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Abstract
This work presents a mixed stitching interferometry method with correction from one-dimensional profile measurements. This method can correct the error of stitching angles among different subapertures using the relatively accurate one-dimensional profiles of the mirror, e.g., provided by the contact profilometer. The measurement accuracy is simulated and analyzed. The repeatability error is decreased by averaging multiple measurements of the one-dimensional profile and using multiple profiles at different measurement positions. Finally, the measurement result of an elliptical mirror is presented and compared with the global algorithm-based stitching, and the error of the original profiles is reduced to one-third. This result shows that this method can effectively suppress the accumulation of stitching angle errors in classic global algorithm-based stitching. The accuracy of this method can be further improved by using high-precision one-dimensional profile measurements such as the nanometer optical component measuring machine (NOM).
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The increasingly high requirements of X-ray mirror figure accuracy for synchrotron and free electron lasers make high-precision two-dimensional (2-D) metrology methods increasingly important. Although different new 2-D metrology methods have been proposed [1], stitching interferometry remains the most commonly used metrology method for measuring the topography of these optics. Stitching the local contour of subapertures into a complete topography has the advantages of high lateral resolution over a 2-D surface and a larger measurable size and surface curvature range [2–4]. And the testing accuracy is also improving. In a recent study, Silva et al. completed stitching with sub-nanometer errors for curved mirrors with an effective length of around 100 mm [5].
Classic stitching interferometry fits the rotation angle, pitch angle, and height correction of each subaperture through the overlapping area of each subaperture [6–8]. Therefore, the shape error in a single subaperture will inevitably lead to the accumulation of errors for the entire stitching surface [9]. This accumulation of errors is mainly reflected in the fitting error of the stitching angle and causes the surface shape after stitching to have an error much larger than the error of a single subaperture [10–12]. To avoid this, auxiliary equipment can be used to correct the stitching angle. The methods that use an auxiliary equipment monitoring angle mainly include relative angle determinable stitching interferometry (RADSI) and the interferometry method based on angle monitoring using other auxiliary equipment. RADSI can obtain a more accurate stitching angle by monitoring the postures of the auxiliary plane mirror that rotates together with the mirror under test. [13–16]. It can reduce the accumulation of angle errors from a single subaperture. Some other methods use high-precision angle measuring equipment to monitor the angle. In 2017, Xue et al. used a three-beam interferometer to monitor the auxiliary flat mirror on the sample stage of a microscope interferometer [17]. In 2018, Huang Lei et al. enhanced the three-beam interferometer into an autocollimator or two tiltmeters to monitor the auxiliary mirror [18–19].
However, stitching with auxiliary monitoring equipment requires a complicated stitching system with extremely high stability [20]. A method such as RADSI monitors the stitching angle of the tested subaperture during stitching interferometry, which is still a measurement method that monitors the angle of the local area of the full aperture. These methods can still produce errors in determining the local angle of the subaperture obtained by the angle monitoring device, which may cause accumulation problems during stitching.
Another method to correct the stitching error uses low-frequency profiles measured by other instruments to aid in stitching, which is called ‘mixed stitching’. This method does not monitor the angle of the local subaperture but directly obtains the angles from the 1-D profile along one direction of the entire tested mirror. Therefore, its accuracy is mainly limited by the accuracy of the selected 1-D profile measuring device and the specific algorithm used to obtain the angles. This means that compared to ordinary 1-D measurements, mixed stitching can extend 1-D profiles to 2-D stitching results without the accumulation of stitching angle error from the local measurements. In principle, this method combines the advantages of high local accuracy of the 2-D interferometry measurement and the global accuracy of a one-dimensional (1-D) profile measurement. This can be particularly useful for the metrology of X-ray mirrors, which typically have a long rectangular shape. M. Bray proposed that profilometry or glancing angle interferometry could be used to measure the overall figure of the mirror under test to correct the stitching profile [21], and a simple simulation was performed to correct the stitching result of eleven subapertures. By using a 1-D profile with a PV (peak-to-valley) of 429 nm, the PV value of the stitching result was corrected from the original 269 nm to 448 nm, which is closer to the 1-D profile used. Assoufid et al. proposed using a long trace profiler (LTP) or glancing angle interferometry for mixed stitching. However, the experimental mixed stitching results were not reported [22]. The experimental feasibility and accuracy of this method were not fully demonstrated, and its possible errors were not studied.
This paper will demonstrate the mixed stitching method based on stitching interferometry using a Fizeau interferometer with correction from 1-D profile measurements for high-precision X-ray mirrors. The contact profilometer was mainly used to provide the measurement of the 1-D profile given its availability in our laboratory. The measurement accuracy of MSI-CP (Mixed Stitching Interferometry with Contact Profilometer) is simulated and experimentally analyzed. The results are compared with classic stitching based on the algorithm to show its improvement in global accuracy. As a high-accuracy profile measurement instrument is used, such as NOM, the stitching algorithm itself achieves a small residual figure error of less than 1 nm (RMS), which demonstrates the potential of this method.
2. Limitation of the global stitching interferometry measurement method
Classic algorithm-based stitching determines the relative angle in the x-direction and y-direction and the relative height of each subaperture by performing algorithm fitting on the measurement data of the overlapping area of the subapertures. Assume that the total number of subapertures for stitching is N. The shape data of each subaperture can be recorded as ${G_i}$ (i = 1,2,3…N). Without considering the subaperture displacement, the i-th subaperture after adjusting the relative angle and height can be expressed as Formula (1):
where the coefficients ${a_i}$ and ${b_i}$ are the angle adjustment coefficients of the i-th subaperture in the x-direction and y-direction, respectively, and the coefficient ${c_i}$ is the adjustment of the i-th subaperture along the height (z-direction). The coefficients ${a_i}$, ${b_i}$, and ${c_i}$ are called stitching coefficients. The coordinate $({x,y} )$ represents the spatial position of a measurement point on the mirror surface, and the height before and after adjustment is represented by ${G_i}$ and $G_i^\ast $.The global subaperture stitching algorithm takes the overlapping areas of all subapertures into the calculation at one time and determines ${a_i}$, ${b_i}$, and ${c_i}$ to minimize the total error of all overlapping areas. However, this algorithm still results in angle error accumulation. As the curvature of the mirror under test increases, the defocusing error [23] and the retrace error [24] within a single subaperture increase, which will cause a serious accumulation of stitching angle error. Therefore, for mirrors with a larger size and larger curvature, it is difficult for global stitching to guarantee the ideal stitching accuracy in the full-size range if the shape error within each subaperture is not strictly limited. A slope threshold [11], which controls the maximum surface slope of each subaperture, can be optimized to reduce the effect of these measurement errors. However, when the accumulation of error is significant, the slope threshold alone is not sufficient.
Through simple stitching of the interferometer data obtained, the above error can be evaluated intuitively. A spherical mirror A with a length of 250 mm, a width of 40 mm, and a radius of curvature (RoC) of 60 m is measured by global stitching using an interferometer with a 12-inch flat reference mirror, and the pixel resolution is approximately 0.267 mm/pixel. Mirror A is typical of mirrors with large tangential RoC required in synchrotron radiation. Its centerline has been measured at Shanghai synchrotron radiation source (SSRF) using NOM, which herein is referred to as ‘NOM’. NOM is a 1-D profile measurement method with a sampling interval of 1 mm. It has a measurement accuracy of ∼0.1% for elliptical parameters and below 50 nrad RMS for the residual figure error after removing the best-fitted ellipse. Therefore, NOM was used as a reference standard for real profiles. The difference between the centerline of the stitching result and NOM is shown in Fig. 1(a), which exhibits an obviously concave error, whose shape approximates an ellipse. This indicates that the curvature measurements between the stitching and NOM at different positions are significantly different.
Fig. 1. (a) Height difference between the global stitching and NOM. (b) Measured RoC of mirror A at different positions.
These curvature differences often originate from retrace and defocusing errors within the subapertures. When the focus situation is only ideal near one subaperture in the central region of the mirror under test, there may be a relatively large shape error in other areas. As shown by the blue dots in Fig. 1(b), the RoC of different subapertures varies significantly, and the profile varies in different local regions itself. The RoC distribution (orange line) of the centerline of the global stitching is very close to the blue dots, which is different from NOM (yellow line). This illustrates that the local curvature measured by stitching is significantly affected by the shape error within each subaperture, which accumulates and ultimately leads to the profile differences shown in Fig. 1.
In summary, when the curvature of the tested mirror is large and the stitching size is large, global stitching needs to strictly control the shape error within each subaperture or use other auxiliary methods to obtain high-precision stitching results.
3. MSI-CP measurement method
3.1 Basic principle
In principle, MSI-CP does not have the accumulation of angle errors caused by the measurement errors in each local area. It mixes two kinds of measurements from interferometers and profilometers. The latter can accurately measure the curvature of the profile of the mirror under test, and with its auxiliary correction, it can greatly suppress the overall profile error in stitching. Similar to algorithm-based stitching, MSI-CP also needs to determine ${a_i}$, ${b_i}$, and ${c_i}$, which can be called ${a_i}^{\prime}$, ${b_i}^{\prime}$ and ${c_i}^{\prime}$. However, among these parameters, ${a_i}^{\prime}$ is determined to minimize the error between the stitching profile and the 1-D profile measured by the contact profilometer. ${b_i}^{\prime}$ and ${c_i}^{\prime}$ are determined by the least square method to minimize the total error of all overlapping areas after ${a_i}^{\prime}$ is corrected. The basic idea of different stitching is shown in Fig. 2. The initial measured geometric relation of the adjacent subapertures A, B, and C is shown in Fig. 2(a). For global algorithm-based stitching, after ${a_i}$, ${b_i}$ and ${c_i}$ are determined, the new geometric relation is obtained, as shown in Fig. 2(b). The geometric relation is adjusted to minimize errors in the overlapping area. For MSI-CP, the corrected relative position is shown in Fig. 2(c) after ${a_i}^{\prime}$, ${b_i}^{\prime}$ and ${c_i}^{\prime}$ are determined. Each subaperture is consistent with the trajectory of the 1-D profile in the x-direction.
Fig. 2. Basic idea of different stitching. (a) Tested subapertures. (b) Subapertures based on classical stitching fitting. (c) 1-D profile and subapertures after correction. (d) Schematic diagram of the algorithm used for calculating the stitching coefficients after 1-D profile correction.
A schematic diagram of the method for calculating the stitching coefficients after 1-D profile correction is shown in Fig. 2(d). For the measured subaperture ${G_{i + 1}}$ (i = 1…N-1), the subaperture after adjusting the angle coefficient of ${G_{i + 1}}$ in the x-direction by MSI-CP is $M_{i + 1}^{\prime}$. The coefficient ${a_{i + 1}}^{\prime}$ is the corrected angle adjustment coefficient of the subaperture in the x-direction. The coefficients ${a_{i + 1}}^{\prime}$ are determined by Formula (2):
With the minimum f as the aim, the coefficients ${a_j}^{\prime}$ and ${h_j}$ can be calculated by the least square method. After subaperture $M_{i + 1}^{\prime}$ with coefficient ${a_{i + 1}}^{\prime}$ is obtained, coefficients ${b_{i + 1}}^{\prime}$ and ${c_{i + 1}}^{\prime}$ can be calculated to minimize the total error of all overlapping areas of each subaperture $M_j^{\prime}$ (j = 1…N). The final corrected subaperture is called ${M_{i + 1}}$, as shown in Fig. 3. The expression of its geometric relation is shown in Formula (4):
Fig. 3. Comparison of SurfA and the results of MSI-CP and the global algorithm-based stitching. (a) Typical shape error of the subaperture. (b) Comparison of the centerlines. (c) Height difference of the 2-D topography between the result of the global algorithm-based stitching and SurfA. (d) Height difference of the 2-D topography between the result of MSI-CP and SurfA.
3.2 Theoretical error simulation
To verify the reliability of the abovementioned stitching principle, errors in the global algorithm-based stitching and MSI-CP are simulated. The simulated spherical mirror is similar to the real mirror A, with a length of 250 mm, an effective width of 20 mm, and a RoC of 60 m. The surface of the ideal sphere generated is called SurfA, and the centerline La of SurfA is taken as the 1-D profile data. Without any other errors, the centerline La is an ideal profile. The MSI-CP simulation result of SurfA can be obtained by using La to correct the simulated subapertures of SurfA. A constant drift angle of 1 × 10−8 rad more for each subaperture than the previous subaperture and a random angle with a standard deviation of 2 × 10−7 rad are added to the simulated test results of subapertures. The final accumulation of drift errors may approach 1 µrad. The simulated angle error is close to the error in angle change of motorized stages in the real stitching system. Gaussian noise (2.5 nm RMS) and systematic errors in the form of (x2 + y2) are also added to the subapertures to evaluate the response of different stitching methods to surface shape errors. The typical shape error is shown in Fig. 3(a), which are errors that may be encountered in the real measurement as induced by defocusing or retrace error.
Figure 3(b) shows the comparison of the centerlines. The centerline of SurfA is the ideal profile La. The centerline of MSI-CP is almost identical to La, and the difference between the two lines is only 0.67 nm RMS. In contrast, the global algorithm-based stitching shows a large second-order profile error. The height difference of the 2-D topography between the result of the global algorithm-based stitching and SurfA is shown in Fig. 3(c), which is 228.49 nm RMS. The fitted RoC of the result of the global algorithm-based stitching is 59.65 m. The error is essentially a sphere with a RoC of about 10 km. This mainly comes from the accumulation of stitching angle errors caused by systematic errors. The 2-D height difference between the result of MSI-CP and SurfA is shown in Fig. 3(d), which is only 1.84 nm RMS. The fitted RoC of the result of MSI-CP is also 60 m. The accumulation of errors has been greatly corrected. The remaining errors come from Gaussian noise and the shape error within each subaperture, and the error introduced by the MSI-CP algorithm itself is almost negligible.
3.3 Setup and procedure
The stitching setup of MSI-CP consists of two components: the stitching platform and the contact profilometer. The stitching platform is composed of a Fizeau interferometer (Zygo make) with a 12-inch flat reference mirror and the corresponding motorized stages. The pixel resolution of the interferometer is usually set to 1200*1200, which corresponds to 0.267 mm/pixel. Since the reference mirror of the interferometer is larger than the size of most of the mirrors under test, there is no translation stage. The motorized stages consist of only a rotating stage and a pitching stage. The stitching interferometry can be completed by rotating mirrors. All stages are placed on a high marble platform to ensure that the mirrors under test are placed at a suitable height and the interferometer can measure the full aperture of the mirror, as shown in Fig. 4(a). Figure 4(b) shows the PGI surface contact profilometer (Taylor-Hobson). The 1-D profile of the mirror under test is measured by this contact profilometer. Its measurement limit length is 200 mm, the minimum measured data interval is 0.25 µm, and the maximum can be set to 50 µm. The PV value of the error measured by the contact profilometer is less than 0.15 µm (based on the standard sphere). With the hardware described above, the procedure of MSI-CP is as follows:
Fig. 4. Actual objects of (a) the stitching interferometry measurement system and (b) the contact profilometry measurement system.
Step 1 Place the mirror under test on the stitching platform and use the tilt/rotation stage to adjust the initial state of the mirror.
Step 2 Use the interferometer to measure the subaperture ${G_j}$ (j = 1…N) of the mirror under test.
Step 3 Place the mirror under test on the sample stage of the contact profilometer and measure the 1-D profile L of the mirror in the area of measured subapertures.
Step 4 Calculate the stitching angle coefficient ${a_j}^{\prime}$ of the subaperture to minimize the error between the stitching profile and the 1-D profile measured by the contact profilometer. Adjust the subaperture ${G_j}$ to $M_j^{\prime}$ with the coefficient ${a_j}^{\prime}$.
Step 5 Calculate the coefficients ${b_j}^{\prime}$ and ${c_j}^{\prime}$ to minimize the error of the overlapping area. Adjust the subaperture $M_j^{\prime}$ to ${M_j}$ with the coefficients ${b_j}^{\prime}$ and ${c_j}^{\prime}.$
After the subapertures ${M_j}$ (j = 1…N) are stitched, the final stitching surface with correction from the 1-D profile measurement can be obtained.
4. Accuracy analysis
4.1 Accuracy of the contact profilometer
Since the stitching angle coefficient ${a_j}^{\prime}$ of MSI-CP is provided by the 1-D profile measured by the contact profilometer, which is not an ideal profile, the measurement accuracy of MSI-CP is greatly affected by the measurement accuracy of the contact profilometer. To evaluate the test accuracy of the contact profilometer, the contact profilometer was used to repeatedly measure the centerline of spherical mirror A in the central 170 mm area five times in two repeated tests (Test A and Test B) on different days. The specifications of mirror A and the relevant stitching parameters are shown in Table 1. The repeatability is analyzed by comparing the difference between repeated measurements, and the measurement reliability is verified by comparing it with the NOM measurement. The comparison between the profilometer measurement results and the NOM results is shown in Fig. 5. The i-th measurement is recorded as Li. The average of five measurements is recorded as L1∼5.
Fig. 5. Comparison of NOM results and multiple measurements by contact profilometer. (a) RoC of the original results. (b) Residual errors after removing the best-fitted circle from the original results. (c) Height difference between different results.
Table 1. The curved mirrors measured by the mixed stitching, and corresponding experimental parameters
Figure 5(a) shows the comparison of the RoC of the original measurement results. The RMS value of RoC obtained by the contact profilometer for five measurements is 0.05 m RMS in Test A and 0.02 m RMS in Test B, and the maximum deviation between the RoC of Test A and Test B is 0.09 m. By simply averaging the results of multiple measurements, the deviation can be reduced. The RoC of the averaged five measurements for Test A and Test B is 59.53 m and 59.51 m, respectively, with a difference of 0.02 m, which has good consistency. The RoC of the averaged five measurements is also closer to the NOM results than the global results. The RoC of the result of the global stitching algorithm is 59.81 m. The RoC of NOM is 59.34 m. The deviation between the RoC of the global stitching algorithm and the RoC of NOM is 0.48 m. The deviation between the contact profiler and SSRF is 0.20 m and 0.18 m, respectively. Compared with the global stitching, the contact profilometer has a higher profile measurement accuracy and can more accurately obtain the curvature data of the surface to be measured.
Contact profilometers still have some repeatability errors even after removing the measured quadratic profile that causes RoC deviation. This repeatability error is largely due to the significant oscillatory residuals after removing the best-fitted circle from the original measurement results, which is between ±50 nm. The RMS value of residual errors of the ten measurements ranges from 14.29 nm to 33.33 nm. The oscillation error originates from the contact profilometer measurement itself and can be reduced by smoothing, interpolation, or other postprocessing methods. After multiple measurements and averaging, this problem can be suppressed. After averaging the five measurements, the residual error is shown as the blue line and black line in Fig. 5(b). The oscillations have been significantly reduced in their amplitude, and the RMS value is 14.36 nm in Test A and 11.95 nm in Test B.
Without averaging, the RMS of the height difference between the residual errors of Test A and Test B ranges from 18.15 nm to 51.76 nm. After averaging, the RMS of the height difference between Test A and Test B is reduced to 13.53 nm, as shown in Fig. 5(c). Figure 5(c) also shows the height difference of the original profiles and residuals between NOM and the averaged result of Test B. The RMS value of the difference between the averaged result and NOM drops to 10.54 nm, while the difference between a single unaveraged line and NOM can reach 30.92 nm RMS. This demonstrates that the difference between the residual of the contact profilometer measurement and NOM is also reduced after averaging.
4.2 Accuracy of MSI-CP
The accuracy of MSI-CP can also be evaluated in two parts: one is the repeatability of the stitching method itself, and the other is the measurement reliability accuracy of the stitching result compared with other measurements. We also use the measurement results of spherical mirror A to evaluate these two parts. The repeatability is divided into the repeatability error caused by the difference in the subapertures provided by the interferometer and the repeatability error caused by the 1-D profile difference provided by the contact profilometer.
4.2.1 Repeatability error caused by the interferometer
By measuring the mirror surface from left to right and measuring the mirror surface from right to left, we can obtain the two sets of subaperture data provided by the two stitching measurements. A 1-D profile with a length of 250 mm is generated to complete the MSI-CP of mirror A. For the same 1-D measurement profile, two sets of stitching results can be obtained. The difference between the two stitching results is shown in Fig. 6. Figure 6(a) shows the 2-D repeatability error of 2.06 nm RMS of the result of the global algorithm-based stitching, which is significantly larger than the 0.24 nm RMS of stitching 80 mm spherical mirrors with a RoC of 100 m [8]. This is because the existing devices are not stable enough for larger mirrors. Figure 6(b) shows that the 2-D repeatability error of MSI-CP caused by the difference in the subapertures reached 1.42 nm RMS, which is slightly smaller than the 2-D repeatability error of the global algorithm-based stitching.
Fig. 6. Repeatability error of the original topography between the results stitching from right to left and from left to right. (a) Global algorithm-based stitching. (b) MSI-CP using the same 1-D profile.
4.2.2 Repeatability error caused by the contact profilometer
The repeatability error caused by the 1-D profile difference provided by the contact profilometer is more important. Even if the repeatability is improved by averaging multiple measurements, the residual error of the 1-D profile provided by the contact profilometer after removing the best-fitted profile still has a repeatability error that oscillates in the range of ±20 nm. This means that the stitching angle coefficient ${a_j}^{\prime}$ provided by the 1-D profile also has a repeatability error of a corresponding magnitude in a local range. A better processing method is to not use only one 1-D profile but to increase the number of 1-D profiles to fit the coefficient ${a_j}^{\prime}$ at the same time. Meanwhile, Formula (3) can be revised as Formula (5).
To verify the above analysis, the contact profilometer was used to measure two 1-D profiles of the spherical mirror on both sides of the centerline of the mirror. Assuming the centerline is at y = 0 in the y direction, the two profiles are at y = ±10 mm. The length of each profile is 170 mm. The two 1-D profiles are called ${L_A}$ and ${L_B}$, respectively. ${L_A}$ and ${L_B}$ are both the mean values of five repeated measurements, and the five measurements are at the same position on the mirror under test. The stitching results with correction from 1-D profiles ${L_A}$ and ${L_B}$ are called ${M_{{L_A}}}$ and ${M_{{L_B}}}$. The stitching result with correction from ${L_A}$ as well as ${L_B}$ is called ${M_{{L_{AB}}}}$. Under the same calculation method, the stitching results obtained by repeated measurement are called $M_{{L_A}}^{\prime}$, $M_{{L_B}}^{\prime}$, and $M_{{L_{AB}}}^{\prime}$. The height difference of different stitching results is shown in Fig. 7. Figure 7(a)(b) shows that when only one 1-D profile is used, there is still a repeatability error of larger than 10 nm RMS before and after removing the best-fitted sphere even if the profile has been averaged. Figure 7(c) shows the repeatability error with correction from both 1-D profiles ${L_A}$ and ${L_B}$, which drops to 6.16 nm RMS after removing the best-fitted sphere. This proves that increasing the number of 1-D profiles for correction is helpful to improve the repeatability error introduced by profiles. When the number of 1-D profiles used for correction increases, the random error will be suppressed. Because the result will be a comprehensive result of more 1-D profiles, which reduces the repeatability difference caused by only a specific profile.
Fig. 7. Height difference of the measurement topography. (a) ${M_{{L_B}}}$ minus $M_{{L_B}}^{\prime}$. (b) ${M_{{L_B}}}$ minus $M_{{L_B}}^{\prime}$ after removing the best-fitted sphere. (c) ${M_{{L_{AB}}}}$ minus $M_{{L_{AB}}}^{\prime}$ after removing the best-fitted sphere. (d) ${M_{{L_{ABs}}}}$ minus $M_{{L_{ABs}}}^{\prime}$ after removing the best-fitted sphere. (e) ${M_{{L_{ABsd}}}}$ minus $M_{{L_{ABsd}}}^{\prime}$ after removing the best-fitted sphere.
The repeatability of the stitching result can be further improved by some specific processing methods for 1-D profiles, such as smoothing the profiles over the 25 mm range after removing the best-fitted circle. The smoothing process is as follows: first, fit the second-order profile (circle or ellipse) of the 1-D profile, and use locally weighted scatterplot smoothing (LOWESS) for the residual after removing the best-fitted circle or ellipse. The smoothing range represents the size of the window when calculating data points. The final smoothing result is the sum of the residual after smoothing and the best-fitted second-order profile. The stitching results after further smoothing the profiles are ${M_{{L_{ABs}}}}$ and $M_{{L_{ABs}}}^{\prime}$. As shown in Fig. 7(d), the repeatability error reached 4.18 nm RMS after removing the best-fitted sphere. When the smoothing range is 50 mm, the stitching results after smoothing are called ${M_{{L_{ABsd}}}}$ and $M_{{L_{ABsd}}}^{\prime}$., and the repeatability error is 3.04 nm RMS, as shown in Fig. 7(e). This demonstrates that expanding the smoothing range is a very effective way to improve repeatability.
4.2.3 Reliability error caused by the contact profilometer
The use of multiple 1-D profiles in stitching for auxiliary correction is also helpful to improve the measurement reliability accuracy. The comparison of the centerline of the stitching results with correction from different 1-D profiles and NOM is shown in Fig. 8(a). The difference between the MSI-CP and NOM is a typical approximate parabolic profile error. This difference is much smaller than that of global stitching and NOM.
Fig. 8. Comparison of the centerlines of stitching results with correction from different 1-D profiles and NOM. (a) Height difference of the original profiles. (b) Residual errors after removing the best-fitted circle or sphere from the original results.
After removing the best-fitted circle or sphere from the original results, the comparison is shown in Fig. 8(b). After using both ${L_A}$ and ${L_B}$ for correction, the difference between ${M_{{L_{AB}}}}$ and NOM is 9.82 nm RMS. This value is smaller than the difference of 11.87 nm RMS between ${M_{{L_A}}}$ and NOM, as well as the difference of 12.47 nm RMS between ${M_{{L_B}}}$ and NOM. Compared with NOM, MSI-CP still shows large oscillatory errors.
When increasing the smoothing range to 25 mm, this oscillatory difference is reduced, as shown by the purple line in Fig. 8(b). The difference in residuals between the stitching results and NOM decreases from 9.82 nm RMS to 7.74 nm RMS. When the smoothing range is doubled to 50 mm, the difference in residuals between the stitching results and NOM is 4.34 nm RMS, which is smaller than the difference of 8.33 nm RMS between the global stitching and NOM. Accuracy is greatly improved with smoothing in the 50 mm range. This is because smoothing greatly reduces random angle errors. The effect of this improvement can be more clearly reflected in Fig. 9.
Fig. 9. Comparison of angles at different subaperture positions obtained from different measurements
As shown in Fig. 9, the accumulation of angle errors in mixed stitching is much smaller than in global algorithm-based stitching, which means that the profile obtained by MSI-CP is closer to NOM. This shows that the MSI-CP has higher accuracy on the overall profile and greatly reduces the accumulation of angle errors. However, the angle differences of ${M_{{L_{AB}}}}$ and NOM show significant oscillations. This demonstrates that MSI-CP still has relatively obvious random angle errors. The red and black dotted lines reflect the decrease in random angle error as the smoothing range increases.
However, the black line has a larger accumulation of angle errors than the red line. The angle difference of the last subaperture of ${M_{{L_{ABsd}}}}$ reaches 7.75 µrad, which is larger than the difference value of 5.59 µrad of ${M_{{L_{ABs}}}}$. This means that a range of smoothing that is too large is likely to introduce an overfitting error to the profile. However, this error is much smaller than the accumulation of angle errors of the global stitching. Therefore, in the general MSI-CP measurement in this work, the common processing method of smoothing the 1-D profile within 25∼50 mm is used.
5. Measurement results and discussions
5.1 MSI-CP on an elliptical mirror
An elliptical cylindrical Si mirror B with a length of 250 mm and a central RoC of 68 m was measured using global algorithm-based stitching and MSI-CP. The specifications of mirror B and the relevant stitching parameters are shown in Table 1. It can be used as a KB mirror for synchrotron radiation. The elliptical parameters are shown as follows: object distance P = 1 m, image distance Q = 8 m, and the grazing incident angle θ; = 1.5°. The stitching interferometry uses 73 subapertures with a slope threshold of 2 × 10−4 rad. The mean overlap rate is 87%. The contact profilometer was used to measure two 1-D profiles of mirror B at y = ±10 mm. Since the size of mirror B is larger than the measurement range of the contact profilometer, the final 1-D profile is formed by stitching the 1-D profiles of two measurements, and the overlap rate of the 1-D profiles of two measurements is 64.86%. Both 1-D profiles are the smoothed averages of repeated measurements. Mirror B also measured the centerline at SSRF using NOM. The profiles of the three measurement results are taken from the central 210 mm for comparison. Figure 10(a) shows the height difference between the original centerline profiles of MSI-CP and global algorithm-based stitching compared with NOM. The RMS of the difference between MSI-CP and NOM is 58.34 nm, and the RMS of the difference between the global algorithm-based stitching and NOM is 185.58 nm, which is three times that of the former. The 1-D profile measured by the contact profilometer at y = 10 mm was also compared to the MSI-CP at this position in Fig. 10(a), and the difference is 7.08 nm RMS, which comes from the oscillation errors inherent in contact profilometers.
Fig. 10. Comparison of the measurement results of the elliptical cylindrical mirror using global algorithm-based stitching, contact profilometer, MSI-CP, and NOM. (a) Height difference of the original profiles. (b) Residual errors of centerlines after removing the best-fitted ellipse.
The measurement results after removing the respective best-fitted ellipse are also compared. When fitting, the shape of the ellipse is fixed (P, Q, and θ are fixed), and only the relative position variable Xo of the center of the ellipse segment is fitted [11]. The fitting range of Xo is ±5 mm. The fitting parameters of the three measurement results are as follows: the Xo of the global algorithm-based stitching is 4.32 mm, the Xo of MSI-CP is 0.34 mm, and the Xo of NOM is -1.35 mm. Comparing the absolute position of the center of the ellipse segment on the coordinate axis, the difference between global algorithm-based stitching and NOM is 0.16%, and the difference between MSI-CP and NOM is 0.05%. The comparison of the fitting parameters is consistent with the comparison of the original profiles, which both reflect that the stitching result with correction from the 1-D profile is closer to NOM than the global algorithm-based stitching.
The comparison after removing the best-fitted ellipse is shown in Fig. 10(b). The RMS of the difference between MSI-CP and NOM is 8.75 nm, and the RMS of the difference between the global algorithm-based stitching and NOM is 16.09 nm. As with the comparison result of the original profiles, the residual error of MSI-CP after removing the best-fitted profile is closer to NOM. However, this result is slightly worse than the difference value of 4.52 nm RMS of the spherical mirror. This may be due to the larger measured length of elliptical mirror B compared to spherical mirror A. The measurement results show that MSI-CP has higher accuracy on the overall profile of the measurement of large-scale and large-curvature curved mirrors than the global algorithm-based stitching.
5.2 MSI-NOM
The measurement results of NOM and the mixed stitching figure corrected by NOM (MSI-NOM) are also presented in Fig. 11. Figure 11(a) shows the 2-D result of MSI-NOM after removing the best-fitted elliptical cylinder. The Xo of MSI-NOM is -1.34 mm, which is almost identical to that of NOM. The difference between NOM and MSI-NOM is only 0.88 nm RMS, as shown in Fig. 11(b). After removing the best-fitted ellipse, the difference is 0.83 nm RMS, as shown in Fig. 11(c). This is consistent with the evaluation of the error in the simulation, which fully reflects the reliability of the current 1-D auxiliary correction method.
Fig. 11. Comparison of the measurement results of mirrors using MSI-NOM, MSI-CP and NOM. (a) 2-D stitching result of mirror B after removing the best-fitted elliptical cylinder. (b) Original centerline profiles of mirror B. (c) Residual errors of centerlines of mirror B after removing the best-fitted ellipse from the centerline. (d) The measured 1-D profile at 10 mm offset from the centerline of mirror A after removing the best-fitted circle.
The difference between the MSI-NOM and NOM is considerably less than the difference between MSI-CP and the contact profilometer shown in Fig. 10(a). It is clear that a measurement method such as the NOM, which can measure 1-D low-frequency profiles more accurately than a contact profiler, is important for high accuracy mixed stitching. There are still some small fluctuations in the profile of the height difference between MSI-NOM and NOM, possibly originating from the repeatability error of subapertures, which can be further reduced by improving the stability of the interferometer measurements. Furthermore, the lateral resolution of the stitching results is close to 25% of that of NOM. This also led to the difference in the two measurements.
To further illustrate the accuracy of MSI-NOM. The spherical mirror A is also stitched and corrected by the centerline measured by NOM. NOM also measured the 1-D profile at 10 mm offset from the centerline of spherical mirror A, which is called Loffset. The RoC of Loffset measured by MSI-NOM and NOM is both 59.34 m. After removing the best-fitted circle, the difference between MSI-NOM and NOM is 3.37 nm RMS, as shown in Fig. 11(d). The error can be related to mismatches in the position of the single 1-D profile and errors in the overlapping areas. The accuracy will be further improved if two or more NOM-tested 1-D profiles are used for correction.
6. Summary
In this paper, mixed stitching interferometry with correction from one-dimensional profile measurements for high-precision X-ray mirrors was studied. A contact profilometer was mainly used to provide the measurement of the 1-D profile given its availability in our laboratory. The corresponding method is called MSI-CP.
The repeatability error of MSI-CP mainly comes from the repeatability error of the subapertures provided by the interferometer and the repeatability error caused by the 1-D profile difference provided by the contact profilometer. The latter type of error can be improved by averaging multiple measurements of the 1-D profile used and using multiple profiles at different measurement positions. Finally, an elliptical cylindrical Si mirror with a length of 250 mm and a central RoC of 68 m was measured using MSI-CP. Compared with NOM, the accuracy of the original profile measured by MSI-CP is more than three times higher than global algorithm-based stitching. After removing the respective best-fitted ellipse, the accuracy of MSI-CP compared with NOM is 8.75 nm RMS, which is better than the 16.09 nm RMS of the difference between the global algorithm-based stitching and NOM. The results show that this mixed stitching can effectively improve the accumulation of stitching angle errors in classical stitching. The accuracy of the mixed stitching method can be further improved by using higher precision 1-D profile measurements such as NOM. The mixed stitching interferometry method demonstrated in this paper can be used for the high-precision metrology of X-ray mirrors through combination with a high-accuracy 1-D profiler. Such metrology methods can also be employed for characterizing strongly curved mirrors with moderate precision requirements but a larger measurable range, as required in other fields.
Funding
National Key Research and Development Program of China (2022YFF0709101); National Natural Science Foundation of China (12235011).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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