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Abstract
Advances in accelerator technologies have enabled the continuous development of synchrotron radiation and X-ray free electron laser (XFEL) sources. At the same time, it has been critical to perform in-situ wavefront sensing to aid delivery of high-quality X-ray beams to the end users of these facilities. The speckle-based scanning technique has obtained popularity due to its high spatial resolution and superior sensitivity compared to other wavefront sensing methods. However, these advantages often come at the expense of longer data acquisition times since multiple images have to be collected to derive the necessary wavefront information. Whereas initial speckle tracking techniques could obtain wavefront information relatively quickly, the installation of additional hardware was routinely required to do so. Here, we propose a novel speckle-based approach, termed Alternating Speckle Tracking (AST), to perform fast wavefront sensing within a conventional beamline setup. The wavefront information derived from the new technique has proven to be valuable for many applications that require temporal resolution. Importantly, both horizontal and vertical wavefront information can be simultaneously derived by moving the speckle generator along the diagonal direction. We expect this method will be widely used by the synchrotron radiation and XFEL community in the future.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Modern synchrotron radiation and X-ray free electron laser (XFEL) sources provide high-brilliance X-rays for cutting-edge scientific and industrial research. The X-ray optics used at these facilities are key to delivering high-quality X-ray beams to the user end stations. Ex-situ optical metrology techniques are routinely used to provide accurate and valuable information about X-ray optics before they are installed [1–5]. Nevertheless, the ultimate performance of an X-ray beamline can be affected by the mechanical and thermal drift, vibration or misalignment of X-ray optics once they have been installed. Therefore, it is often highly desirable to perform in-situ X-ray wavefront sensing to help achieve diffraction-limited and coherence-preserving X-ray beams.
Over the last two decades, a variety of X-ray wavefront sensing techniques have been developed and excellent advances have been made to characterize and optimize X-ray beam performance at synchrotron and XFEL facilities [6–18]. However, most of the existing X-ray wavefront sensing techniques suffer from having either complex experimental setups or stringent experimental requirements, which have restricted their application more widely. In contrast, speckle-based techniques [19–22] have gained popularity due to their relatively simple experimental setups and ease of use. Currently employed speckle-based techniques can be divided into two categories according to their data acquisition strategy: 1) the X-ray speckle scanning (XSS) wavefront sensing method and 2) the X-ray speckle tracking (XST) wavefront sensing method and its variations. For the XSS method, the speckle generator is scanned either one-dimensionally (1D) [23,24] or two-dimensionally (2D) [25–28] perpendicular to the beam. The wavefront gradient can be calculated by comparing the stacks of speckle images collected with and without the test optics in the beam. The local curvature of the wavefront can be derived from the stack of speckle images with the test optics in the beam alone. It should be noted that the local curvature of the wavefront at the speckle generator plane corresponds to the irregular intensity measured in the far-field images [29]. The XSS method has both high sensitivity and high spatial resolution irrespective of which physical quantity is derived. However, the scanning and image collection can be time-consuming and, therefore, average values of the wavefront curvature are obtained over the data acquisition period. The long data acquisition times for the XSS technique can be overcome by using the conventional XST method and its variations [30–32]. For XST the reference speckle images are collected before the test optics are moved into the beam, or when two speckle images are collected at two different detector positions. However, it is often impractical to move the optics in and out of the beam or to translate the detector in fast wavefront sensing applications. It is also difficult to use customized double cameras [31,32] for the data acquisition considering the careful alignment and calibration required for these two detectors. These drawbacks limit the usability of the conventional XST method, and its variations, in a broader range of applications.
In many applications where high spatial resolution is not critical, fast wavefront sensing can be highly desirable. Here, we propose a fast speckle-based wavefront sensing technique, the Alternating Speckle Tracking (AST) method, to overcome some of the limitations of earlier speckle-based techniques. In this novel method, each image collected can be compared with the one collected immediately preceding it, in an alternating fashion, so that temporal wavefront information can be calculated from every image in the scan. Thus, when applied to fast wavefront sensing applications, the new method can be regarded as a single-shot measurement because each new image can provide immediate information about the local curvature of the wavefront at that moment in time. Either the horizontal or the vertical local curvature of the wavefront can be obtained by moving the speckle generator along the corresponding direction. Furthermore, both horizontal and vertical local curvatures can be simultaneously acquired if the speckle generator is translated along the diagonal direction. A schematic comparison of the self-reference XSS method and our proposed AST method is shown in Fig. 1.
Fig. 1. Above: The data acquisition and processing procedure for the self-reference XSS (left) and AST wavefront sensing methods (right). ixXSS/AST, iyXSS/AST are the results from cross-correlation of the speckle pattern in the x and y directions, respectively. In the self-reference XSS wavefront sensing method, two rows of pixels from each image in the image stack are extracted and stitched together to form two new images. The two images are then used in a cross-correlation calculation to obtain the local wavefront information. In the AST wavefront sensing method, only two images, taken at two different times, (t1 and t2, for example) are required. Small subregions are selected and cross-correlated to provide the local wavefront information. Below: Experimental schemes for the two methods within a single time frame. The self-reference XSS method (left) processes all the data acquired within a scan and, therefore, takes a certain amount of time depending on the exposure time and the scan range. Conversely, the AST method compares each adjacent image in turn, as shown by the arrows. As images can be collected at high frame rates, the local wavefront information can be reconstructed from each pair of images and processed on the fly.
At first glance, our proposed method looks like a simplified version of the self-reference XSS method by simply reducing the number of images necessary for the wavefront reconstruction and because the methods share the same basic geometric relationship. However, the two methods differ significantly in their data processing procedures. Likewise, the way the physical theory is developed in our proposed AST method also differs from the conventional XST technique, as discussed in section 2. The standard self-reference XSS technique is difficult to apply to situations requiring fast wavefront sensing. Therefore, we will describe various fast wavefront sensing applications of the novel AST technique in section 3. We will demonstrate that it can provide similar information to the self-reference XSS technique but at much faster speeds.
2. Principles of the Alternating Speckle Tracking (AST) wavefront sensing method
The novel AST method we propose shares the same underlying geometric relationship as the self-reference XSS technique. Also, like the conventional XST technique, our proposed method only requires two images for reconstruction of the wavefront information. For conventional XST techniques, the reference image is taken without the test optics in the beam, or with the detector at a different longitudinal position along the beamline. In the AST method, the reference image is taken at a different speckle generator position without needing to move the test optics. The main difference between the two methods is that conventional XST techniques provide the gradient (i.e. the first-derivative) of the wavefront [30–32] whereas the AST method reconstructs the local curvature (i.e. the second-derivative) of the wavefront. A discussion of the classification of the various X-ray speckle-based techniques is given in [33].
As both the self-reference XSS method and the AST method output the local curvature of the wavefront it will be helpful to directly compare the theory for each of the two techniques when applied to characterizing X-ray focusing optics. Figure 2(a) shows a typical case for a strongly focusing curved mirror test optic. The speckle generator is placed downstream of the focal plane of the test optic. R is the local radius of curvature of the wavefront at the detector plane, r is the local radius of curvature at the speckle generator plane, D is the distance between the speckle generator and the detector, δτ is the measured incremental movement of the speckle generator and Δp is the corresponding displacement of the speckle patterns which are cross-correlated. We have the following geometric relationship, as depicted in Fig. 2(a):
Fig. 2. (a) The geometrical setup for the self-reference XSS and the AST wavefront sensing methods. R is the local radius of curvature of the wavefront at the detector plane, r is the local radius of curvature at the speckle generator plane, D is the distance between the speckle generator and the detector, δτ is the measured incremental movement of the speckle generator and Δp is the corresponding displacement of the speckle pattern, i and j represent the ith and jth rows of pixels in the raw image stack. (b) Comparison of the results from the two wavefront sensing methods using a plane mirror as an example optic. The main features obtained from these two methods correspond well with each other.
The description above is based on the strong focusing case. Sign changes may be necessary when describing the weak focusing case, although these should be easy to determine and are, subsequently, not included here. A fuller description of the physical meaning of Eq. (1) can be found in [22] and the relationship can also be derived from a different route [29]. The self-reference XSS and the AST methods are distinguished from each other in the way that δτ and Δp in Eq. (1) are calculated empirically as discussed below.
The XSS method enables pixel-by-pixel analysis of the curvature of the wavefront along the scan direction making it suitable for at-wavelength measurements with high sensitivity and precision [21,22]. The XSS wavefront sensing method requires the speckle generator to be scanned in at least one direction. For the self-reference mode, the optics under test are kept in a fixed position. Hereafter, we will assume that the speckle generator is translated vertically so that the scan direction is along the y axis of the beamline. However, for the case of scanning horizontally, that is along the x axis, we need only to replace the rows of pixels in the analysis with columns. As shown in Fig. 1, two rows of pixels in each of the images in a dataset are extracted and stitched together to form a pair of new images which are then cross-correlated. Each image in the pair is produced from one specific row of pixels in the raw image stack. The cross-correlation results from the stitched image pairs are used to calculate the displacement of the speckle pattern, which can then be converted to give the 1D local curvature of the wavefront. In the self-reference XSS technique, the stitched images formed from each row of pixels can be regarded as probes for the wavefront measurement. nstep is the interval between the two rows that are cross-correlated, so that nstep = j – i, where i and j represent the ith and jth rows of pixels in the raw image stack. In the self-reference XSS method, the cross-correlated speckle patterns always have a shift of nstep on the detector plane. As a result, the displacement of the speckle pattern is determined by Δp = p × nstep, where p is the detector pixel size. Similarly, the speckle pattern cross-correlation result in the y direction, iyXSS is in the units of the speckle generator scan step size, s. As a result, the corresponding incremental movement of the speckle generator is given by δτ = iyXSS × s.
It is difficult to apply this self-reference XSS technique to fast wavefront sensing because it requires collection of enough scans to form the stitched images which can then be cross-correlated. In the XSS technique, short scans of just two images will not be sufficient to form the stitched image pairs necessary for accurate determination of the speckle pattern displacement. However, the basic geometric relationship of Eq. (1) can still be used to derive the equation to calculate the local curvature of the wavefront for the AST technique.
The AST wavefront sensing technique we propose also uses self-reference and can be applied to achieve temporal wavefront information. As can be seen in the schematic in Fig. 1, instead of scanning the speckle generator in one direction, it is quickly moved between two alternating positions with a new image collected at each one. The adjacent images can be simultaneously cross-correlated in the data processing step. Therefore, information about the wavefront can be reconstructed from every new image collected.
To reconstruct the wavefront information using the AST method, only two adjacent speckle images are necessary. The data processing method used for the self-reference XSS technique cannot be applied to the AST method. To be able to apply the same geometric relationship described in Eq. (1), different representations of Δp and δτ are necessary. Firstly, the raw speckle images are divided into subregions along the scan direction, as shown in Fig. 1, and each subset of images is then cross-correlated. The dimensions of the subregion selected are dependent on the nature of the speckle generator and the required spatial resolution. iyAST is the output of the cross-correlation of the selected subregion in the y (vertical) direction. The displacement of the speckle pattern is Δp, so that Δp = iyAST × p, where p is the pixel size. In contrast to the self-reference XSS method, the subregions selected to be cross-correlated are the same in the two images collected at adjacent times. The observed shift of the speckle pattern is caused by the movement of the speckle generator and the incremental movement of the speckle generator δτ equals the scan step s. According to Eq. (1), the local curvature of the wavefront at the detector plane, 1/R, is therefore given by:
The radius of curvature r at the speckle generator plane can then be calculated from R by subtracting the distance D between the speckle generator and the detector.
The two images that are cross-correlated in the self-reference XSS method are formed by stitching together of single rows (or columns) of pixels from the acquired image stack. As shown in Fig. 1, the XSS method enables pixel-wise speckle tracking along the scan direction. In contrast, the AST method requires several pixels along the scan direction to achieve good speckle tracking accuracy. Therefore, there is some compromise of spatial resolution inherent in the AST technique that is necessary to achieve fast wavefront sensing from only two speckle images. Figure 2(b) shows an example comparing the AST and self-reference XSS results for a plane mirror. The plane mirror was 450 mm long and positioned to reflect the X-ray beam horizontally with a grazing angle of 3 mrad. The mirror surface was coated with nickel. The main features in the wavefront local curvature obtained by the two methods correspond well with each other.
Table 1 provides a comparison between the self-reference XSS and the novel AST methods. In order to obtain high-resolution results, approximately 50-100 images per scan are typically required for the XSS method. Although two images are necessary for the data processing steps in the AST method, a single image is acquired in each time frame. Therefore, the AST technique is at least ×50 faster than the XSS method.
Table 1. Comparison between the self-reference XSS and the AST wavefront sensing techniques
For the XSS method, the speckle patterns are tracked in the pairs of stitched images and, therefore, the displacement is in the units of the scan step size. Conversely, for the AST method, the tracked speckle patterns are derived from the subregions selected in the raw images acquired and the tracked displacement is in the units of the detector pixel size. While the basic geometric relationship between the two methods is the same, the reasoning for choosing the scan step size empirically is different in each case. As a general rule, to obtain an accurate measurement of the tracked speckle displacement, the selected subregions of collected images should cover more than one speckle grain. Typically, one speckle grain will cover several pixels on the detector. For the XSS technique, in order to obtain subpixel resolution, the scan step must be smaller than the dimensions of each pixel, although this also increases the number of images required in a single scan. As a result, the scan step size chosen for the XSS method is a compromise between resolution and data acquisition time.
For the AST technique, the speckle pattern displacement is correlated with the movement of the speckle generator. In order to achieve accurate speckle tracking results, the speckle generator should be translated across several speckle grains. Therefore, the larger translation movement necessary for AST requires selection of larger image subregions. However, this also limits the spatial resolution. Typically, therefore, the scan step size for the AST method is often much larger than the XSS method. As a result, the requirement for highly accurate translation stages is relaxed for the AST method compared with the XSS method. The maximum curvature of the wavefront that the AST method can achieve depends on the required spatial resolution, the distances between the components, the detector size and the pixel size in a particular experiment setup.
Table 2 lists the scan step sizes used for all the experiments described in this paper. For both the AST method and the conventional XST method, any aberrations from the detector will introduce errors in measurement of the speckle pattern displacement. However, the same algorithm used to correct these aberrations in the XST technique [31] can also be used for the AST method.
Table 2. Main experimental parameters used for comparing the XSS and AST methods
One of the advantages of using AST for fast wavefront sensing of test optics is the removal of the influence of any positional drift of the X-ray beam during the measurements. It is often important to measure solely the temporal changes due to the test optics, as we will show in the example of dynamic monitoring of a bimorph mirror in the next section.
All the experiments described in this work were conducted on the B16 Test Beamline at the Diamond Light Source [34] using several different experiment modes. Figure 3 shows the layout of the B16 beamline and a summary of these experiments is given in Table 2. We used sandpaper with different grain sizes (in the range of 1 to 20 µm) as the speckle generators.
Fig. 3. Layout of the B16 Test Beamline at the Diamond Light Source. Either a double crystal monochromator (DCM) or a double multilayer monochromator (DMM) was used to provide the monochromatic beam to the experiment. The speckle generator was placed either upstream or downstream of the optical element under test. In this work, the speckle generator was placed downstream of the optical element for strongly focusing optics. The speckle generator was mounted on high-precision translation stages which provided 2D movement.
3. Applications of Alternating Speckle Tracking (AST) wavefront sensing
There are many X-ray metrology applications that can benefit from faster data acquisition and processing, or that require temporal resolution, rather than high spatial resolution. We describe three simple applications of the AST wavefront sensing method below.
3.1 Dynamic monitoring of incident X-ray beams
The AST wavefront sensing technique can be used to monitor the incident X-ray beam stability dynamically. The variation in the wavefront local curvature with time can be obtained using this technique. By combining the traditional X-ray speckle tracking (XST) technique for measuring incident beam stability, in which the speckle generator is fixed while acquiring images [35], both the global and local changes of the incident wavefront can be characterized.
We monitored the incident beam from both the double multilayer monochromator (DMM) and the double crystal monochromator (DCM) on the B16 beamline. Figure 3 shows the layout of the beamline [36] and the two monochromators that were characterized. By translating the speckle generator diagonally, the wavefront local curvature in both the vertical and horizontal directions could be obtained simultaneously. Figure 4 shows the calculated wavefront local curvature plotted against time. Due to the different flux for the two monochromatic modes, different exposure times were used. The images of the DMM beam (with an aluminum filter to protect the camera) were collected with an exposure time of 500 ms and the images of the DCM beam used an exposure time of 100 ms. Due to the movement of the speckle generator and the detector data writing time, consecutive images in the scan are taken approximately every 2 s. The RMS value of the wavefront local curvature of the DMM beam in the vertical direction is much larger than in the horizontal direction. It is also significantly larger than the local variations of the wavefront in the X-ray beam from the DCM. There are two possible reasons for this result. Firstly, the vibration of the DMM in the vertical direction is much larger than the corresponding displacement of the DCM. This was observed in the AST wavefront sensing results by calculating the mean values of the 1D wavefront local curvature at every timestamp. The variations of the mean curvature for each monochromator are shown in Fig. 4 above each image. The RMS value of the variation (RMS_vib) can be used as a metric of beam stability. The DMM stability in vertical direction is substantially larger than in the horizontal direction and larger than the DCM values by one order of magnitude. Secondly, the smoothness of the wavefront after the DMM is more heterogeneous than that after the DCM because of different manufacturing processes involved in the fabrication of the optical components. For the AST method, the smoothness will impact the wavefront variations in the vertical direction due to the corresponding motion of the DMM according to the geometric layout of the beamline. As a result, for improved data quality, we used the DCM beam as the incident wavefront for the following optic characterization applications.
Fig. 4. Dynamic monitoring of the X-ray beam from the DMM (left) and the DCM (right) on the B16 beamline. The calculated 1D wavefront local curvatures are plotted against time for both vertical (V) and horizontal (H) directions. The exposure time for a single image was 500 ms for the DMM and 100 ms for the DCM and images were sampled approximately every 2 s. The RMS values of local curvature are also given. The mean value of the local curvature at every timestamp were calculated and are plotted above each image. The RMS value of this variation (RMS_vib) is also presented.
3.2 Auto-alignment of Kirkpatrick-Baez mirrors
Precise alignment is crucial for optics that focus X-rays down to the submicron scale. Kirkpatrick-Baez (KB) mirror pairs are routinely used to achieve submicron focusing at X-ray sources. The accurate positioning of the pitch angles of the two mirrors in the KB system plays an important role in achieving the optimal wavefront and focus profile.
There are several different approaches used for the alignment of the pitch angle of KB mirror pairs [6,8,11,13,14,37–43] and the X-ray near-field speckle-based method [44] stands out among them due to its simple experimental setup and the easily obtained hardware, including the low-cost speckle generators. The theory for determining the mirror pitch angle using a speckle-based technique has been described previously [44]. It can be calculated that the wavefront local radius of curvature varies along the mirror length if the mirror is misaligned. Only when the mirror is set to its nominal pitch angle does the wavefront have the same local curvature along the detector coordinate, as described in the simulated results in Fig. 5(a). If we use R(x) to represent the wavefront local radius of curvature along the detector coordinate x, then the R(x) can be fitted with a first-order polynomial of x. The more the mirror deviates from its nominal angle, the larger the slope of the fitted line. Further analysis shows that the slopes of the fitted lines have a linear relationship with the deviation from the nominal angle Δθ, as shown in the simulated result in Fig. 5(b). These results can be simulated using ray-tracing software developed for synchrotron radiation or XFEL applications. We used the xrt [45] software to calculate the misalignment for an elliptical mirror. A point source was used and suitable geometric parameters for the experiment were chosen. To account for the positional error, the mirror was intentionally set 2 m further downstream from its designated p value.
Fig. 5. (a) The simulated wavefront local radius of curvature at different mirror pitch angles relative to the nominal pitch angle for the mirror (0 µrad). (b) The linear relationship between the pitch angles and the fitted slopes of the simulated wavefront local radius of curvature. The zero value corresponds to the nominal pitch angle where the mirror is perfectly aligned. (c) Several 1D wavefront local radius of curvature curves obtained using the AST technique. For the vertically focusing mirror (VFM), a total of 13 pitch angles around the nominal value were measured, although only three are shown here. Similarly, 10 angles were measured for the horizontally focusing mirror (HFM). (d) Linear fits of the slope of the AST measurements (circles) used to determine a nominal mirror pitch angles of ±3 mrad for VFM (blue) and HFM (red). The self-reference XSS measurements are also plotted for comparison (triangles).
The general strategy for mirror alignment is to find the pitch angle for which the wavefront local curvature is constant along the mirror length. This method has been verified in previous work [44,46]. The previously described speckle-based method used the self-reference XSS technique to characterize the distorted wavefront from the mirror and calculated the pitch angle from the measured wavefront. Due to the scanning required for the XSS-based method, mirror alignment cannot be achieved very fast. However, the determination of mirror pitch angle does not need high spatial resolution and the advantage of using the AST method is the much shorter data acquisition times involved.
The KB system discussed here consisted of two elliptical mirrors with parameters of pv = 45.11 m, qv = 0.125 m and θv = 3 mrad for the vertically focusing mirror (VFM) and ph = 45 m, qh = 0.235 m and θh = 3 mrad for the horizontally focusing mirror (HFM). The two mirrors were both 100 mm long. Figure 5(c) shows six 1D wavefront local radius of curvature curves obtained using the AST technique. Results for three pitch angles from a total of 13 collected for the VFM and three from 10 for the HFM are plotted. These 1D curves were fitted with first-order polynomials and the slopes calculated. The nominal pitch angle of 3 mrad for these KB mirrors is achieved when the fitted slope is equal to zero, as shown in Fig. 5(b). Therefore, the optimum pitch angle can be determined by the linear fitting the AST results. Figure 5(d) shows results from one AST measurement. The solid lines in Fig. 5(d) depict the linear fits of the slopes from the AST results with varying pitch angle. The blue linear fit corresponds to the VFM and the red linear fit to the HFM.
We also obtained the wavefront local curvature curves using the self-reference XSS method and they are also plotted in Fig. 5(d). As seen in the figure, the self-reference XSS results are very similar to the linear fitting of the AST results. The self-reference XSS method can also be used to determine the nominal pitch angles although the linear fits are not shown in the figure. The difference in the optimum pitch angle determined by the self-reference XSS method and the AST method were 9.2 µrad for the VFM and 6.0 µrad for the HFM. The residual linear fitting error for the self-reference XSS method was 0.62×10−6 (VFM) and 1.02×10−6 (HFM) while for the AST method it was 1.13×10−6 (VFM) and 1.83×10−6 (HFM).
Although the self-reference XSS results have smaller residual fitting errors than the AST method, the time taken for alignment of the KB mirror pair can be reduced from over one hour using the self-reference XSS technique to just a few minutes using the AST technique. As shown above, the fitting to the linear relationship between the pitch angles and the slopes of the wavefront local radius of curvature from the AST method can be used to obtain satisfactory results while dramatically reducing the data acquisition times.
3.3 Fast determination of the piezo response functions and dynamic monitoring of a bimorph mirror
Bimorph mirrors [47–50] can be used to correct wavefront errors, or provide variable profiles, of X-ray beams at synchrotron radiation and XFEL beamlines. The mirror surface is modulated by multiple piezoelectric ceramic blocks that are either embedded within the polished optical substrate or bonded to the side of the optic. Although valuable information on the behavior of the piezo components can be acquired using various ex-situ optical metrology methods [2], it is essential to obtain the in-situ response of each piezoelectric component when a bimorph mirror is installed on a beamline. Each individual piezoelectric ceramic is characterized by its piezo response function (PRF). The PRF describes the zonal control of a single piezoelectric ceramic. Only after the measurement of the PRFs can the bimorph mirror be used to correct the distorted wavefront [17]. The PRFs are used to calculate the appropriate voltages applied to the piezoelectric (PZT) actuator driver. A photograph and schematic diagram of a bimorph mirror are shown in Fig. 6(a).
Fig. 6. (a) Photograph and diagram of a typical bimorph mirror. A single piezoelectric ceramic has zonal control of the mirror shape in its local vicinity. When all the piezo components are combined together, the deformation of the mirror can correct the incident wavefront to an ideal one. (b) The PRFs determined by the self-reference XSS method (top) and the AST method (bottom). As with the KB mirror experiment, these two methods obtain almost identical information. The AST wavefront sensing technique substantially reduces the data acquisition time.
Several in-situ methods [16,17,51] have already been employed to characterize the bimorph mirror used in this experiment, including a speckle-based technique [22]. Although the speckle-based method has been used in previous studies of bimorph mirrors, it generally requires several minutes to complete each scan for determining one PRF, so that it is time-consuming to determine all of them. The piezoelectric ceramic modulates the low spatial frequencies of the mirror shape and there is no need to accurately measure the high spatial frequencies using the self-reference XSS technique. In contrast, the AST wavefront sensing technique can provide fast PRF measurements within a few seconds. The dynamical monitoring of the piezo behavior becomes possible due to the temporal resolution achieved with the AST method.
The bimorph mirror characterized in this work had eight piezoelectric ceramic components as shown in Fig. 6(a). The PRFs determined by this method are expressed in terms of the wavefront local curvature. Figure 6(b) shows the PRFs determined using the self-reference XSS and AST methods and there is, again, good agreement between both techniques. High-frequency features are visible in the PRFs derived from the AST method due to errors in the speckle tracking algorithm, although these are not significant enough to impact on the practical use of the bimorph mirror.
Once the PRFs were obtained, an inverse-matrix method was used to correct the measured wavefront distortion within a few iterations [22]. We can use the AST method to measure the wavefront distortions and apply the necessary correction with the PZT actuator driver. The procedures are identical to those described in [22] and are outside the scope of this paper. By comparing the PRFs obtained with the AST method and the self-reference XSS method, we demonstrate that the AST technique is valid for use in correcting the wavefront errors deriving from bimorph mirrors.
The ability to characterize changes in the wavefront within just a few seconds enables the behavior of each piezo to be monitored. Hence, we demonstrated the use of the AST wavefront sensing technique for monitoring the change of the local curvature of the wavefront over time. Five different voltages were applied to the bimorph mirror. At each voltage, i, 400 V was applied to the (i+3)th piezoelectric ceramic by the PZT actuator driver and the other seven voltages were set to zero. The measured 1D wavefront local curvature at each voltage with elapsed time is shown in Fig. 7. Each voltage was measured for approximately 40 mins before switching to the next one. Images for the AST measurement were taken approximately every 2 s with an exposure time of 500 ms. The zonal response of the bimorph mirror to the applied voltages on the piezoelectric components can immediately be seen in Fig. 7. While the corresponding scan using the self-reference XSS technique requires approximately five minutes at each voltage. During this time, the piezo behavior will have already changed significantly, as can be seen in Fig. 7. These fast changes of the piezo cannot be discerned using the self-reference XSS technique but are observed using the AST method.
Fig. 7. Dynamic monitoring of a bimorph mirror performance at five different applied voltages. As with the measurement of the incident X-ray beam, the calculated 1D wavefront local curvatures are plotted against time. At voltage i, 400 V was applied to the (i+3)th piezoelectric ceramic by the PZT actuator driver and all of the other voltages were set to zero. The zonal control of the piezoelectric ceramics can be seen clearly from the resulting plot of the wavefront local curvature. Images were taken every 2 s with an exposure time of 500 ms. The case for Voltage 3 is shown on the righthand side of the graph. The local curvatures are different at the start (red) and the end (black) of the data acquisition period. As time elapsed, the constraints applied by the piezoelectric ceramics relaxed.
It has been reported that after applying a target voltage, a bimorph mirror will slowly deviate from its initial shape [48,49] and this is clear in the AST results in Fig. 7. Previous research [47,48] used only a single parameter, such as the average wavefront curvature or the resultant beam size, to characterize the behavior of the piezo. The AST technique was used to measure the overall effect on the X-ray beam of the piezo changing. Two 1D plots of the wavefront local curvature are shown on the righthand side of Fig. 7. These plots represent the wavefront shape at the start (red line) and the end (black line) of the data acquisition labelled Voltage 3. For each piezoelectric ceramic, the effect of the applied voltage on the bimorph mirror relaxed as time elapsed. The 1D wavefront information provided by the AST method can be employed to actively control the bimorph mirror more effectively [52].
4. Conclusion
We have presented a novel speckle-based X-ray metrology technique called the Alternating Speckle Tracking method. In contrast to some commonly employed speckle-based methods, this approach has shown great potential for applications requiring fast measurements of the wavefront. From the theoretical derivation, we have shown that although the physical quantity obtained from the AST method is identical to the self-reference XSS wavefront sensing method, the conceptual transition from the self-reference XSS method to the AST method is not trivial. We have successfully utilized the new AST approach for 1) monitoring the incident X-ray beam from two different monochromators, 2) the fast pitch angle optimization for a KB mirror system, 3) the fast determination of the piezo response functions for a bimorph mirror and 4) the dynamical monitoring of bimorph mirror profiles after applying voltages. At present, the time resolution of the AST method is limited by the speed of the speckle generator motion and by the detector readout speed. Commercial high-precision piezo stages can switch between two positions several micrometers apart in a few tens of milliseconds, and detector readout times are faster than this. The current time resolution achievable by the AST method is, therefore, in the order of tens of milliseconds. The time resolution may be further improved by using motion stages that can move or oscillate between two alternating positions at high frequency and employing fast detectors. The proposed method in this paper has potential for XFEL time-resolved wavefront sensing. Although the applications shown are limited to 1D determination of the wavefront curvature, there are no constraints on extending the method to 2D analysis. We expect the AST wavefront sensing technique will be widely used at XFELs and synchrotron radiation facilities in the near future.
Funding
Diamond Light Source.
Acknowledgements
The authors would like to thank Dr Simon Alcock and Dr Ioana Nistea (DLS) for fruitful discussions and technical assistance with the mirror installation. Andrew Malandain (DLS) is also thanked for his expert technical support for all of the experiments. Diamond Light Source Ltd. is acknowledged for providing funding for this work. This work was undertaken on the B16 Test Beamline at the Diamond Light Source.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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