1. Introduction

With the technical improvement of the diffraction-limited storage ring and free-electron lasers, the produced X-ray beam quality is getting higher. Correspondingly, to perfectly transport or focus the X-ray beam at the diffraction limit, the requirement on the X-ray optics becomes tighter. The figure error of X-ray mirrors is typically getting down to sub-100 nrad RMS in slope or sub-nm RMS in height to preserve the high quality of the beams [1].

At many synchrotron radiation facilities, like NSLS-II, X-ray mirrors are generally inspected with the point-wise scanning deflectometric profilers [2–13]. These metrology tools have been developed in the late 1980s from the Long Trace Profiler (LTP) by Takacs et al. [2], to the autocollimator (AC) based Nanometer Optical component measuring Machine (NOM) by Siewert et al. in 2004 [6]. These two options have been widely implemented in the synchrotron facilities around the world [7,9,10]. With some improvements, the Nano-accuracy Surface Profiler (NSP) was recently developed by Qian et al. [11]. As the metrology requirement is getting stricter, the intrinsic instrument error of the optical head or the AC needs to be characterized and calibrated by a more accurate small angle generator to further improve the measurement accuracy [12,14–16].

Since the proposal [17] and the suggestion [3] of using a moving pentaprism, the pentaprism-based LTP/NOM has become the dominant system in X-ray mirror metrology [4,6,8,10]. Using this geometry to measure an X-ray mirror surface as the Surface Under Test (SUT), the Sensor-to-SUT Distance (SSD) changes during the pentaprism motion. Because the instrument error is distance dependent [14,18,19], it is not technically sound to apply a distance-fixed calibration to such an SSD-varying scan. On the other hand, the SSD is fixed in the NSP principle, typically between 5 mm to 10 mm. This very short distance brings practical difficulties in the AC calibration. One way to overcome this issue was proposed by Lacey et al. to transfer the calibration from one calibrated reference AC to the scanning AC mounted on the gantry [16].

Another way to compensate the instrument error was proposed by Polack et al. while they were developing an LTP stitching procedure with the estimation of the instrument error [20,21]. Nicolas et al. applied a similar instrument error suppression algorithm in the calibration of mechanical mirror benders [22] and the continuous scanning ALBA-NOM [23]. When applying this instrument error suppression algorithm with multi-pitch redundant angular dataset to a moving-pentaprism-based LTP/NOM system, it is assumed that the instrument error is the same in all scans with different pitches.

However, due to the distance dependency of the instrument error [14,18,19], this assumption may not be always "true" in a moving-pentaprism-based LTP/NOM case, especially when the pitch range is large. This is illustrated in Fig. 1. A 2D instrument error map is simulated as a function of the angle $\alpha \in [-5, 5]$ mrad and the SSD $\in [200, 600]$ mm in Fig. 1(d). On this error map whose amplitude is $\pm 1\mu$rad, the measurement paths for NSP and LTP/NOM are superimposed. The instrument error curves in Fig. 1(c) at distances SSDs = 250 mm, 350 mm, 440 mm, and 560 mm is simulated referring to the Fig. 4(b) in Ref. [14]. Figure 1(e) displays the instrument error curves of a moving-pentaprism-based LTP/NOM in multi-pitch measurement have large discrepancies in the overlapping angular ranges with a 2 mrad pitch step and 310 mm varying working distance. As a contrast, the instrument error curves of an NSP in multi-pitch measurement coincide well with a 2 mrad pitch step and 5 mm distance variations as shown in Fig. 1(f). Since the SSD of the sample-beam AC in NSP is fixed as illustrated in Figs. 1(b), the instrument error can reasonably be assumed the same in all scans with different pitches. For instance, with an $L=1$ m long mirror and $R_p = 10$ mrad active pitch range, if NSP SSD $= 300$ mm (In real measurement, the NSP SSD should be around 10 mm. Here we have simulated instrument error map with SSD from 200 mm to 600 mm only, so we assume NSP SSD is fixed as 300 mm for illustration purpose.), the SSD only changes $L/2\cdot R_p = 5$ mm, and as illustrated in Figs. 1(d) and 1(f), the instrument error curves are very close to each other with 5 mm distance variation. It indicates that the NSP is an ideal instrument to employ the multi-pitch instrument error reduction algorithm. Therefore, in this work, we apply the multi-pitch method to the NSP to improve the measurement accuracy by estimating the instrument error from the redundant angular dataset with the linear least squares method. In addition, the ambiguities in the multi-pitch algorithm are addressed by applying corresponding constraints to the pitch angle and instrument error function. The feedback from an extra pitch-beam AC is compared with the estimated pitch angles. Moreover, to get a smaller matrix condition number in the linear least squares estimation, parametric settings are discussed to keep the result being less sensitive to the measurement noise.

 

Fig. 1. Due to the distance dependency of the instrument error, the NSP is more suitable than the LTP/NOM to implement the multi-pitch measurement. Sketches of NOM (a) and NSP (b) in a multi-pitch measurement scheme, (c) the simulated instrument error curves at different distances referring to Fig. 4(b) in Ref. [14], (d) the simulated 2D instrument error as a function of the angle and the SSD, (e) the instrument error vs the angle for a varying-SSD system, and (f) the instrument error vs the angle for a fixed-SSD system.

Download Full Size | PDF

2. Principle of multi-pitch self-calibration measurement with NSP

The multi-pitch NSP measurement system layout is shown in Fig. 2. Three ACs are used to measure the angles of interest. The 15-mm-aperture reference-beam AC couples with a flat mirror to track the pitch angle of the carriage, on which the sample-beam AC scans the SUT through a 2.5-mm-diameter pinhole. The 50-mm-aperture pitch-beam AC coupled with another flat mirror measures the pitch angle of the pitch stage which holds the sample mirror.

 

Fig. 2. The layout of the multi-pitch self-calibration NSP. The reference-beam AC coupled with a flat mirror measures the pitch angle of an air-bearing carriage. The SUT placed on the sample stage is scanned by the sample-beam AC moving along the carriage. The pitch angle of the sample stage can be actively adjusted, and a pitch-beam AC coupled with a flat mirror can monitor the pitch angle of the sample stage.

Download Full Size | PDF

With a commercially available air-bearing stage, the pitch error can remain below $10\ \mu$rad during 1-m-long motion. The pitch-beam AC, which is optional, provides assistant information of the active pitch angles. The $x$ and pitch motions as well as the angle and temperature data acquisition are controlled by a home developed software written in C++ and Qt. The instrument error of the sample-beam AC is estimated by using the multi-pitch algorithm described in the following section.

2.1 Mathematical model and optimization

With the $n$th active pitch angle $\theta _n$ in total $N$ pitches $\boldsymbol {\theta }=[\theta _1, \theta _2,\ldots , \theta _N]$, combining the measured angles at NSP coordinates $x$ from the sample-beam AC, $m_n(x)$, and the reference-beam AC, $r_n(x)$, the local surface tilt measurement $m_n(x)+r_n(x)$ (the sign depends on actual AC orientations, and here the sign is "+" in our NSLS-II NSP setup) can be described by the expression

In the real experiment, the center of rotation of the active pitch stage is usually not in the plane of the mirror surface. This can introduce a pitch-dependent horizontal shift between the actual sampling locations $x^s_n$ in SUT coordinates and the NSP stepping position $x$ in NSP coordinates which is consistent for all pitch angles. By applying the rotation relation, we have

Two independent uniform cubic B-splines $\boldsymbol {B}(\boldsymbol {x^s})$ and $\boldsymbol {E}(\boldsymbol {m})$, where $\boldsymbol {x^s} = [\boldsymbol {x^s_1}, \boldsymbol {x^s_2},\ldots , \boldsymbol {x^s_N}]$ and $\boldsymbol {m} = [\boldsymbol {m_1}, \boldsymbol {m_2},\ldots , \boldsymbol {m_N}]$, are used to represent two unknown functions, the SUT slope $s(\boldsymbol {x^s})=\boldsymbol {B}(\boldsymbol {x^s})\boldsymbol {c}^\top$ and the instrument error $e(\boldsymbol {m})=\boldsymbol {E}(\boldsymbol {m})\boldsymbol {f}^\top$. The problem is converted to estimate the unknown coefficient vectors $\boldsymbol {c}$ and $\boldsymbol {f}$. Therefore, Eq. (1) becomes

In this equation, the stepping position in NSP coordinates $x$, the sample-beam reading $m_n(x)$, the reference-beam reading $r_n(x)$, and the distance between SUT and pitch rotation axis $d$ are known in our measurement. Therefore, the basis of B-splines $\boldsymbol {B}(x+d\cdot \theta _n)$ and $\boldsymbol {E}(m_n(x))$ are known as well. During the optimization process, we need to find the best coefficients $\boldsymbol {c}$ and $\boldsymbol {f}$ as well as the pitch angles $\boldsymbol {\theta }$ that best explain the measured data in a least square sense as described with the follow expression.

Before minimizing the residuals, the ambiguities among solutions need to be addressed. Some proper regularization is necessary to avoid the measurement data to be well explained with different $\boldsymbol {c}$, $\boldsymbol {f}$ and $\boldsymbol {\theta }$.

2.2 Ambiguities and regularization

The angular bias ambiguity. It is not difficult to find that the bias values of the SUT slope, instrument error, and active pitch are coupled together with their sum equaling to the bias value of $m_n(x)$. In order to get rid of this ambiguity, we need to regularize two of these three angular bias values. This can be done by defining the origins of the angles.

First, we define the first pitch angle $\theta _1 = 0$ for simplicity. It is also a natural choice for the slope stitching process, as we can let the stitching start from the sample-beam AC reading at the left end of the SUT close to 0, and after all the pitch scans, the sample-beam AC reading at the right end of the SUT coming back to 0 to make sure the stitching covers the whole space and slope ranges with redundancy.

Second, it is preferable to define the instrument error as 0 while its reading is 0, i.e. $e(0)=0$ as implemented as $\boldsymbol {E}(0)\boldsymbol {f}^\top =0$. It is equivalent to that the sample-beam AC and the calibration tool are well aligned with both readings are 0 during an AC calibration.

In this case, the resultant SUT slope offset will be fixed to a unique solution with $e(0)=0$ and $\theta _1 = 0$. Since the slope offset is actually a tilt in height which is not of interest for X-ray mirror surface shape measurement, the above regularization is acceptable.

The angular linear term ambiguity. If we consider to separate the instrument error $e(m) = \varepsilon \cdot m + e_h(m)$ as its linear term $\varepsilon \cdot m$ and other higher orders $e_h(m)$, Eq. (3) becomes

After some modifications, we have

Considering the linear factor $\varepsilon$ of a high-quality AC, which is nothing else but the error committed on evaluating its focal length, is usually small with $|\varepsilon |< 0.1\%$, e.g. $\varepsilon \approx \pm 2 \mu rad/10000 \mu rad = \pm 0.02\%$ in Fig. 1(c), and the reference-beam AC reading $r_n(x)$ is within $\pm 5\ \mu$rad, the term $\varepsilon \cdot r_n(x)$ is within $\pm 5$ nrad which is negligible while the noise $n_a$ is at 100 nrad RMS level.

That is to say, for any $n$ and $x$, there are infinite combinations of $\varepsilon$ with $s(x^s)+e_h(m_n(x))+\theta _n$ that can match the measured value $m_n(x) + r_n(x)$. Since this ambiguity is a linear factor on angles and the X-ray mirror is usually in a concave or convex shape, this linear slope ambiguity is also called as curvature ambiguity. To regularize this ill-posed problem, we regularize the linear factor $\varepsilon =0$, then the solution of SUT slope $s(x^s)$, the high-order instrument error $e_h(m)$, and the pitch $\theta _n$ will be unique. Of course, by doing so, we will suffer from a curvature error in the X-ray mirror measurement, close to $\varepsilon <0.1\%$. With this very small error, this regularization is acceptable for X-ray mirror shape measurement.

Putting both regularization together, Eq. (4) becomes the following constrained linear least-squares optimization problem.

The angular periodic ambiguity with uniform pitch steps. If a uniform pitch step $\Delta \theta$ is used, with the previously constrain $\theta _1=0$, the $n$th pitch angle is $\theta _n=(n-1)\cdot \Delta \theta$. If we consider there is a periodic function $p(s) = p(s+P)$ with period of $P = \Delta \theta /k, \ k\in \mathbb {N}$, we have

Considering $r_n(x)$ and $e(m_n(x))$ are commonly much smaller than $m_n(x)$ when measuring curved mirrors, we have the following approximation

Applying Eq. (9) and Eq. (10) to Eq. (1), we get

To avoid this ambiguity, similar to the solutions in the self-calibration stitching interferometry [24,25], we can introduce random variations to the pitch angles in data acquisition to "break" their periodicity. We demonstrate its effectiveness with a simulation result shown in Fig. 3. Weighted cosine fluctuations with a period around 322 $\mu$rad are used to simulate the regional middle-frequency waviness of the instrument error. The angular measurement noise is set as 100 nrad RMS. The global "angular tilt" is not estimated from this multi-pitch approach as mentioned above. By maintaining the same averaged pitch step $\overline {\Delta \theta }=200\ \mu$rad and the spacing of slope B-spline control point $d_{cp}=40\ \mu$rad, we compare the results of using different standard deviations of the pitch step randomization $\sigma _{\Delta \theta }=0$, 1, and 30 $\mu$rad. This clearly demonstrates that the periodic error can be significantly suppressed if enough random variations exist in pitch angles.

 

Fig. 3. Enough random variations on the pitch steps can suppress the periodic errors on both the reconstructed SUT slope and the reconstructed instrument error due to the angular periodic ambiguity. (a) the SUT slope reconstruction under different conditions, (b) the reconstruction error of the SUT slope without the linear term, (c) the instrument error reconstruction under different conditions, and (d) the reconstruction error of the instrument error without the linear term.

Download Full Size | PDF

Although the pitch angle steps are typically performed with certain intrinsic randomization at the sub-$\mu$rad to $\mu$rad RMS level due to the mechanical limitation, we want to highlight that an extra randomization is still necessary. As compared in the Power Spectral Density (PSD) curves of the reconstructed instrument errors in Fig. 4, even though $\sigma _\theta = 1 \mu$rad pitch step variations exist, the two error peaks of the angular period ($P = \Delta \theta /k, \ k\in \mathbb {N}$) at 200 $\mu$rad ($k=1$) and 100 $\mu$rad (k=2) still appear, while there is no obvious such peaks with $\sigma _\theta =30\ \mu$rad pitch step variations.

 

Fig. 4. The PSD plots of the instrument error reconstructions of Fig. 3(c) also indicate the error from periodic ambiguity can be reduced by applying enough randomization on pitch steps.

Download Full Size | PDF

2.3 Density of slope sampling and control points

Apart from the above ambiguities, some restrictions for the slope sampling and control points spacing need to be noted as well. Based on the Nyquist-Shannon sampling theorem, to properly reconstruct the instrument error with uniform cubic B-splines, the slope sampling frequency $f_s$ should be at least two times higher than the highest frequency $f_h=1/P_h$ component to reconstruct, i.e. $f_s\geq 2f_h$. In practice, considering the angular measurement noise, the samplings should be denser. The control points of B-splines should have a proper density to represent the highest frequency but without over-fitting artifacts. Usually, we set the spacing of control point $d_{cp}$ as one third of the smallest period to reconstruct, i.e. $d_{cp}=P_h/3$.

For instance, we simulate an instrument error with two periods, $P_1=322\ \mu$rad and $P_2=13.33\ \mu$rad, as shown with the black curve in Fig. 5(a). If the slope samplings period $P_s = 3.33\ \mu$rad along the x scanning on a spherical cylindrical SUT, the nonuniform pitch step with average step size of $\overline {\Delta \theta } = 200\ \mu$rad, and the spacing of slope control points $d_{cp}$ is set as $d_{cp} = P_2/3 \approx 4.44\ \mu$rad, both variations with periods of $P_1$ and $P_2$ are reconstructed as shown in Fig. 5.

 

Fig. 5. Reconstruction of the instrument error with two major periods, $P_1=322\ \mu$rad and $P_2=13.33\ \mu$rad, under the conditions of different sampling period $P_s$, the average pitch step $\overline {\Delta \theta }$, and the spacing of control points $d_{cp}$ (a) within 10 mrad range, and (b) a close view from 3 mrad to 4 mrad (The dots stand for the slope samplings.) shows that large reconstruction errors occurs at periods with insufficient sampling points.

Download Full Size | PDF

If the slope samplings are not dense enough to represent the highest-frequency $f_h$ fluctuation of the instrument error, it is impossible to retrieve the finest periodic variations in the instrument error, but we still can reconstruct a lower-frequency trend of the instrument error with a suitable $d_{cp}$ selection. Using the previous example, if we used an $x$-step providing $66.67\ \mu$rad/step along the $x$ scan on SUT, and the other conditions maintain the same, i.e. $\overline {\Delta \theta } = 200\ \mu$rad and the slope control point spacing $d_{cp} = 4.44\ \mu$rad, we will then get large artificial errors due to the insufficient samplings on the slope for some periods as illustrated by the red curves in Fig. 5. However, despite that the finest period $P_2=13.33\ \mu$rad are not reconstructed, the lower-frequency trend and $P_1=322\ \mu$rad periodic variations of the instrument error can be reconstructed by setting the spacing of the control points as $d_{cp}=P_1/3=107.33\ \mu$rad (the yellow curve) in Fig. 5.

3. Experimental determination of the instrument error

In order to validate our model and method, we have performed a series of experiment by using the NSLS-II NSP instrument [11] with the multi-pitch self-calibration technique. The SUT used in the experiment is a 150-mm-long spherical cylindrical mirror fabricated by JTEC Corp. with a tangential radius of curvature $R \approx 15$ m and a 10 mrad total slope range. The measured residual slope error derived from the stitching height data from the manufacturer is about 70 nrad RMS. This is a perfect match for the self-calibration study of the 10-mrad-measuring-range Möller-Wedel Elcomat 3000 AC. The mirror is placed on a motorized pitch stage, and the SUT is about 10 mm under the sample-beam AC with a 2.5-mm-diameter pinhole.

3.1 Repeatability of the instrument error determination in two continuous sets of measurement

The first two continuous sets of multi-pitch measurement are scanned with the SUT in A2B orientation (Here the "A2B" orientation means that the SUT is measured from its one end "A" to the other end "B". By physically rotating the mirror $180^{\circ }$ we will have the SUT in the opposite orientation named "B2A".). The raw slope data of these two sets of measurement are shown in Figs. 6(a) and 6(b). The temperature variations shown in Fig. 6(c) are less than 0.05 $^\circ$C PV for both scans. The active pitch range is about 10 mrad and the average pitch step $\overline {\Delta \theta } = 100\ \mu$rad. In total, 101 pitch angles are scanned to provide the redundant slope data. At each pitch angle, the SUT is scanned in 147 mm with a sampling step $\Delta x =1$ mm. To reduce the influence from the drift, "forward and backward" scans are performed and averaged as the result of each $x$ scan.

 

Fig. 6. Two continuous multi-pitch self-calibration measurements in 101 pitch angles with $\overline {\Delta \theta } = 100\ \mu$rad are performed with slope data acquisition (a) from 01 Nov, 2019 to 02 Nov, 2019 and (b) from 02 Nov, 2019 to 03 Nov, 2019. The temperature (c) measured during these two scans are stabled with 0.046 $^\circ$C PV and 0.047 $^\circ$C PV variation, respectively.

Download Full Size | PDF

In the reconstruction, the spacing of the slope control points is set as $d_{cp} = 100\ \mu$rad to reconstruct the low-frequency instrument error only. The reconstructed SUT slope errors, instrument errors, and the pitch angles are plotted in blue and orange curves in Figs. 7(a), 7(c), and 7(e).

 

Fig. 7. Two multi-pitch self-calibration scans in 101 pitch angles with $\overline {\Delta \theta } = 100\ \mu$rad (blue: the 1st scan, red: the 2nd scan). (a) Reconstructed SUT slope errors with the same sphere removed (the red regions indicating the $\pm 2\sigma$ uncertainty range obtained by analysis of the variance (ANOVA) of the datasets), (b) the discrepancy of two SUT slope reconstruction, (c) the reconstructed instrument error of the sample-beam AC (the red regions indicating the $\pm 2\sigma$ uncertainty range), (d) the discrepancy of two reconstruction of instrument errors, (e) the measured pitch angles by pitch-beam AC and the estimated pitch angles, and (f) the differences between the estimations and measurements. Some curves are vertically shifted for better visual comparison.

Download Full Size | PDF

These two sets of scans are repeatable. As shown in Fig. 7(b), the difference of two reconstructed SUT slope errors is 12 nrad RMS, and the discrepancy of the reconstructed instrument error is 12 nrad RMS in Fig. 7(d). The differences between the estimated pitch angles and the pitch-beam AC measured angles are shown in Fig. 7(f).

3.2 Reproducibility of the instrument error determination

The SUT was then measured in A2B orientation with 51 pitch angles over 10 mrad with $\overline {\Delta \theta } = 200\ \mu$rad, and then the SUT was rotated $180^{\circ }$ to take B2A 51-pitch scans with $\overline {\Delta \theta } = 200\ \mu$rad. The resultant SUT slope errors and instrument errors are shown in Figs. 8(a) and 8(c).

 

Fig. 8. Taking the 101-pitch A2B scans with $\overline {\Delta \theta } = 100\ \mu$rad as the reference, 51-pitch A2B and B2A scans with $\overline {\Delta \theta } = 200\ \mu$rad are compared. (a) Reconstructed SUT slope (the red regions indicating the $\pm 2\sigma$ uncertainty range), (b) their discrepancies from the reference SUT slopes, (c) reconstructed instrument errors (the red regions indicating the $\pm 2\sigma$ uncertainty range), (d) the reconstruction discrepancy of instrument errors from the reference results, and (e) the PSD of the instrument errors.

Download Full Size | PDF

If we take the average of the two previous $\overline {\Delta \theta } = 100\ \mu$rad scans in Fig. 7 as our reference in the following comparison, the discrepancies of $\overline {\Delta \theta } = 200\ \mu$rad A2B and B2A scans from the reference are only 20 nrad RMS and 53 nrad RMS for the SUT slope and 20 nrad RMS and 15 nrad RMS for the instrument error as shown in Figs. 8(b) and 8(d), respectively. From the PSD of the reconstructed instrument errors shown in Fig. 8(e), we can see the main period of the low-frequency fluctuation in the instrument error is about 322.6 $\mu$rad.

To reconstruct high-frequency fluctuations in the instrument error, small steps are preferred to avoid the issue of insufficient samplings. For example, another A2B multi-pitch scan with $\Delta x =0.1$ mm and $\overline {\Delta \theta } = 20\ \mu$rad are performed in a pitch range of [-0.5, 0.5] mrad. The resultant instrument error is illustrated in Fig. 9(a) and enlarged in Fig. 9(b), in which the reference low-frequency instrument error of Fig. 8 is also plotted to show the self-consistency in trend. Figure 9(c) displays the PSDs of the three instrument errors, which indicates the period of the high-frequency fluctuations of the sample-beam AC is about 13.33 $\mu$rad.

 

Fig. 9. The 20-$\mu$rad-step multi-pitch scans can reconstruct the high-frequency instrument error. (a) Comparison of the reconstructed instrument errors with $\pm 2\sigma$ uncertainty regions indicated, (b) is a zoom-in view of (a), and (c) the PSD of the instrument errors.

Download Full Size | PDF

3.3 Effective compensation with the estimated instrument error

To check the effectiveness of applying the reconstructed instrument error from Fig. 8 for slope error compensation, another two sets of A2B and B2A usual scans are repeated 50 times with no active pitch. For a clear comparison, their averaged SUT slope residuals before and after the instrument error compensation are shown in Fig. 10(a). The result shows effective compensation of the "waviness" in green ovals for both scans. If we take the SUT slope error of the previous 101-pitch A2B scan as the reference for comparison, the discrepancies from the reference and the slope compensation amount according to the instrument error "look-up" curve are plotted in Fig. 10(b). Since the instrument error is related on the angle, not $x$-position, the "waviness" in green ovals on A2B and B2A slope errors locate at different $x$-positions.

 

Fig. 10. The A2B and B2A normal scans are compensated by the reconstructed instrument errors. (a) SUT slope error before and after compensation, and (b) the compensation amount and the discrepancies of the slope errors before and after compensation with respect to the 101-pitch A2B scans.

Download Full Size | PDF

4. Discussion

As demonstrated above in both simulation and experiment, the multi-pitch self-calibration technique applied to the NSP instrument can reconstruct the SUT slope, systematic instrument error, and the introduced pitch angles at the same time. Here we would like to address some features when using multi-pitch method with an NSP measurement system.

This technique is useful to decouple the mirror surface slope and the NSP systematic error from the acquired redundant measurement data. The experimental results indicates the reconstruction of the SUT slope and NSP instrument error is repeatable and reproducible. In addition, the instrument error extracted from the multi-pitch measurement can be used to correct single-pitch scans as was demonstrated in section 3.3. Furthermore, the multi-pitch method allows slope stitching ability to extend the measuring range of the NSP instrument.

It is important to emphasize that the temperature change induces angle drifts between the SUT and the measuring equipment. During a multi-pitch scan, its main effect, on the short term, is to distort the individual profiles. On the long term it also produces a drift of the average slope value of one profile with respect to others. Since pitch angle $\theta _n$ does not need to be measured and is a parameter of the fit, this drift is taken into account in the actual $\theta _n$ value. Therefore, stability should be only assured on the acquisition time of one profile which is in our case 5.5 mK over 15 minutes.

We would also want to highlight that equidistant pitches introduce a periodic error despite that the slope sampling points are not equidistant. Random variations on pitch angles can be employed to depress this periodic error. Moreover, the slope sampling points should be dense enough in order to fully reconstruct the instrument error.

In addition to the above-mentioned merits of applying the multi-pitch method to the NSP instrument, we would also like to discuss some of the limitations and concerns of this approach.

The linear term of the instrument error cannot be recovered with this method, but can be calibrated separately by using a curved mirror with a known curvature. This is a general issue of all slope measuring instruments.

The distance $d$ between the pitch rotation axis and the mirror surface needs to be determined before taking the multi-pitch calculation. Owing to this distance $d$, the sampling points on the SUT are not exactly at the same locations for different tilts. It will be better if the pitch rotation axis can be placed close to the SUT to reduce the lateral shift of the sampling positions. The error of this distance is one of the error sources of the multi-pitch self-calibration method.

Moreover, due to the lateral shift caused by the distance $d$, some of the local high-frequency spikes on the SUT may not be sampled in the scans at different pitch angles. The model-based estimation algorithm (B-spline in our case) will treat these "flickering" signals as the measurement noise. As a result, these surface details may not be resolved as illustrated in the red ovals in Fig. 10(b). At certain level, there is always a question mark for a model-based fitting method to judge whether it is a signal or noise.

To get the high-frequency instrument errors, the stability of the roll axes needs to be considered, as the angle readings in the other direction of the sample-beam AC may have similar high-frequency errors. A possible improvement on this is to make multi-pitch-and-roll self-calibration scans to get the two-dimensional instrument error map like in Ref. [26].

5. Conclusion

We present the application of multi-pitch self-calibration method on the NSLS-II NSP instrument for X-ray mirror nanoradian slope metrology. During its $x$-scan and pitch-scan, the NSP has a distance-fixed sample-beam AC. It is consequently an ideal application scenario for the multi-pitch method. Ambiguities of the multi-pitch method are addressed and several practical solutions and regularization are proposed to solve this ill-posed problem. In our experiment with a 10-mrad total slope spherical mirror, the SUT slope and the low-frequency instrument errors (period larger than 300 $\mu$rad) can be repeatably reconstructed with different pitch settings and mirror orientations. This multi-pitch self-calibration method is a good approach to measure mirrors whose total slope is larger than the measuring range of the sample-beam AC or the moving optical head.