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  • Editor: Xi-Cheng Zhang
  • Vol. 39, Iss. 8 — Apr. 15, 2014
  • pp: 2518–2521
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Fast optimization of a bimorph mirror using x-ray grating interferometry

Hongchang Wang, Kawal Sawhney, Sebastien Berujon, John Sutter, Simon G. Alcock, Ulrich Wagner, and Christoph Rau  »View Author Affiliations


Optics Letters, Vol. 39, Issue 8, pp. 2518-2521 (2014)
http://dx.doi.org/10.1364/OL.39.002518


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Abstract

An x-ray grating interferometer was employed for in situ optimization of an x-ray bimorph mirror. Unlike many other at-wavelength techniques, only a single interferogram image, captured out of the focal plane, is required, enabling the optical surface to be quickly optimized. Moiré fringe analysis was used to calculate the wavefront slope error, which is proportional to the mirror’s slope error. Using feedback from grating interferometry, the slope error of a bimorph mirror was reduced to <200nrad (rms) in only two iterations. This technique has the potential to create photon beams with spatially homogeneous intensities for use in synchrotron and free electron laser beam lines.

© 2014 Optical Society of America

X-ray mirrors are widely used at synchrotron facilities for micro- and nano-focusing due to the achromatic nature of total external reflection. In response to demand and increased metrology capabilities, the optical quality of mirrors has gradually improved, with sub-microradian optical slope errors now commonplace. The recent introduction of “super-polishing” techniques has dramatically improved slope errors to only tens of nanoradians [1

1. K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, Rev. Sci. Instrum. 73, 4028 (2002). [CrossRef]

]. Nonetheless, even using such state-of-the-art mirrors, the focal spot size is often limited by wavefront aberrations from upstream optics or transmission windows caused by photon heat loads, vibrations, or mechanical clamping distortions.

At modern synchrotron facilities, increasing numbers of active optics, including piezo bimorph mirrors and mechanically bendable mirrors, are employed to correct wavefront errors or provide variable sizes of x-ray beams [2

2. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, Nat. Phys. 6, 122 (2009). [CrossRef]

4

4. R. Signorato, O. Hignette, and J. Goulon, J. Synchrotron Rad. 5, 797 (1998).

]. For example, the diameter of a hard x-ray focal spot has been reduced to <10nm using a deformable mirror to restore the wavefront shape with phase retrieval methods [2

2. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, Nat. Phys. 6, 122 (2009). [CrossRef]

]. At-wavelength metrology techniques are leading candidates to achieve distortion-free and coherence-preserving x-ray optics, and push the resolution and sensitivity to the nanometer scale [3

3. K. Sawhney, S. Alcock, J. Sutter, S. Berujon, H. Wang, and R. Signorato, J. Phys. Conf. Ser. 425, 052026 (2013). [CrossRef]

,5

5. H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, Opt. Express 19, 16550 (2011). [CrossRef]

,6

6. J. Sutter, S. Alcock, and K. Sawhney, J. Synchrotron Rad. 19, 960 (2012).

].

Over the last two decades, various at-wavelength metrology methods have been developed and extensively used to characterize and optimize x-ray mirrors. The pencil beam technique is frequently used for optimizing active optics, due to its simplicity [7

7. O. Hignette, A. K. Freund, E. Chinchio, P. Z. Takacs, and T. W. Tonnessen, Proc. SPIE 3152, 188 (1997).

], but its angular resolution suffers for mirrors with short focal lengths [6

6. J. Sutter, S. Alcock, and K. Sawhney, J. Synchrotron Rad. 19, 960 (2012).

]. Hartmann x-ray wavefront sensors have been demonstrated for the characterization of active x-ray optics, but the ultimate resolution is inherently limited by the pitch of the microlens array [8

8. M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, Nucl. Instrum. Meth. A 616, 162 (2010).

]. Although Fresnel propagation iterative algorithms and ptychography have proven useful for the alignment and characterization of x-ray mirrors, several hundred images need to be collected for a successful reconstruction [9

9. H. Yumoto, H. Mimura, S. Matsuyama, S. Handa, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, Rev. Sci. Instrum. 77, 063712 (2006). [CrossRef]

,10

10. 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, Opt. Express 18, 23420 (2010). [CrossRef]

]. Recently, the near-field speckle-based technique has been used to perform the on-line characterization of hard x-ray reflective optics with high angular accuracy and a simple setup. However, the membrane has to be scanned to acquire dozens of images and accurately recover the mirror shapes [11

11. S. Berujon, H. Wang, S. Alcock, and K. Sawhney, Opt. Express 22, 6438 (2014). [CrossRef]

,12

12. S. Berujon, H. Wang, and K. Sawhney, Phys. Rev. A 86, 063813 (2012). [CrossRef]

]. In addition, an interferometer using a single grating was employed to measure the wavefront distortion from a deformable mirror [13

13. S. Matsuyama, H. Yokoyama, R. Fukui, Y. Kohmura, K. Tamasaku, M. Yabashi, W. Yashiro, A. Momose, T. Ishikawa, and K. Yamauchi, Opt. Express 20, 24977 (2012). [CrossRef]

]. However, it can be difficult using a single grating with smaller beam divergence from the focal plane of the mirror, due to the trade-off between sufficient illumination area on the grating and the detectable interferograms.

In contrast, a more sophisticated x-ray grating interferometer (GI) consists of a phase grating and an absorption grating, which can be freely placed downstream and out of the focal plane. Moiré fringes, generated from superimposition of the phase grating self-image and the absorption grating and set to the fractional Talbot distance, can be clearly resolved by x-ray detectors. This GI has been widely used for x-ray phase contrast imaging [14

14. C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, Appl. Phys. Lett. 81, 3287 (2002). [CrossRef]

18

18. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, Phys. Rev. Lett. 105, 248102 (2010). [CrossRef]

], as well as for online metrology [5

5. H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, Opt. Express 19, 16550 (2011). [CrossRef]

,15

15. T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, Appl. Phys. Lett. 86, 054101 (2005). [CrossRef]

]. In this Letter, we use the GI to optimize the shape of a bimorph mirror by minimizing slope errors using Moiré fringe analysis. Taking advantage of the rotating shearing interferometer technique, the optimal focal size was quickly achieved in only two iterations.

Experiments were performed with Diamond Light Source’s I13 beamline (Coherence branchline), using x rays from an undulator source on the 3 GeV storage ring. The vertical source size was 18.8 μm FWHM, and an x-ray energy of 8 keV was selected using a Si(111) pseudo-channel-cut crystal monochromator. As shown in Fig. 1, the bimorph mirror and the GI were each mounted on independent motorized towers.

Fig. 1. Optical layout of the in situ optimization of a bimorph mirror with 8 piezo actuators A1, A2,,A8 using a GI. The phase grating is noted G1 and the absorption grating G2.

The novel mirror under investigation was produced by combining bimorph technology and Elastic Emission Machining “super-polishing” [1

1. K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, Rev. Sci. Instrum. 73, 4028 (2002). [CrossRef]

,3

3. K. Sawhney, S. Alcock, J. Sutter, S. Berujon, H. Wang, and R. Signorato, J. Phys. Conf. Ser. 425, 052026 (2013). [CrossRef]

,4

4. R. Signorato, O. Hignette, and J. Goulon, J. Synchrotron Rad. 5, 797 (1998).

]. The 8 piezo electrodes enable the substrate to be bent to a large range of cylinders or ellipses, with a residual figure error of <1nm (rms) [3

3. K. Sawhney, S. Alcock, J. Sutter, S. Berujon, H. Wang, and R. Signorato, J. Phys. Conf. Ser. 425, 052026 (2013). [CrossRef]

]. The mirror’s figure was pre-polished as a tangential ellipse, whose foci are located at distances p=41.5m (source) and q=0.4m (image) from the bimorph mirror when no voltages are applied to its electrodes. The grazing angle of incidence at the center of the mirror is θ=3mrad. When the mirror was mounted on the coherence branchline of the I13 beamline, the distance L1 between the source and the mirror was 217m, and the focal distance L2 was set to 380 mm. The radiation source is, therefore, not located at the focus of the mirror’s pre-polished elliptical surface; however, the mirror’s shape can be corrected by the application of suitable voltages to its electrodes.

The GI consists of a first phase grating acting as a beam splitter and a second absorption grating acting as a transmission mask. Although transmission is reduced by adding the absorption grating, the combined effects of the two gratings on an x-ray beam produce Moiré fringes that can be resolved using a 2D detector with pixels much larger than the grating pitch. The phase grating G1, with a pitch of d1=3.99μm, was mounted on a rotation stage. The absorption grating G2, with a pitch of d2=2.00μm, was fabricated by depositing electroplated gold onto a silicon grating [19

19. C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, Microelectron. Eng. 84, 1172 (2007). [CrossRef]

]. The silicon substrates for both gratings were etched down to a 50 μm thickness to minimize x ray absorption. The grating periodicity can be manufactured with nanometer accuracy; hence, distortion of the x-ray wavefront from grating defects was minimal and, thus, neglected. The grating G1 was placed at a distance L3=400mm downstream of the mirror focal plane, and the inter-grating distance G1 to G2 was set to L4=123mm, corresponding to the 7th fractional Talbot order.

Interferograms were recorded with a high-resolution imaging x-ray microscope, consisting of a microscope objective, a Ce-doped YAG scintillator producing a visible light image of the x-ray beam intensity, a mirror reflecting at 90°, and a PCO 4000 CCD camera, onto which the 10× objective focuses light. The pixel size of the CCD camera is 6.5μm×6.5μm.

Before measuring the absolute wavefront slope of the reflected beam, a calibration scan was performed by rotating the phase grating angle β along the z axis. Such a scan permits recovery of the angle of the absorption grating α relative to the detector’s horizontal x axis. In Fig 2, the inset interferogram image corresponds to one of the acquisitions taken at various values of β. The Moiré fringe periods dx and dy along x and y for each image, were extracted using Fourier analysis [5

5. H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, Opt. Express 19, 16550 (2011). [CrossRef]

]. The values γx and γy correspond to the ratio between the absorption grating pitch d2 and the components dx and dy, respectively:
γx=d2dx,γy=d2dy.
(1)
The inclination angle and horizontal period of the Moiré fringes varies with angle β. The angle of the absorption grating α was found to be 2.5° by fitting γy as a function of γx [5

5. H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, Opt. Express 19, 16550 (2011). [CrossRef]

]. Once α is calculated, one can accurately retrieve the wavefront slope S along the vertical direction y (which corresponds to the direction along the mirror’s length) from a single interferogram. It can be derived that the wavefront slope error ΔS is approximately twice the mirror’s tangential slope error Δδ for a focusing mirror [9

9. H. Yumoto, H. Mimura, S. Matsuyama, S. Handa, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, Rev. Sci. Instrum. 77, 063712 (2006). [CrossRef]

,15

15. T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, Appl. Phys. Lett. 86, 054101 (2005). [CrossRef]

]. Since the relationship between the mirror slope and the wavefront gradient in a given plane is linear, analysis and optimization of the wavefront slope error can be used to directly minimize the mirror’s slope error.

Fig. 2. Experimental data and fitted model for γy as a function of γx. Moiré fringe periods dx and dy along the x and y directions are also shown in the inset image.

To determine the optimum voltages to minimize slope errors, an initial interferogram was collected, with all piezo voltages set to 0 V. For each subsequent image, the voltage on electrode j was incremented by ν (200 V), until, by the 9th image, all piezo voltages were at 200 V. The wavefront slopes Sij were calculated using the method described in reference [5

5. H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, Opt. Express 19, 16550 (2011). [CrossRef]

] from the measurements at n=691 points, corresponding to an illuminated mirror length of 107mm, and stored in an n×j matrix. As shown in Fig. 3, the response of the jth electrode is calculated by subtracting the slope extracted from the jth and (j+1)th image. Mirror surface height changes are calculated by numerically integrating the slope differences and scaling these to the surface length. It can be seen in Fig. 3 that the height change induced by each piezo actuator is below 1nm/V. An n×8 interaction matrix M, which defines the response of each electrode per unit voltage change, is constructed from:
Mij=(Si,j+1Si,j)/v,
(2)
where Sij is the x-ray beam wavefront slope in the ith pixel of the mirror in the jth scan.

Fig. 3. Piezo response function of the bimorph mirror: (Top) the slope and (bottom) height changes induced by applying a fixed voltage to each piezo actuator from A1 to A8, in sequence. Here, line 1–0 means the slope response of the 1st electrode by subtracting the 0th image (no voltages applied) from the 1st image (only apply voltage for 1st electrode). The height change caused by each piezo actuator was calculated by numerical integration of the slope change.

Once the wavefront slope Si0 is derived, the slope error ΔSi0 can be calculated by subtracting the best fit linear term. As plotted in Fig. 4 (black, dashed line), the initial wavefront slope error ΔSi0 was 3.11μrad (rms).

Fig. 4. Wavefront slope error at zero volts (without correction), and the 1st and 2nd iteration corrections.

The relationship between the mirror’s deformation and the applied piezo actuator’s voltage is sufficiently linear to permit the interaction matrix method to be applied without modification. Hence, the required voltage Vj can be calculated by multiplying the Moore–Penrose pseudo-inverse matrix of M+ with the wavefront slope error ΔSi0 [4

4. R. Signorato, O. Hignette, and J. Goulon, J. Synchrotron Rad. 5, 797 (1998).

,20

20. R. K. Tyson, Principles of Adaptive Optics, 3rd ed. (CRC Press, 2010).

]:
[V1V2V8]=[M11M12M18M21M22M28Mn1Mn2Mn8]1[ΔS1ΔS2ΔSn].
(3)
Note that the Moore–Penrose pseudo-inverse of a general matrix M (which need not be square) is unique and can be calculated by standard linear algebra routines. The matrix M inversion was performed by singular value decomposition after using the transpose matrix MT:
M+=(MTM)1MT.
(4)
After applying correction voltages, Fig. 4(a) (red, dotted line) shows the dramatic improvement to the wavefront slope error, which was reduced to 0.42μrad (rms). The wavefront slope error was further improved to 0.35μrad (rms) after a 2nd iteration (blue, solid line), corresponding to a mirror slope error of 0.17μrad (rms). Figure 4(b) highlights the wavefront slope errors for the 1st and 2nd iterations. Further iterations did not improve the slope error, as the 8 piezos cannot correct higher spatial frequencies. Data collection time for each interferogram image was only 0.5 s. Therefore, once the piezo response functions are derived, the minimal slope error can be achieved within minutes (limited only by the time taken to apply the voltages).

To validate the optimized slope error, the focused beam size was measured by performing “knife edge” scans with a 200 μm diameter Au wire, as shown in Fig. 5. By setting the piezos of the bimorph mirror to the optimal voltages, determined using the GI, the measured FWHM size of the focused beam was reduced to 0.34 μm. The theoretical beam size at the focal position is 0.32 μm, accounting for the source size contribution (0.03 μm) and the slope error contribution (0.31 μm). This demonstrates that the measured beam size is consistent with the theoretical calculation, assuming a mirror slope error of 0.17μrad (rms).

Fig. 5. Transmission signal from a gold wire scan (blue squares) in the focal plane of the bimorph mirror. A Gaussian fit (red line) of the derivative of the raw data (black dots) indicates a FWHM of 0.34 μm.

We have shown that a GI can be used for fast optimization of an active optic, bimorph mirror. The GI can be freely located out of the focal plane, and the requirement of divergence matching of the two gratings pitches is relaxed by using Moiré fringe analysis [5

5. H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, Opt. Express 19, 16550 (2011). [CrossRef]

]. For this experimental setup, the angular sensitivity of the GI is less than 50 nrad, and can be further improved by choosing a larger inter-grating distance L4 [15

15. T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, Appl. Phys. Lett. 86, 054101 (2005). [CrossRef]

]. Moreover, the technique can easily be extended to focus the beam in two dimensions (2D) by rotating the 1D grating 90°, or using a 2D grating [16

16. H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, Opt. Lett. 38, 827 (2013). [CrossRef]

18

18. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, Phys. Rev. Lett. 105, 248102 (2010). [CrossRef]

]. This technique has the potential to create a user-defined level of defocus or beam-shaping for active optics. In addition, it can also be used to speed up mirror alignment by combining with the conventional knife edge scan technique. As only a single shot is required, the technique will be highly advantageous for x-ray free electron laser mirror commissioning and dynamical wavefront sensing.

This work was carried out with the support of Diamond Light Source Ltd., U.K. The authors are also grateful to Simon Rutishauser and Christian David from PSI for fabricating the gratings used in this experiment. The authors would also like to thank Riccardo Signorato from Cinel for his contribution to the design and development of the super-polished bimorph mirror, and to beamline technician Andrew Malandain for his technical assistance.

References

1.

K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, Rev. Sci. Instrum. 73, 4028 (2002). [CrossRef]

2.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, Nat. Phys. 6, 122 (2009). [CrossRef]

3.

K. Sawhney, S. Alcock, J. Sutter, S. Berujon, H. Wang, and R. Signorato, J. Phys. Conf. Ser. 425, 052026 (2013). [CrossRef]

4.

R. Signorato, O. Hignette, and J. Goulon, J. Synchrotron Rad. 5, 797 (1998).

5.

H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, Opt. Express 19, 16550 (2011). [CrossRef]

6.

J. Sutter, S. Alcock, and K. Sawhney, J. Synchrotron Rad. 19, 960 (2012).

7.

O. Hignette, A. K. Freund, E. Chinchio, P. Z. Takacs, and T. W. Tonnessen, Proc. SPIE 3152, 188 (1997).

8.

M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, Nucl. Instrum. Meth. A 616, 162 (2010).

9.

H. Yumoto, H. Mimura, S. Matsuyama, S. Handa, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, Rev. Sci. Instrum. 77, 063712 (2006). [CrossRef]

10.

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, Opt. Express 18, 23420 (2010). [CrossRef]

11.

S. Berujon, H. Wang, S. Alcock, and K. Sawhney, Opt. Express 22, 6438 (2014). [CrossRef]

12.

S. Berujon, H. Wang, and K. Sawhney, Phys. Rev. A 86, 063813 (2012). [CrossRef]

13.

S. Matsuyama, H. Yokoyama, R. Fukui, Y. Kohmura, K. Tamasaku, M. Yabashi, W. Yashiro, A. Momose, T. Ishikawa, and K. Yamauchi, Opt. Express 20, 24977 (2012). [CrossRef]

14.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, Appl. Phys. Lett. 81, 3287 (2002). [CrossRef]

15.

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, Appl. Phys. Lett. 86, 054101 (2005). [CrossRef]

16.

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, Opt. Lett. 38, 827 (2013). [CrossRef]

17.

S. Berujon, H. Wang, I. Pape, K. Sawhney, S. Rutishauser, and C. David, Opt. Lett. 37, 1622 (2012). [CrossRef]

18.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, Phys. Rev. Lett. 105, 248102 (2010). [CrossRef]

19.

C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, Microelectron. Eng. 84, 1172 (2007). [CrossRef]

20.

R. K. Tyson, Principles of Adaptive Optics, 3rd ed. (CRC Press, 2010).

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(120.3940) Instrumentation, measurement, and metrology : Metrology
(120.4640) Instrumentation, measurement, and metrology : Optical instruments
(340.7450) X-ray optics : X-ray interferometry

ToC Category:
Imaging Systems

History
Original Manuscript: December 18, 2013
Revised Manuscript: March 17, 2014
Manuscript Accepted: March 18, 2014
Published: April 15, 2014

Citation
Hongchang Wang, Kawal Sawhney, Sebastien Berujon, John Sutter, Simon G. Alcock, Ulrich Wagner, and Christoph Rau, "Fast optimization of a bimorph mirror using x-ray grating interferometry," Opt. Lett. 39, 2518-2521 (2014)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-39-8-2518


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References

  1. K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, Rev. Sci. Instrum. 73, 4028 (2002). [CrossRef]
  2. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, Nat. Phys. 6, 122 (2009). [CrossRef]
  3. K. Sawhney, S. Alcock, J. Sutter, S. Berujon, H. Wang, and R. Signorato, J. Phys. Conf. Ser. 425, 052026 (2013). [CrossRef]
  4. R. Signorato, O. Hignette, and J. Goulon, J. Synchrotron Rad. 5, 797 (1998).
  5. H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, Opt. Express 19, 16550 (2011). [CrossRef]
  6. J. Sutter, S. Alcock, and K. Sawhney, J. Synchrotron Rad. 19, 960 (2012).
  7. O. Hignette, A. K. Freund, E. Chinchio, P. Z. Takacs, and T. W. Tonnessen, Proc. SPIE 3152, 188 (1997).
  8. M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, Nucl. Instrum. Meth. A 616, 162 (2010).
  9. H. Yumoto, H. Mimura, S. Matsuyama, S. Handa, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, Rev. Sci. Instrum. 77, 063712 (2006). [CrossRef]
  10. 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, Opt. Express 18, 23420 (2010). [CrossRef]
  11. S. Berujon, H. Wang, S. Alcock, and K. Sawhney, Opt. Express 22, 6438 (2014). [CrossRef]
  12. S. Berujon, H. Wang, and K. Sawhney, Phys. Rev. A 86, 063813 (2012). [CrossRef]
  13. S. Matsuyama, H. Yokoyama, R. Fukui, Y. Kohmura, K. Tamasaku, M. Yabashi, W. Yashiro, A. Momose, T. Ishikawa, and K. Yamauchi, Opt. Express 20, 24977 (2012). [CrossRef]
  14. C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, Appl. Phys. Lett. 81, 3287 (2002). [CrossRef]
  15. T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, Appl. Phys. Lett. 86, 054101 (2005). [CrossRef]
  16. H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, Opt. Lett. 38, 827 (2013). [CrossRef]
  17. S. Berujon, H. Wang, I. Pape, K. Sawhney, S. Rutishauser, and C. David, Opt. Lett. 37, 1622 (2012). [CrossRef]
  18. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, Phys. Rev. Lett. 105, 248102 (2010). [CrossRef]
  19. C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, Microelectron. Eng. 84, 1172 (2007). [CrossRef]
  20. R. K. Tyson, Principles of Adaptive Optics, 3rd ed. (CRC Press, 2010).

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