X-ray wavefront characterization using a rotating shearing interferometer technique

Hongchang Wang; Kawal Sawhney; Sébastien Berujon; Eric Ziegler; Simon Rutishauser; Christian David

doi:10.1364/OE.19.016550

1. Introduction

The combination of at-wavelength (i.e. using x-rays) and in situ metrology methods is considered the best pathway for surpassing the present optics performance and to realize the true nano-focusing optical elements needed for future synchrotron experiments [1]. Nevertheless, the need to push the resolution and sensitivity to nanometer scale still requires improvements that are the object of significant attention from the X-ray community. Of the many methods that exist in visible optics for optical components metrology, only a few are applied to X-Rays. One such method is the grating based shearing interferometer (SI), nowadays widely used by the synchrotron community for X-Ray wavefront characterization, tomography and radiography [1–6]. Indeed, the X-ray shearing interferometer presents the advantages of compactness, robustness, and a low requirement on longitudinal coherence [7]. The instrument consists of a first phase grating as a beam splitter and a second absorption grating acting as a transmission mask. The combined effects of the two gratings on an X-ray beam produce Moiré fringes that can be resolved by a 2D detector, and the calculation of their distortion from the straight lines allows one to characterize wavefront aberrations in the order of the wavelength [3]. A common method used with the SI is the phase stepping method in which at least one period of a grating is scanned while the two gratings are set at the same angle with respect to the beam. To reduce the artifact effects when using the phase stepping method for radiography and tomography, it is important to consider the divergence of the beam and accordingly use gratings with perfectly matching periods [6]. However, for routine wavefront characterization, changing the gratings several times to select the adequate one is time consuming, and then, this becomes a limiting factor when working with different optics or different configurations on a beamline. Moreover, the careful calibration needed in the phase stepping method to set the gratings parallel to each other is a stringent requirement that can lead to non- negligible systematic errors. The Moiré fringes analysis method for wavefront characterization is based on the assumption that the two gratings are inclined symmetrically in the opposite direction [8], leading usually to unexpected variations in the wavefront curvature measurements [9]. We show that these limitations are overcome when applying the rotation method: from several images collected at different rotation angles, the relevant grating angles are accurately extracted, and then the X-ray wavefront radius of curvature and its distortions can be obtained from any of these images.

2. Theoretical considerations

The pitch of the phase grating and the absorption grating are respectively noted 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:

η = \frac{d_{0}}{d_{2}}

Working with a collimated beam, the Talbot distance [10] beyond the π-shifting phase grating is defined by

L_{m} = m \frac{d_{0}^{2}}{8 λ}

where λ is the wavelength, and 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]:

L_{m}^{*} = \frac{R}{R - L_{m}} L_{m}

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:

M = \frac{L_{m}^{*} + R}{R}

And so the relation linking d ₁ to d ₂ can be written as

d_{1} = \frac{η d_{2}}{2} M = \frac{d_{2}}{κ}

where κ is the revised demagnification factor defined as:

κ = \frac{2 R}{(L_{m}^{*} + R) η}

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]. As illustrated in Fig. 1 , the observed Moiré fringes on the detector make an angle 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.

Fig. 1 Moiré fringes formation. The Moiré fringes will have an inclination angle θ with respect to the vertical axis when the two gratings are tilted from the horizontal axis by angles β and α, respectively

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The fundamental harmonics of the interference pattern intensity created by the phase grating I ₁(x, y), and the intensity transmitted through the absorption grating I ₂(x, y) can be written as:

{\begin{cases} I_{1} (x, y) = \cos^{2} [\frac{π}{d_{1}} (y \cos β - x \sin β)] \equiv \cos^{2} B \\ I_{2} (x, y) = \cos^{2} [\frac{π}{d_{2}} (y \cos α - x \sin α)] \equiv \cos^{2} A \end{cases}

where B and A respectively represent the quantities in square brackets in the above equation. Following the methodology of reference [8], we calculate the intensity distribution on the detector which is equal to the product of I ₁(x, y) and I ₂(x, y).

\begin{array}{l} I (x, y) = I_{1} (x, y) \times I_{2} (x, y) = \cos^{2} B \times \cos^{2} A \\ = \frac{1}{8} [\cos (2 B + 2 A) + \underset{I_{m} (x, y)}{\underset{︸}{\cos (2 B - 2 A)}} + 4 \underset{I_{1} (x, y)}{\underset{︸}{\cos^{2} B}} + 4 \underset{I_{2} (x, y)}{\underset{︸}{\cos^{2} A}} - 2] \end{array}

The Moiré fringes information recorded on the detector is generated by the term I _m(x, y):

I_{m} (x, y) = \cos [\frac{2 π}{d_{1}} (y \cos β - x \sin β) - \frac{2 π}{d_{2}} (y \cos α - x \sin α)]

Substituting the expression of d ₁ from Eq. (5), Eq. (9) can be written as:

I_{m} (x, y) = \cos [\frac{2 π}{d_{2}} (\sin α - κ \sin β) x - \frac{2 π}{d_{2}} (\cos α - κ \cos β) y]

As described in reference [8], the inclination angle θ of the Moiré fringes (see Fig. 1) can be written as:

\tan θ = \frac{\cos α - κ \cos β}{\sin α - κ \sin β}

and the components d _x and d _y of the Moiré fringe period along x and y are:

{\begin{cases} d_{x} = \frac{d_{2}}{\sin α - κ \sin β} \equiv \frac{d_{2}}{γ_{x}} \\ d_{y} = \frac{d_{2}}{\cos α - κ \cos β} \equiv \frac{d_{2}}{γ_{y}} \end{cases}

where

{\begin{cases} γ_{x} = \sin α - κ \sin β \\ γ_{y} = \cos α - κ \cos β \end{cases}

For 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:

I_{j} (x) = a_{0} + b_{0} \cos (\frac{2 π}{d_{x}} x + φ)

Using the Fourier method [3], the phase 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 is

\frac{2 π}{d_{x}} x + φ = c

Therefore the inclination θ of the fringes can be derived as:

\tan θ = \frac{\partial x}{\partial y} = - \frac{d_{x}}{2 π} \frac{\partial φ}{\partial y}

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):

γ_{y} = \cos α - \sqrt{κ^{2} - {(γ_{x} - \sin α)}^{2}}

Here, the parameters γ_x, γ_y and κ are average values over one image. Once the angle α has been calculated, Eq. (16) can be rewritten as:

κ (y) = \sqrt{{(γ_{y} (y) - \cos α)}^{2} + {(γ_{x} (y) - \sin α)}^{2}}

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):

R (y) = \frac{L_{m}^{*} η κ (y)}{2 - η κ (y)}

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].

S_{y} = \int_{0}^{y} R {(y)}^{- 1} d y

One can see that both 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

The experiments were carried out at the Diamond Light Source beamline B16 [12] where X-rays are produced by a bending magnet on a 3 GeV storage ring. A schematic of the rotating shearing interferometer at B16 beamline is given in Fig. 2 . The phase grating G₁ was made from a silicon wafer with a designed pitch of d ₀ = 4.0 µm. The absorption grating G₂ with a design pitch d ₂ = 2.0 µm was made by covering the lines of a silicon grating with electroplated gold [13]. The rotating shearing interferometer was mounted on a versatile optics test bench, with each grating placed on a motorized tower capable of independent three axis translations. The distance between the two gratings was changed by moving the absorption grating stages. The distance from the source to the first grating G₁ 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.

Fig. 2 A schematic of the rotating shearing interferometer at B16 beamline.

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In the first instance, the direct beam wavefront was characterized by retracting the test mirror from the beam path. An inter-grating distance L*_m = 0.075m, corresponding to the 3rd order Talbot distance, was employed, which gave an average visibility [9] of 0.45 for the Moiré fringes. Figure 3 shows four interferograms recorded at different phase grating tilt angles 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).

Fig. 3 Interferograms recorded at four different angles: (a) β = 0.019 rad, (b) β = 0.025 rad, (c) β = 0.041 rad, (d) β = 0.048 rad.

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Figure 4 shows some data collected at the 3rd, 9th, 15th and 21st Talbot order distances. For each Talbot order, Eq. (16) was used to calculate the inclination angle 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.

Fig. 4 Experimental data and fitted model from Eq. (16) for γ _y as function of γ _x at the 3rd, 9th, 15th and 21st order Talbot distances. The points corresponding to the four images shown in Fig. 2 are circled.

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The calculated 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.

Table 1. Radius of Curvature of DCM direct beam at B16 with RSI

View Table

Fig. 5 Linear regression of the κ factors calculated at the different Talbot orders. The 3rd, 9th, 15th and 21st Talbot orders are circled.

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It can be seen from Table 1 that the α values calculated from different Talbot orders are non-zero but identical, as they should be, since the absorption grating is held fixed, and only 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.

The value extracted from the regression of 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.

Once 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 ΔS(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.

Fig. 6 Data extracted from the images presented in Fig. 3: (1) Radius of curvature as function of y, (2) Wavefront slope error obtained after subtraction of the linear fit. Different line types correspond to calculated data from different images (a-d) of Fig. 3.

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Figure 7 illustrates a second application of the RSI method, using the same pair of gratings as for the direct beam study, but now for measuring the beam wavefront reflected from a curved focusing mirror. The average values of 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.

Fig. 7 Experimental data and its fitting to Eq. (16): γ _y as function of γ _x at the 3rd Talbot order distance (L = 93 mm) for a wavefront reflected from a focusing mirror.

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7. Conclusions

The rotating shearing interferometer method has been established at DLS B16 beamline enabling different wavefronts to be accurately characterized. The experimental results are in good agreement with the physical expectations. Full wavefront characterization can be determined from a single image, once the transmission grating angle offset (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 [14,15]; allowing the Fourier analysis to be performed in two orthogonal planes simultaneously. The presented RSI method will also work with such 2D-interferometers.

Acknowledgments

This work was carried out with the support of the Diamond Light Source Ltd UK. We thank Timm Weitkamp from Synchrotron Soleil for useful discussions. We also gratefully acknowledge Andrew Malandain for his support during the setting up of the experiment. The research leading to these results has received partial funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under the grant agreement nº 226716.

References and links

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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]

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11. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1993).

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]

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]

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Order	*L_m (m)**	α (rad)	κ	R ^’ (m)	R (m)
21st	0.507	0.0351	0.9900	50.4	49.8
15th	0.363	0.0350	0.9929	50.7	49.8
9th	0.219	0.0350	0.9958	51.3	49.8
3rd	0.075	0.0350	0.9986	55.0	50.4
3rd Fig. 3(a)	0.075	0.0000	0.9981	39.2	36.7
3rd Fig. 3(b)	0.075	0.0000	0.9983	44.1	41.1
3rd	0.074	0.0350	0.9986	54.2	49.7
21st	0.506	0.0351	0.9900	50.3	49.7

X-ray wavefront characterization using a rotating shearing interferometer technique

Abstract

1. Introduction

2. Theoretical considerations

3. Experiments and results

7. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (20)

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