of the beam in the plane of the diffuser is noted W(−d, y) = Wi(y), and the wavefront upstream of the detector with W(z, y) = Wo(y). We associate to each wavefront, the intensity function wi(y, τ) and wo(y, τ) generated by the speckle interferences which depends on the position of the diffusor. Assuming a rigid translation (no distortion) of the membrane, downstream of the diffusor we have the relation:
wi(yδy,τ)=wi(y,τ+δy)
(2)

Considering two rays passing through the point Pk−1 and Pk such that after reflection upon the mirror surface they fall on the detector at the point Nk−1 and Nk, corresponding to the location of two adjacent pixel rows Nk1Nk¯=p, we have:
wi(Pk1,τ)=wo(Nk1,τ)wi(Pk,τ)=wo(Nk,τ)
(3)

Because the near-field speckle pattern is not distorted over a distance much larger than a few pixels, we deduce:
wo(Nk1,τ)=wo(Nk,τ+δτk)
(4)

This equation means that the pixel rows k − 1 and k of the detector will record the same signal but at a different time of the scan τ and τ + δτk.

Combining Eq. (3) and (4), we obtain:
wi(Pk1,τ)=wi(Pk,τ+δτk)
(5)

And by identification using Eq. (2):
δy=Pk1Pk¯=δτk
(6)

In practice, we can recover δτk, the delay between patterns seen from two points of the detector (Fig. 1(b)), using a cross-correlation operation ★:
δτk=argmaxt(wo(Pk,τ)wo(Pk1,τ))
(7)

More specifically, Eq. (6) and (7) state that from localizing the maximum in the signal cross-correlation map (Fig. 1(c)), one can recover Pk1Pk¯, the distance separating the rays falling on adjacent pixel rows at the entrance of the system.

The position of the points Pk is retrieved by integration:
y(Pk)=kδτk+cst
(8)

The constant is chosen such that the central ray of the beam with respect to the mirror is at y(P0) = 0.

The local magnification of the beam can be written as:
Mk=Nk1Nk¯Pk1Pk¯=pδτk
(9)

In the case of wavefront sensing where no reflection is introduced, this relation permits calculation of the local curvature 1Rk of the wavefront in the plane of the detector located at distance zt from the membrane [21

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

]:
1Rk2W(zt,y)y2=1δτpzt
(10)

However, further assumptions are necessary for the correct characterization of a reflective optic surface, as described in Ref. [12

12. S. Berujon and E. Ziegler, “Grating-based at-wavelength metrology of hard X-ray reflective optics,” Opt. Lett. 37, 4464–4466 (2012). [CrossRef] [PubMed]

] where an iterative method is employed. Considering the paths of the rays, we have:
yk0=y(Pk),xk0=y(Pk)tan(Θinc),y(Nk)=Yk=pk+2ztanΘ
(11)
where the zero superscript denotes the first iteration of the method. The corresponding initial mirror slope is calculated using:
Slk012Ykyk0zxk0
(12)

An assumption in this reconstruction method is made regarding the divergence of the incoming beam. One acceptable approximation is to consider the probe beam with a spherical wavefront of radius R, the distance from the source point to the membrane location. The quantitative slope correction applied for each point and iteration is then yR.

4. Experiment

4.1. Setup

The above method was experimentally applied for at-wavelength characterization of a pair of focusing mirrors mounted in a KB configuration. The experiment was performed at the Test beamline B16, at Diamond Light Source (DLS), which is devoted to optics, metrology, and instrumentation developments [22

22. K. J. S. Sawhney, I. P. Dolbnya, M. K. Tiwari, L. Alianelli, S. M. Scott, G. M. Preece, U. K. Pedersen, and R. D. Walton, “A test beamline on Diamond Light Source,” AIP Conference Proceedings 1234, 387–390 (2010). [CrossRef]

]. The experimental setup is shown in Fig. 2(a) and consists of a scattering membrane mounted on a two-dimensional piezo translation stage located at a distance d ≈ 400 mm upstream of the mirror system. The membrane is a piece of filter usually employed for microfiltration in laboratory. This one was made of cellulose acetate, with a thickness of ∼ 120 μm and a nominal pore size of 1.2 μm.

Fig. 2 (a) Sketch of the experimental setup with zh = 3015 mm, zv = 3125 mm and d ≈ 400 mm. (b–c) Single reflected beams from respectively the vertical and horizontally focusing mirrors.

The pair of elliptically polished mirrors consisted of two 100 mm long silicon substrates, each with a 90 mm long platinum coating. The design parameters were HFM = {ph = 47 m, qh = 0.235 m, Θh = 3 mrad} and VFM = {pv = 47.11 m, qv = 0.125 m, Θv = 3 mrad} where pv/h represents the distance from the source to the centre of each mirror, pv/h the distance from the mirror center to the focus, and Θv/h is the incidence angle of operation. After calibrating the Θ = 0 angle of the mirror, the mirror was titled to its operational incidence angle and fully illuminated with a beam larger than the optics aperture. Thus, upon reflections in the KB system, the incoming beam was split into four parts as shown in Fig. 2: the direct beam, a double reflection beam and two single reflection beams.

Images were first processed by selecting the relevant single reflected beam corresponding to each mirror (Fig. 2(b–c)). Since the separation between the two beams was larger than the detector field of view, the mirrors were characterized only one at a time by translating the camera between the scans to intercept each single reflected beam.

Data were acquired during translation of the scattering membrane in both transverse directions. Vertical scans permit recovery of the tangential figure of the VFM and sagittal figure of the HFM, and reciprocally, horizontal scans recover the tangential figure of the HFM and sagittal figure of the VFM. Scans with 80 points and a step size of s = 250 nm were used to accurately recover the mirror shapes. For sagittal measurements, the scan step was increased to s = 1 μm.

4.2. Results

Mirror slopes were calculated using the method detailed in Sec. 3 and fitted to the best ellipse (Θ as the free parameters). A comparison of slope errors measured using the Diamond-NOM are shown in Fig. 3. The online X-ray method found best fit ellipses with parameters Θh = 2.90 mrad for the HFM and Θv = 3.04 mrad for the VFM, which are in a perfect agreement with the data extracted from the Diamond-NOM. The VFM slope deviation from the perfect ellipse was measured to be 0.25 μrad rms with the Diamond-NOM and 0.16 μrad rms with the present online technique. For the HFM, the slope deviation was of 0.60 μrad rms and 0.64 μrad rms calculated respectively from the offline and online measurements. Moreover, the deviation between the slope error measured online and the one measured with the Diamond-NOM is <0.25 μrad rms for both the VFM and HFM.

Fig. 3 Slope errors of the (a) vertical mirror and (b) horizontal mirror measured online and with the Diamond-NOM.

Distortion of the mirror in the sagittal direction (illumination aperture ≈ 1.4 mm) was also extracted using the speckle scanning technique of Sec. 3. In this case, it was assumed that x(Pk) = cst in Eq. (12) and no iterative processing was employed. The results of the sagittal distortions, calculated at three different position of the VFM along the lines drawn in Fig. 2(b), are plotted in Fig. 4. A sagittal slope error of up to 0.25 μrad rms would noticeably degrade the horizontal focusing performance after double reflection in the KB system.

Fig. 4 Sagittal measurements of the VFM: green, blue and red lines show line profiles across the width of the mirror in Fig. 2. (a) Mid-spatial frequency slope errors. (b) Corresponding figure errors.

5. Discussion

A strong contrast in the speckle pattern does not automatically imply accuracy in the calculation of δτ. One factor of merit of the cross-correlation algorithm is given by the maximum of the peak correlation divided by the average absolute value of the correlation map (Fig. 1(c)). Such factors can be optimized by making sure of the randomness of the scatterers and by matching the speckle grains size and the step size. The optimal average size of speckle grains should correspond to a few detector pixels. Despite not being a critical factor, as many filtering membranes are available on the market with pore ranging from 0.1 μm to 10 μm, it is easy to tune the statistical scatterer size to the experimental setup.

The visibility of the speckle is higher when the scattering membrane generates a maximum dephasing greater than π. It ensues that several membranes may have to be stacked to ensure a sufficient thickness of the scattering material, especially when working at higher photon energies.

Assuming that the piezo position step errors are described by a Gaussian distribution, it can be shown that the error on the delay calculation of Eq. (7) decreases at a rate 1N, where N is the number of scan points [25

25. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886–2894 (2001). [CrossRef]

]. Hence, long scans minimize the influence of piezo motor errors.

Using the parameters of our experiment, the theoretical angular sensitivity to the wavefront calculated using Eq. (13) is expected to be σ(α) = 0.01s/z < 1 nrad.

This theoretical limit would hold true in the case of absolute wavefront measurements. However, in the case of mirror characterization, this limit is not reached due to the time stability and deviation from a sphere of the perfect incoming wavefront assumed in the method. Modern beamlines are able to offer beams with wavefront slope errors of < 50 nrad over a millimeter square aperture. This assumption could be verified experimentally on the Diamond B16 beam-line using this method. In order to extend the accuracy of the present method, one has to correct for defects of the X-ray probing wavefront, for instance measuring it beforehand using Eq. (10) and then taking it into account to adapt the necessary local corrections to the reconstruction.

The spatial resolution on the mirror is non-linear due to the reconstruction process. However, the order of magnitude of the spatial sampling is <δτk> / tanΘ, where <.> denotes the average operator. For the present experiment, this gives an average footprint on the mirror of <0.25 mm in the focusing directions and p = 6.4 μm in the sagittal direction.

There are three major advantages of the proposed technique compared to methods using absorption gratings such as in Ref. [12

12. S. Berujon and E. Ziegler, “Grating-based at-wavelength metrology of hard X-ray reflective optics,” Opt. Lett. 37, 4464–4466 (2012). [CrossRef] [PubMed]

]: membranes are easier and cheaper to purchase than X-ray gratings; phase unwrapping, which is particularly difficult when the mirror is strongly curved, is not required; and finally the outgoing wavefront suppresses the sensitivity to the errors introduced by grating defects in phase shifting methods. Our approach can also be used with a grating, and can be considered as a special case of the intensity pattern [21

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

]. Ongoing work focuses on the extension of the technique that will bring it the capability of fully mapping 2D reflective optics. Two-dimensional scan schemes have already been demonstrated to be feasible [21

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

], opening important future possibilities for the characterization of, for instance, long and sagitally focusing mirrors or multilayer-coated surfaces.

6. Conclusion

We have presented an online metrology technique based on near-field speckle, permitting the characterization of hard X-ray reflective optics with high accuracy. This technique offers the advantages of a mechanically simple setup, combined with non-stringent requirements for the beam coherence. Theoretical calculations and the experimental data both show that an angular sensitivity of the order of a few tens of nanoradians can be achieved. Excellent agreement is observed with ex-situ, non-X-ray techniques. We hope that characterization of the optical properties of reflective optics during X-ray operation will lead to further improvements in the development of new X-ray optics and their optimization on X-ray synchrotron beamlines. The technique is expected to find increased interest for measurements of strongly curved or toroidal mirrors.

Acknowledgments

This work was carried out at and with the support of Diamond Light Source Ltd, UK. Stewart Scott, Igor Dolbnya and David Laundy are acknowledged for developing the KB mirror system for the B16 Test beamline. The authors thank Andrew Malandain for his help during the setup of the experiments.

References and links

1.

S. Berujon, At-Wavelength Metrology of Hard X-Ray Synchrotron Beams and Optics (University of Grenoble, Online, 2013).

2.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: A non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res., Sect. A 616, 224–228 (2010). [CrossRef]

3.

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, 17–22 (2013). [CrossRef]

4.

K. Sawhney, S. Alcock, J. Sutter, S. Berujon, H. Wang, and R. Signorato, “Characterisation of a novel super-polished bimorph mirror,” J. Phys: Conf. Ser. 425, 052026 (2013).

5.

J. Sutter, S. Alcock, and K. Sawhney, “In situ beamline analysis and correction of active optics,” J. Synchrotron Radiat. 19, 960–968 (2012). [CrossRef] [PubMed]

6.

O. Hignette, A. K. Freund, E. Chinchio, P. Z. Takacs, and T. W. Tonnessen, “Incoherent X-ray mirror surface metrology,” Proceedings of SPIE 3152, 188–199 (1997). [CrossRef]

7.

M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instrum. Methods Phys. Res., Sect. A 616, 162–171 (2010). [CrossRef]

8.

H. Yumoto, H. Mimura, S. Matsuyama, S. Handa, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “At-wavelength figure metrology of hard X-ray focusing mirrors,” Rev. Sci. Instrum. 77, 063712 (2006). [CrossRef]

9.

S. Berujon, H. Wang, E. Ziegler, and K. Sawhney, “Shearing interferometer spatial resolution for at-wavelength hard X-ray metrology,” AIP Conference Proceedings 1466, 217–222 (2012). [CrossRef]

10.

S. Rutishauser, A. Rack, T. Weitkamp, Y. Kayser, C. David, and A. T. Macrander, “Heat bump on a monochromator crystal measured with X-ray grating interferometry,” J. Synchrotron Radiat. 20, 300–305 (2013). [CrossRef] [PubMed]

11.

C. M. Kewish, M. Guizar-Sicairos, C. Liu, J. Qian, B. Shi, C. Benson, A. M. Khounsary, J. Vila-Comamala, O. Bunk, J. R. Fienup, A. T. Macrander, and L. Assoufid, “Reconstruction of an astigmatic hard X-ray beam and alignment of K-B mirrors from ptychographic coherent diffraction data,” Opt. Express 18, 23420–23427 (2010). [CrossRef] [PubMed]

12.

S. Berujon and E. Ziegler, “Grating-based at-wavelength metrology of hard X-ray reflective optics,” Opt. Lett. 37, 4464–4466 (2012). [CrossRef] [PubMed]

13.

J. W. Goodman, Speckle Phenomena in Optics; Theory and Applications, 1st edition (Roberts & Company Publishers, Greenwood Village, 2006).

14.

R. S. Sirohi, Speckle Metrology, Optical Science and Engineering (Marcel Dekker, In., 1993).

15.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008). [CrossRef]

16.

S. Berujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional X-ray beam phase sensing,” Phys. Rev. Lett. 108, 158102 (2012). [CrossRef] [PubMed]

17.

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

18.

J. Rizzi, T. Weitkamp, N. Gurineau, M. Idir, P. Mercre, G. Druart, G. Vincent, P. da Silva, and J. Primot, “Quadri-wave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett. 36, 1398–1400 (2011). [CrossRef] [PubMed]

19.

R. Cerbino, “Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem,” Phys. Rev. A. 75, 053815 (2007). [CrossRef]

20.

E. Born and E. Wolf, Principle Of Optics, 7th edition (Cambridge University, 2008).

21.

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

22.

K. J. S. Sawhney, I. P. Dolbnya, M. K. Tiwari, L. Alianelli, S. M. Scott, G. M. Preece, U. K. Pedersen, and R. D. Walton, “A test beamline on Diamond Light Source,” AIP Conference Proceedings 1234, 387–390 (2010). [CrossRef]

23.

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, 062001 (2009). [CrossRef]

24.

B. Pan, H.-m. Xie, B.-q. Xu, and F.-l. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615 (2006). [CrossRef]

25.

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886–2894 (2001). [CrossRef]

OCIS Codes
(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure
(340.0340) X-ray optics : X-ray optics
(340.7470) X-ray optics : X-ray mirrors
(120.6165) Instrumentation, measurement, and metrology : Speckle interferometry, metrology

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: November 18, 2013
Revised Manuscript: January 26, 2014
Manuscript Accepted: January 27, 2014
Published: March 12, 2014

Virtual Issues
Vol. 9, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Sebastien Berujon, Hongchang Wang, Simon Alcock, and Kawal Sawhney, "At-wavelength metrology of hard X-ray mirror using near field speckle," Opt. Express 22, 6438-6446 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6438


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References

  1. S. Berujon, At-Wavelength Metrology of Hard X-Ray Synchrotron Beams and Optics (University of Grenoble, Online, 2013).
  2. S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, H. Lammert, “The Diamond-NOM: A non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res., Sect. A 616, 224–228 (2010). [CrossRef]
  3. K. Sawhney, H. Wang, J. Sutter, S. Alcock, S. Berujon, “At-wavelength metrology of X-ray optics at Diamond Light Source,” Synchrotron Radiat. News 26, 17–22 (2013). [CrossRef]
  4. K. Sawhney, S. Alcock, J. Sutter, S. Berujon, H. Wang, R. Signorato, “Characterisation of a novel super-polished bimorph mirror,” J. Phys: Conf. Ser. 425, 052026 (2013).
  5. J. Sutter, S. Alcock, K. Sawhney, “In situ beamline analysis and correction of active optics,” J. Synchrotron Radiat. 19, 960–968 (2012). [CrossRef] [PubMed]
  6. O. Hignette, A. K. Freund, E. Chinchio, P. Z. Takacs, T. W. Tonnessen, “Incoherent X-ray mirror surface metrology,” Proceedings of SPIE 3152, 188–199 (1997). [CrossRef]
  7. M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instrum. Methods Phys. Res., Sect. A 616, 162–171 (2010). [CrossRef]
  8. H. Yumoto, H. Mimura, S. Matsuyama, S. Handa, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, K. Yamauchi, “At-wavelength figure metrology of hard X-ray focusing mirrors,” Rev. Sci. Instrum. 77, 063712 (2006). [CrossRef]
  9. S. Berujon, H. Wang, E. Ziegler, K. Sawhney, “Shearing interferometer spatial resolution for at-wavelength hard X-ray metrology,” AIP Conference Proceedings 1466, 217–222 (2012). [CrossRef]
  10. S. Rutishauser, A. Rack, T. Weitkamp, Y. Kayser, C. David, A. T. Macrander, “Heat bump on a monochromator crystal measured with X-ray grating interferometry,” J. Synchrotron Radiat. 20, 300–305 (2013). [CrossRef] [PubMed]
  11. C. M. Kewish, M. Guizar-Sicairos, C. Liu, J. Qian, B. Shi, C. Benson, A. M. Khounsary, J. Vila-Comamala, O. Bunk, J. R. Fienup, A. T. Macrander, L. Assoufid, “Reconstruction of an astigmatic hard X-ray beam and alignment of K-B mirrors from ptychographic coherent diffraction data,” Opt. Express 18, 23420–23427 (2010). [CrossRef] [PubMed]
  12. S. Berujon, E. Ziegler, “Grating-based at-wavelength metrology of hard X-ray reflective optics,” Opt. Lett. 37, 4464–4466 (2012). [CrossRef] [PubMed]
  13. J. W. Goodman, Speckle Phenomena in Optics; Theory and Applications, 1st edition (Roberts & Company Publishers, Greenwood Village, 2006).
  14. R. S. Sirohi, Speckle Metrology, Optical Science and Engineering (Marcel Dekker, In., 1993).
  15. R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bosecke, M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nat. Phys. 4, 238–243 (2008). [CrossRef]
  16. S. Berujon, E. Ziegler, R. Cerbino, L. Peverini, “Two-dimensional X-ray beam phase sensing,” Phys. Rev. Lett. 108, 158102 (2012). [CrossRef] [PubMed]
  17. K. S. Morgan, D. M. Paganin, K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100, 124102 (2012). [CrossRef]
  18. J. Rizzi, T. Weitkamp, N. Gurineau, M. Idir, P. Mercre, G. Druart, G. Vincent, P. da Silva, J. Primot, “Quadri-wave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett. 36, 1398–1400 (2011). [CrossRef] [PubMed]
  19. R. Cerbino, “Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem,” Phys. Rev. A. 75, 053815 (2007). [CrossRef]
  20. E. Born, E. Wolf, Principle Of Optics, 7th edition (Cambridge University, 2008).
  21. S. Berujon, H. Wang, K. Sawhney, “X-ray multimodal imaging using a random-phase object,” Phys. Rev. A 86, 063813 (2012). [CrossRef]
  22. K. J. S. Sawhney, I. P. Dolbnya, M. K. Tiwari, L. Alianelli, S. M. Scott, G. M. Preece, U. K. Pedersen, R. D. Walton, “A test beamline on Diamond Light Source,” AIP Conference Proceedings 1234, 387–390 (2010). [CrossRef]
  23. B. Pan, K. Qian, H. Xie, A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20, 062001 (2009). [CrossRef]
  24. B. Pan, H.-m. Xie, B.-q. Xu, F.-l. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615 (2006). [CrossRef]
  25. K. A. Goldberg, J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886–2894 (2001). [CrossRef]

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