## Abstract

We report a novel method for in situ metrology of an X-ray bimorph mirror by using the speckle scanning technique. Both the focusing beam and the “tophat” defocussed beam have been generated by optimizing the bimorph mirror in a single iteration. Importantly, we have demonstrated that the angular sensitivity for measuring the slope error of an optical surface can reach accuracy in the range of two nanoradians. When compared with conventional ex-situ metrology techniques, the method enables a substantial increase of around two orders of magnitude in the angular sensitivity and opens the way to a previously inaccessible region of slope error measurement. Such a super precision metrology technique will be beneficial for both the manufacture of polished mirrors and the optimization of beam shaping.

© 2015 Optical Society of America

## 1. Introduction

Modern third-generation synchrotron radiation sources provide coherent and extremely bright X-ray radiation beams. Apart from beam focusing to a micrometer or nanometer beam size, the generation of a “tophat” beam is highly desired by many X-ray synchrotron users because it can significantly reduce the radiation damage on fragile samples [1]. Hence, active optics such as bimorph and mechanically bendable mirrors are widely used for beam shaping since they permit a wide choice of focal lengths and beam sizes [2–4]. Inevitably, the optimization of such mirrors is often complicated and time consuming. Although valuable information from ex-situ characterization of the mirrors can be provided by using traditional visible light techniques, including Fizeau interferometry and deflectometry [5], the ultimate measure of an X-ray optic’s performance is when it is installed on the beamline, subject to distortion from mechanical clamping and illumination with an intense X-ray beam. Hence, in situ metrology methods are considered the best pathway to overcome this limitation and surpass the present optics performance [6–9]. Over the last two decades, many in situ metrology methods [9–12], such as the pencil beam technique, Hartmann X-ray wavefront sensors, Fresnel propagation iterative algorithms and X-ray grating interferometer have been developed for aligning or optimizing the X-ray mirrors. Nevertheless, they inherently suffer from low spatial or angular sensitivity, slow optimization process or complicated setup.

In this letter, we present a novel method for in situ characterization and optimization of a bimorph mirror by using the speckle scanning technique. We have demonstrated that the unprecedented angular sensitivity of two nanoradians has been achieved. This is about two orders of magnitude better than conventional ex-situ metrology techniques [5]. We have shown that a bimorph mirror can be optimized for various desired beam shapes in just one iteration.

## 2. Principle

Our method presented here is developed from the speckle tracking technique, which has been successfully used for X-ray phase contrast imaging and metrology [13, 14]. The speckle tracking technique operates by either changing the relative distance between the phase object, such as biological filtering membranes or abrasive paper, and the detector along the beam direction or by recording the speckle pattern with and without the sample under test in the beam. The first derivative of the wavefront phase, the wavefront slope *φ*, can be extracted by tracking the displacement of the speckles between the two images.

However, the speckle tracking technique suffers from lower spatial resolution because the fine structure of the test object will be smeared within the speckle pattern. In addition, it will be difficult to apply the digital image correlation algorithm for tracking the speckle movement if the beam at the test sample is highly divergent, since the field of view of the x-ray beam will vary dramatically. In contrast, these limitations can be overcome by scanning the phase object across the beam [8]. Here, we would like to emphasize that the technique proposed here has a different nature and purpose from the one described in [8]. In the previous work, the membrane was placed *upstream* to measure the first derivative of mirror surface, namely the mirror’s slope. This earlier technique was therefore most suitable for direct measurements of the mirror’s slope errors. On the other hand, in this study the membrane was placed *downstream* of the mirror’s focal position and the second derivative of the wavefront, namely the inverse of its local radius of curvature *R*, was measured. This takes account of imperfections introduced to the wavefront not only by the mirror, but also by any of the optics upstream from the point of measurement. Also, the direct measurement of the spatial variation of the wavefront is vital for tailoring the X-ray beam to a specified size and shape while minimizing structure in its profile. It is due to this that the mirror’s shape could be optimized so rapidly. The remaining errors in the wavefront are thus due not to shortcomings of the technique, but to the mirror’s limited ability to correct the wavefront errors.

The optical layout and the operation principle of the speckle scanning technique are illustrated in Fig. 1. The calculations are performed entirely in the vertical plane because that is the tangential plane for all the beam-deflecting optics (monochromator and mirror). Sagittal slope errors have much less effect on the beam structure, partly because the radiation source size is much larger horizontally than vertically, but mainly because the extreme grazing incidence on the mirror makes the sagittal deflection negligible. Figure 1(a) shows that the mirror’s slope error introduces errors to the wavefront, thus broadening the beam size at the focal point F. The bimorph mirror’s slope error and hence the wavefront error can be minimized by applying appropriate voltages to the mirror’s piezo actuators. The membrane phase object’s position *y _{m}* transverse to the beam was scanned with a step size

*μ*using a piezo stage. At each

*y*a speckle pattern was recorded, and Fig. 1(c) shows an example. Once the membrane scan was complete, Figs. 1(d) was built up from the

_{m}*i*

^{th}rows of all the speckle images in order of increasing

*y*. Figures 1(e) was built up the same way from the

_{m}*j*

^{th}rows of all the speckle images. The 2-D cross-correlation of Figs. 1(d) and Figs. 1(e) is shown in Figs. 1(f). By localizing the peak in Fig. 1f, the shift in the speckle pattern between Figs. 1(d) and Figs. 1(e) could be determined with sub-pixel accuracy [15]. If

*i*=

*j*, Fig. 1(d) and Figs. 1(e) will be identical and therefore the correlation peak in Fig. 1f will be at (

*x*= 0,

*y*= 0). Otherwise, the correlation peak will appear at some offset

_{m}*y*= ε

_{m}*dependent on*

_{ij}μ*i*and

*j*. It is clear that for a given

*i*and

*j*the maximum correlation between Figs. 1(d) and Figs. 1(e) will be obtained if the speckle pattern at the detector shifts by Δ

*y*= (

*j*−

*i*)

*p*, where

*p*is the detector’s pixel size. Meanwhile, the speckle pattern at the membrane has shifted by ε

*. For small*

_{ij}μ*φ*, the local wavefront radius of curvature

*R*at the detector is approximately spherical.

*R*

^{−1}can be written as

*y*is along the vertical direction, which corresponds to the direction along the mirror’s length. The wavefront’s radius at the membrane will be

*R*–

*d*, where

*d*is the distance from the membrane to the detector position. Therefore,If the wavefront is collimated, then

*R*is infinite. Equation (2) shows that the motion of the speckle at the membrane and at the detector will then be equal as expected. From Eq. (2),Hence, the local wavefront radius of curvature can be measured precisely by analyzing the speckles from neighbouring pixels. Once R

^{−1}is calculated, one can calculate the wavefront slope

*φ*along the vertical coordinate

*y*of the detector from Eq. (1). It can be derived that the wavefront slope error Δ

*φ*is approximately twice the mirror’s tangential slope error Δ

*δ*for a focusing mirror [12]. Hence, the analysis and optimization of the wavefront radius of curvature can be used to directly minimize the mirror’s slope error.

The sensitivity of the measurement of the wavefront slope may be estimated as follows. First, by the definition of ε* _{ij}*, the quantity

*j*−

*i*). Therefore,

*η*will depend only on

*y*. Substituting Eq. (1) and Eq. (4) into Eq. (3) yieldsIntegrating Eq. (5) from

*y*=

*y*

_{1}to

*y*=

*y*

_{2}, one obtains

*y*, a wavefront slope error $\Phi (y)=\phi (y)-y/d$ can be defined. Expressing Eq. (6) in terms of this error and taking the absolute value of both sides yields

*y*

_{2}–

*y*

_{1}) will of course be an integer multiplied by

*p*; therefore, the spatial resolution of the measurement of Φ(

*y*) is the detector’s pixel size. If

*η*(

*y*) varies on length scales much longer than

*p*, and if

*y*

_{2}=

*y*

_{1}+

*p*, then Eq. (7) is approximated byand this is the sensitivity of the slope error measurement. It shows, surprisingly, that the sensitivity is independent of the detector’s pixel size, but can be improved by a smaller membrane step size

*μ*or larger membrane-detector distance

*d*. Here, the minimum detectable

*η*(

*y*) is dependent on the tracking accuracy in the cross-correlation process.

## 3. Experiment

Experiments were performed on the Diamond Light Source bending magnet Test beamline B16. The effective vertical source size is about 40 μm (FWHM) due to vibration. An X-ray energy of 9.2 keV was selected using a silicon double-crystal monochromator (DCM). The bimorph adaptive mirror under test was mounted on an independent motorized tower which was located 47m away from the source. 8 piezo electrodes were cemented underneath the silica substrate of the mirror to allow correction of the figure error [3]. In order to minimize the small- and large-angle scattering, which is dominated by the middle- and high-frequency roughness, the state-of-the-art super-polishing technique of elastic emission machining was employed [16]. The length of the silica substrate is 150mm with the central part of 130mm pre-polished as a tangential ellipse whose foci are located at distances ${p}_{0}$ = 41.5 m (source) and ${q}_{0}$ = 0.4 m (image) from the mirror. The grazing angle of incidence at the centre of the mirror is *θ* = 3 mrad. The membrane used to introduce speckle was a biological filter membrane with average pore size of 1 um, mounted on a piezo stage placed downstream from the mirror. The sensitivity for measuring the local wavefront radius of curvature can be increased by putting the membrane close to the focal plane F, and accordingly we installed it 100mm downstream of F. The random intensity pattern (speckle) was produced from the high-spatial frequency features contained in the membrane and was recorded by a CCD camera with an effective pixel size of *p* = 6.4 μm. In order to increase the spatial resolution and resolve the speckle features on the detector, the detector was located at a large distance (*L* = 3636 mm) from the mirror. The tangential spatial resolution for slope error for our measurement system is about 0.27mm, which is determined by the pixel size *p*, geometrical magnification ratio ($(L-{q}_{0})/{q}_{0}$and the mirror grazing incidence angle *θ*.

To determine the optimum voltages which minimize the wavefront errors, firstly 60 speckle images were recorded with an exposure time of 1.2 s as the membrane was scanned with all bimorph voltages set to 0 V. This is scan *k* = 0. For each subsequent scan *k* ≥ 1, the voltage on electrode *k* was incremented by *ν* (400 V), until by the 9th scan (*k* = 8) all piezo voltages were at 400 V. The calculated inverse of the wavefront radius at pixel *i* in the *k*th scan, *R*^{−1}* _{ik}*, was stored in an

*n*×

*m*matrix, where

*n*is the total number of pixels along

*y*and

*m*is the total number of scans. Here,

*i*is linked to the position along the mirror. As shown in Fig. 2, the response of the

*k*

^{th}electrode is calculated by subtracting the values of

*R*

^{−1}extracted from the (

*k*−1)

^{th}and

*k*

^{th}image. An

*n*× 8 interaction matrix

*M*, which defines the response of each electrode per unit voltage change, is constructed from:

*R*

^{−1}rather than the slope δ, which was commonly used in previous work by the other ex-situ and in situ metrology methods [9, 12, 17]. The direct measurement of the wavefront’s local radius of curvature is simpler, faster and more efficient than the measurement of the wavefront slope [18], because the curvature is a scalar field whereas the slope is a vector field. In addition, the PRF when measured in local curvature may be approximated as a Gaussian function, whereas the PRF as measured in slope more closely resembles the non-analytical error function [19]. Therefore, when the Poisson Eq. (1) is solved numerically to obtain the optimal voltages, it will converge much more rapidly if the PRFs are expressed in

*R*

^{−1}than in δ. Once the wavefront radius of curvature

*R*

^{−1}

_{i}_{0}is derived, its error Δ

*R*

^{−1}

_{i}_{0}can be calculated by subtracting the best-fit linear term.

## 4. Results and discussion

The wavefront radius of curvature for four cases, (a) bimorph voltages at 0 V, (b) bimorph voltages for focused operation after 1st iteration, (c) first optimization for bimorph defocused, (d) second optimization for bimorph defocused, are plotted in Fig. 3. The initial wavefront radius of curvature error Δ*R* for case (a) (black line) shows the biggest oscillation, with an rms value of 14 mm. The required voltage *V _{k}* can be calculated with the same procedure described in reference [4, 12]. After applying correction voltages, the rms value of Δ

*R*for case (b) (red dotted line) has been dramatically decreased to 4 mm. The corresponding rms mirror slope error was reduced from 1.172 μrad in (a) to 0.167 μrad in (b) after only one iteration.

The bimorph mirror can also provide different beam sizes at the focal plane F by changing its surface to a different ellipse. In this case, the focal length *q* will be changed. If the distance *L* between the mirror and the detector is fixed, the measured wavefront radius *R* at the detector plane will be changed with *q*. The merit function for the wavefront radius of curvature optimization can be written as

*R*

^{-}^{1}

*and*

_{t}*R*

^{-}^{1}

*are, respectively, the target and initial*

_{i}*R*

^{−1}. Obviously, the focus optimization is one special case when the

*R*

^{-}^{1}

*is equal to the mean value of*

_{t}*R*

^{-}^{1}

*. In order to generate a beam with uniform intensity across a specified width, the target*

_{i}*R*

^{−1}_{t}will be set uniform over the detector. Then, the defocused beam size can be calculated by multiplying the beam divergence and the defocus distance. Figure 3(c) and Fig. 3(d) show the moderate and strong defocus optimization results for the wavefront radius of curvature.

The optimization results for four cases are summarized in Table 1. The focused beam size (*BS*) was optimized from 1.2μm down to 0.6μm, and it was then defocused up to 25μm and 50μm. The average *R* was increased from 3.224m (b) to 3.268m (c) and 3.293m (d), and the corresponding rms variation Δ*R* is 6mm for both cases.

To verify the focus and defocus optimization, a series of images was taken by the high-resolution camera PCO 4000 CCD camera along the beam direction with 1 mm step size. Figure 3 shows the intensity profiles around the focal plane of the bimorph mirror for the four cases. There are strong oscillations on the edge of the beam profile for case (a) that are mainly due to the large mirror slope error before optimization. The beam intensity profile became much more uniform for case (b) at the focal plane after the optimization. Figure 3(c) and Fig. 3(d) show the focal position moving upstream towards the mirror while the intensity profile remains uniform. The focal position changes *q* from 406mm (b) to 366mm and 339mm for case (c) and (d), respectively. As summarized in Table 1, the changes of focal position *q* measured by the PCO 4000 camera are consistent with the focal position derived from the speckle measurement by subtracting the measured *R* from *L*. For example, the focal position should change to 337 mm for case (d) according to the speckle measurement, and indeed the offset from the focus found by the PCO 4000 camera is only 2 mm. This offset is within the 6 mm rms error of the radius of curvature (Δ*R*). If the measured offset of 2 mm is compared with the total radius of curvature *R* = 3293 mm, the relative error for tracing the focal position is less than 0.06%.

To validate the optimization results, the beam size was also measured by performing ‘knife edge’ scans with a 200 μm diameter gold wire, which is mounted on the same piezo for scanning the membrane. As shown in Fig. 4, the bimorph mirror was optimized from an initially focused beam size of 0.6μm FWHM to defocused beam up to 25μm FWHM and 50μm FWHM in a controlled way. Even though the beam size is defocused more than 40 time, the fluctuation of structures on the plateau from case (c) is only 11% (standard deviation/mean intensity), thanks to the extremely small slope error polished by EEM technique. This is quite favourably to the beamline users compared to a typical X-ray focusing mirror where defocusing by a factor of two or three times the focal size produces unacceptable structures in the beam shape. Nevertheless, the noticeable ripple on the beam shape still exist for such small remaining slope error on the mirror surface, and it indicates that the mirror slope error should be further reduced in order to minimize the structure in the defocused beam profiles.

Figure 5 shows the derived mirror slope error immediately after the optimized voltages were applied to the piezo actuators and then every 3 minutes afterward. The rms mirror slope error is 167 nrad for the first scan and increased to 218 nrad after five scans. The slope error changes due to the hysteresis of the piezo actuators, since it will take some time for the deformation to be stabilized after applying the voltage. From the 3rd scan to the 4th scan, the rms slope error changes by 7 nrad; however, even such a tiny change is clearly noticeable in the zoomed region in Fig. 5. From Eq. (8), the sensitivity of the slope error measurement is dependent on tracing accuracy *η*(*y*), step size *μ* and distance between membrane and detector *L*. It should be mentioned that the performance tracing accuracy *η*(*y*) will be influenced by the quality of the speckle image, which can be affected by the vibration of the source and optics, detector efficiency and data acquisition time. By considering all these factors, the tracing accuracy *η*(*y*) is about 0.05 pixel in this study. With *μ* = 0.25 μm and *d* = 3.1 m, the minimum detectable mirror’s tangential slope error Δ*δ* is about half the 4 nrad wavefront slope error, or 2 nrad. It should be pointed out that some advanced sub-pixel registration algorithms can further improve the accuracy to 0.001 pixel [15]. With *η*(*y*) ≈0.01 and same *d* = 3.1m for a conservative estimate, the angular sensitivity for this advanced metrology technique can be further increased 25 times by decreasing the membrane step size to 0.05um.

## 5. Conclusion

In summary, we have demonstrated that the speckle scanning technique can be used for the optimization of a bimorph mirror. This technique has the advantages of simple experimental arrangement and modest requirements of mechanical stability and transverse coherence. Since the membrane can be freely located out of the focal plane, the sample position does not need to be accessed. Only a single iteration is required to achieve the desired shape by optimizing with the second derivative of the wavefront. In addition, the speckle scanning technique can provide the focal position by measuring the wavefront radius of curvature at a given distance between the mirror and detector. The angular sensitivity of this technique can be brought to sub-nrad levels by employing advanced tracking algorithms and decreasing the membrane step size. Such a high-precision metrology technique will be extremely beneficial for the manufacture and in situ alignment / optimization of x-ray mirrors for next-generation synchrotron radiation sources and astronomical telescopes.

## Acknowledgments

This work was carried out with the support of Diamond Light Source Ltd. The authors are grateful to Sebastien Berujon from ESRF and Yogesh Kashyap from Diamond Light Source for fruitful discussions.

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