## X-ray wavefront characterization using a rotating shearing interferometer technique |

Optics Express, Vol. 19, Issue 17, pp. 16550-16559 (2011)

http://dx.doi.org/10.1364/OE.19.016550

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### Abstract

A fast and accurate method to characterize the X-ray wavefront by rotating one of the two gratings of an X-ray shearing interferometer is described and investigated step by step. Such a shearing interferometer consists of a phase grating mounted on a rotation stage, and an absorption grating used as a transmission mask. The mathematical relations for X-ray Moiré fringe analysis when using this device are derived and discussed in the context of the previous literature assumptions. X-ray beam wavefronts without and after X-ray reflective optical elements have been characterized at beamline B16 at Diamond Light Source (DLS) using the presented X-ray rotating shearing interferometer (RSI) technique. It has been demonstrated that this improved method allows accurate calculation of the wavefront radius of curvature and the wavefront distortion, even when one has no previous information on the grating projection pattern period, magnification ratio and the initial grating orientation. As the RSI technique does not require any *a priori* knowledge of the beam features, it is suitable for routine characterization of wavefronts of a wide range of radii of curvature.

© 2011 OSA

## 1. Introduction

1. E. Ziegler, L. Peverini, I. V. Kozhevnikov, T. Weitkamp, and C. David, “On-line mirror surfacing monitored by X-ray shearing interferometry and X-ray scattering,” AIP Conf. Proc. **879**, 778–781 (2007). [CrossRef]

1. E. Ziegler, L. Peverini, I. V. Kozhevnikov, T. Weitkamp, and C. David, “On-line mirror surfacing monitored by X-ray shearing interferometry and X-ray scattering,” AIP Conf. Proc. **879**, 778–781 (2007). [CrossRef]

6. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **13**(16), 6296–6304 (2005). [CrossRef] [PubMed]

7. F. Pfeiffer, O. Bunk, C. Schulze-Briese, A. Diaz, T. Weitkamp, C. David, J. F. van der Veen, I. Vartanyants, and I. K. Robinson, “Shearing interferometer for quantifying the coherence of hard x-ray beams,” Phys. Rev. Lett. **94**(16), 164801 (2005). [CrossRef] [PubMed]

3. T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. **86**(5), 054101–054103 (2005). [CrossRef]

6. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **13**(16), 6296–6304 (2005). [CrossRef] [PubMed]

8. T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, M. Stampanoni, E. Ziegler, and C. David, “Moire interferometry formulas for hard x-ray wavefront sensing,” Proc. SPIE **5533**, 140–144 (2004). [CrossRef]

9. A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. **17**(3), 299–307 (2010). [CrossRef] [PubMed]

## 2. Theoretical considerations

*d*

_{0}and

*d*

_{2}, and the period of the interference fringes created by the phase grating, at the position of the absorption grating, is noted

*d*

_{1}. One can define the ratio

*η*of the two grating pitches by:Working with a collimated beam, the Talbot distance [10

10. J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synchrotron Radiat. **11**(6), 476–482 (2004). [CrossRef] [PubMed]

*m*an odd integer which corresponds to the order of the fractional Talbot distance. For a diverging beam with a radius of curvature

*R*, these distances rescale to [11]:The interference pattern produced by the phase grating at the distance

*L*

_{m}* where the second grating is located, is magnified from the expected self-referenced Talbot periodic grating pattern, by a factor:And so the relation linking

*d*

_{1}to

*d*

_{2}can be written aswhere

*κ*is the revised demagnification factor defined as:So far, the mathematical relations used in the literature for hard X-ray wavefront analysis of Moiré fringes were derived assuming that the small inclination angle

*β*of the phase grating from the detector axis is equal to the inclination angle

*α*of the absorption grating [8

8. T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, M. Stampanoni, E. Ziegler, and C. David, “Moire interferometry formulas for hard x-ray wavefront sensing,” Proc. SPIE **5533**, 140–144 (2004). [CrossRef]

*θ*with the vertical detector axis. In the illustration, the ratio

*η*of the two grating pitch is set to 1.3, and the two gratings are inclined to

*α*= 17.2° and

*β*= −5.7°, respectively. The periods of the recorded Moiré fringe along the

**x**and

**y**axis directions of the detector are noted

*d*

_{x}and

*d*

_{y}.

*I*

_{1}(x, y), and the intensity transmitted through the absorption grating

*I*

_{2}(x, y) can be written as:where B and A respectively represent the quantities in square brackets in the above equation. Following the methodology of reference [8

8. T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, M. Stampanoni, E. Ziegler, and C. David, “Moire interferometry formulas for hard x-ray wavefront sensing,” Proc. SPIE **5533**, 140–144 (2004). [CrossRef]

*I*

_{1}(

*x*,

*y*) and

*I*

_{2}(

*x*,

*y*).

*I*

_{m}(

*x*,

*y*):Substituting the expression of

*d*

_{1}from Eq. (5), Eq. (9) can be written as:As described in reference [8

**5533**, 140–144 (2004). [CrossRef]

*θ*of the Moiré fringes (see Fig. 1) can be written as:and the components

*d*

_{x}and

*d*

_{y}of the Moiré fringe period along

**x**and

**y**are:whereFor each horizontal line

**j**in one image, the intensity of the fundamental component of the Moiré fringes along the horizontal

**x**direction can be written as:Using the Fourier method [3

3. T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. **86**(5), 054101–054103 (2005). [CrossRef]

*ϕ*as well as the period

*d*

_{x}can be extracted for each row

**j**. The phase along the same fringe is constant

*c*, that isTherefore the inclination

*θ*of the fringes can be derived as:Once

*θ*and

*d*

_{x}are derived, the parameters

*γ*

_{x}and

*γ*

_{y}can be calculated using Eqs. (11) and (12). Although the parameters

*α*and

*κ*can be determined from Eq. (13) using only two different images with two different angles β, a much better way to proceed is to extract both at the same time by fitting

*γ*

_{y}as a function of

*γ*

_{x}when varying the phase grating angle

*β*. One can deduce the relation between

*γ*

_{x}and

*γ*

_{y}from Eq. (13):Here, the parameters

*γ*,

_{x}*γ*and

_{y}*κ*are average values over one image. Once the angle

*α*has been calculated, Eq. (16) can be rewritten as:The revised demagnification ratio

*κ*(y) is determined for each line

**j**. Once

*η*and

*κ*are known, the radius of curvature can be determined from the rearranged form of Eq. (6):Finally, the wavefront slope can be recovered by integration of the inverse of the radius of curvature R

^{−1}along the vertical direction

**y**[8

**5533**, 140–144 (2004). [CrossRef]

*γ*

_{x}and

*γ*

_{y}depend on the phase grating angle

*β*. The numerical extraction of

*β*is however not required for the wavefront slope and radius of curvature reconstruction process.

## 3. Experiments and results

12. 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]

_{1}was made from a silicon wafer with a designed pitch of

*d*

_{0}= 4.0 µm. The absorption grating G

_{2}with a design pitch

*d*

_{2}= 2.0 µm was made by covering the lines of a silicon grating with electroplated gold [13

13. C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. **84**(5-8), 1172–1177 (2007). [CrossRef]

_{1}was 46.5 m and the angle of this grating was changed using a rotation motor. The phase grating was designed to produce a phase shift of π

*rad*at the X-ray energy of 14.8 keV, thus allowing maximum fringe visibility. The exact energy value was selected by a silicon double-crystal monochromator (DCM). An X-ray 2D CCD detector with indirect illumination and with an effective pixel size of 6.4 μm was used to record the Moiré fringe pattern. The wavefronts before and after insertion into the beam of a focusing curved test mirror were characterized using the presented RSI method.

*L**

_{m}= 0.075m, corresponding to the 3rd order Talbot distance, was employed, which gave an average visibility [9

9. A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. **17**(3), 299–307 (2010). [CrossRef] [PubMed]

*β*. As expected, the inclination angle

*θ*and the horizontal period

*d*

_{x}of the Moiré fringes vary with this angle. Interferograms were taken at various odd Talbot orders (distance between gratings), and up to 25th Talbot orders were used. The absorption grating tilt angle

*α*was kept the same for all data sets. For each inter-grating distance

*L**

_{m}, the average

*γ*

_{x}and

*γ*

_{y}values were extracted and fitted using Eq. (16).

*α*of the absorption grating and the revised demagnification ratio

*κ*. The good fitting of the experimentally extracted points

*γ*

_{y}as a function of

*γ*

_{x}confirm the validity of the model described by Eq. (16). Only some values close to

*γ*

_{x}= 0 do not fit well. The reason for this is that for

*γ*

_{x}≈0 the Moiré fringes are almost horizontal, rendering the phase extraction complex and inaccurate.

*γ*

_{x}≈0 corresponds to the two gratings being parallel to each other i.e. β = α. RSI therefore provides a quantitative method to more accurately find the required zero angle of

*β*and α, for use in the phase stepping method.

*α*and

*κ*values for the 3rd, 9th, 15th and 21st Talbot orders are tabulated in Table 1 . The approximations

*d*

_{0}= 4.0 μm and

*d*

_{2}= 2.0 μm (

*η*=

*d*

_{0}/

*d*

_{2}= 2) only give approximate radii of curvatures

*R*, as derived from Eq. (18). The

^{’}*R*values are also tabulated in Table 1 and are seen to depend on the Talbot order used. For instance, a difference of 4.6 m is observed between the values derived from measurements at the 3rd and 21st orders. Thus, there is a need to know the period ratio η very accurately. This has been done by fitting the model for

^{’}*κ*described by Eq. (6), with the

*κ*values calculated using Eq. (16), as shown in Fig. 5 . A value of

*η*= 1.99975, with an accuracy of 10

^{−5}, has been extracted from the fit. It is therefore possible to compare the periods of the two gratings with sub-nanometer accuracy with the RSI technique. The average radii of curvatures

*R*calculated for each image using this new value of

*η*are given in Table 1. The difference in the radii of curvatures between 3rd and 21st orders is only 0.6m now, which comes about from the measurement error (~1mm) of inter-grating distance

*L**

_{m}. For instance, if the inter-grating distance is changed by 1 mm, then both 3rd and 21st orders return a value of 49.7m for the radius of curvature of the wavefront.

*β*is scanned. If, however, one had assumed the absorption grating angle

*α*to be equal to zero, the radius of curvature

*R*would have varied depending on the

^{’}*β*angle. For instance, for the 3rd Talbot order,

*R*would have changed from 39.2m to 44.1m, as

^{’}*β*changed from 0.025 rad to 0.019 rad (images a and b in Fig. 3), thus demonstrating the sensitivity of the radius of curvature calculation to the correct knowledge of the α angle value.

*κ*(Fig. 5) for the radius of curvature is 49.8 m, which is 5% higher compared to the expected value of 47.5 m. This discrepancy may be due to the heat bump caused by the non-uniform X-ray illumination of the water-cooled DCM. Moreover, we will like to emphasize that these detailed measurements at different Talbot distances, which might seem time consuming, have only to be performed once, in order to determine the exact period ratio

*η*. This value can be then directly used for all subsequent wavefront measurements.

*η*and

*α*are accurately known, the local radii of curvature

*R*(y) can be recovered using Eqs. (17) and 18. As shown in Fig. 6 (1), the calculated radii of curvature of the wavefronts, extracted from the images (a-d) of Fig. 3, are consistent. The mean value of

*R*is 51.4m for image (a) and 51.1m for image (b). The plots show that the wavefronts are not spherical but their curvatures vary with a standard deviation of 9.2 m and 7.4 m respectively. Next, the wavefront slope

*S*(y) was calculated by integration of

*R*

^{−1}along

**y**, as per Eq. (19). The slope error Δ

*S*(y), defined as the difference between

*S*(y) and its best linear fit, is plotted as a function of the vertical position in Fig. 6 (2). The standard deviations of the slope error are 0.18 μrad for image (a) and 0.20 μrad for image (b), and hence are in good agreement with each other.

*γ*

_{x}and

*γ*

_{y}were extracted from different images taken during a rotation scan of the phase grating and the same procedure as described previously was applied. This time the demagnification ratio

*κ*was determined to be 0.9723 and the wavefront radius of curvature to be 3.3 m. The wavefront from the focusing mirror has a strong curvature, and its precise retrieval by the RSI therefore shows that the wavefronts can be characterized by RSI even when the pitches of the two gratings are far from satisfying the criteria of divergence matching.

## 7. Conclusions

*α*) is known (derived from a simple rotation scan). The RSI technique allows accurate derivation of the projected fringes period, without any previous knowledge on the beam characteristics. Unlike the phase stepping method, the RSI technique with the 1D interferometer provides averaged information in one direction, it however relaxes the requirement of divergence matching of the two gratings. The RSI method has been demonstrated to be routinely usable for X-ray wavefront characterization and at-wavelength metrology for beams with various radii of curvatures. Recently, genuine 2D gratings have been successfully fabricated and used for two-dimensional Talbot interferometry [14

14. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. **105**(24), 248102 (2010). [CrossRef] [PubMed]

15. H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express **19**(4), 3339–3346 (2011). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | E. Ziegler, L. Peverini, I. V. Kozhevnikov, T. Weitkamp, and C. David, “On-line mirror surfacing monitored by X-ray shearing interferometry and X-ray scattering,” AIP Conf. Proc. |

2. | C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. |

3. | T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. |

4. | A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. |

5. | F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. |

6. | T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express |

7. | F. Pfeiffer, O. Bunk, C. Schulze-Briese, A. Diaz, T. Weitkamp, C. David, J. F. van der Veen, I. Vartanyants, and I. K. Robinson, “Shearing interferometer for quantifying the coherence of hard x-ray beams,” Phys. Rev. Lett. |

8. | T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, M. Stampanoni, E. Ziegler, and C. David, “Moire interferometry formulas for hard x-ray wavefront sensing,” Proc. SPIE |

9. | A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. |

10. | J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synchrotron Radiat. |

11. | M. Born and E. Wolf, |

12. | 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. |

13. | C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. |

14. | I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. |

15. | H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express |

**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:**

X-ray Optics

**History**

Original Manuscript: June 17, 2011

Revised Manuscript: July 25, 2011

Manuscript Accepted: August 10, 2011

Published: August 12, 2011

**Citation**

Hongchang Wang, Kawal Sawhney, Sébastien Berujon, Eric Ziegler, Simon Rutishauser, and Christian David, "X-ray wavefront characterization using a rotating shearing interferometer technique," Opt. Express **19**, 16550-16559 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16550

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### References

- E. Ziegler, L. Peverini, I. V. Kozhevnikov, T. Weitkamp, and C. David, “On-line mirror surfacing monitored by X-ray shearing interferometry and X-ray scattering,” AIP Conf. Proc. 879, 778–781 (2007). [CrossRef]
- C. David, B. Nöhammer, 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]
- T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86(5), 054101–054103 (2005). [CrossRef]
- A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]
- 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]
- T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef] [PubMed]
- F. Pfeiffer, O. Bunk, C. Schulze-Briese, A. Diaz, T. Weitkamp, C. David, J. F. van der Veen, I. Vartanyants, and I. K. Robinson, “Shearing interferometer for quantifying the coherence of hard x-ray beams,” Phys. Rev. Lett. 94(16), 164801 (2005). [CrossRef] [PubMed]
- T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, M. Stampanoni, E. Ziegler, and C. David, “Moire interferometry formulas for hard x-ray wavefront sensing,” Proc. SPIE 5533, 140–144 (2004). [CrossRef]
- A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. 17(3), 299–307 (2010). [CrossRef] [PubMed]
- J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synchrotron Radiat. 11(6), 476–482 (2004). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1993).
- 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]
- C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. 84(5-8), 1172–1177 (2007). [CrossRef]
- I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105(24), 248102 (2010). [CrossRef] [PubMed]
- H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express 19(4), 3339–3346 (2011). [CrossRef] [PubMed]

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