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In this work, we proposed for stitching interferometry to use a triple-beam interferometer to measure both the distance and the tilt for all sub-apertures before the stitching process. The relative piston between two neighboring sub-apertures is then calculated by using the data in the overlapping area. Comparisons are made between our method, and the classical least-squares principle stitching method. Our method can improve the accuracy and repeatability of the classical stitching method when a large number of sub-aperture topographies are taken into account. Our simulations and experiments on flat and spherical mirrors indicate that our proposed method can decrease the influence of the interferometer error from the stitched result. The comparison of stitching system with Fizeau interferometry data is about 2 nm root mean squares and the repeatability is within ± 2.5 nm peak to valley.
© 2017 Optical Society of America
1. Introduction
A high accuracy profile mirror measurement for example for synchrotron mirror, lithography optics or telescope mirror is still a very challenging task. For high accuracy profile mirror measurements, it is mandatory to use accurate metrology tools or advanced algorithms, such as large aperture interferometers or various stitching methods. Interferometers with a large aperture are usually very complicated to calibrate. To bypass the tradeoff between spatial resolution and field of view on a single measurement, stitching interferometry was proposed in the 1980s [1, 2], which enables combining large measuring range with high spatial resolution. Stitching algorithm can offer good measurement results for freeform mirrors [3] even if the overlapping sub-apertures contain large relative tilt [4]. Stitching interferometers were therefore extensively studied [1–17] and commercialized [5].
The classical stitching algorithms [2, 6] calculate information on the relative tilt and piston between two sub-aperture topographies from their overlapping region. Even though the tilt influence is negligible for small apertures and small tilts, the tilt estimation error still exist and will propagate during the stitching process [6]. The accumulation of the errors could be insignificant if only a few sub-aperture topographies are stitched together, but this effect cannot be ignored when a large number of sub-aperture topographies are taken into account. Furthermore, reconstruction accuracy may become even worse in the presence of additional error sources such as retrace error or temperature effect [7]. The random stitching error will significantly decrease by increasing either the resolution of each sub-aperture or the overlapping area on the Surface Under Test (SUT) [8]. However, in real measurements, large overlaps between sub-apertures increase the total measuring time significantly. This long measuring time may cause drift on the measuring system mainly due to the change of environment (including temperature, humidity, and vibration) [9–11]. In addition to the random errors, the loss of common path in the interferometer induces further aberrations in a non-null test. The relative angle between the surface under test and reference will introduce a specific fringe pattern in the interferometry data. These so-called “retrace errors” are difficult to calibrate and can often exceed the required accuracy (and thus cannot simply be ignored) [12]. The angle problem is mainly caused by the stage translation and small wedge of mirror body [10]. With the different error sources mentioned above, a stitching algorithm alone cannot provide reliable results [13]. The challenge is even more difficult with strong curvatures mirrors; this will result in obvious stitching errors if a pure software stitching method is used, unless assisted by other instruments, such as a multi-axis positioning device [14] or RADSI system [15] and MSI system [16]. During SUT or interferometer motion for the stitching process, the stage imperfection is another source of errors. The repeatability and accuracy of the linear stage are crucial for ultimate stitching performances. The positioning error of the linear stage should be less than 1/10th of the physical size of the interferometer pixel [17]. The traceable multi-sensor method was proposed [6, 18] which can reconstruct flat and slightly curved surface profiles by a combination of sub-aperture measurements. However, the motion error of the stage was not considered into the stitching process and therefore included in the final result. To overcome this issue, a similar approach with the combination of a distance sensor and an autocollimator was proposed in [19].
In this work, we propose a simple and effective one-dimensional (1D) stitching interferometry assisted by a Triple-Beam Interferometer (TBI) to measure both angle and distance simultaneously. Using the measured tilt angle and position, the stitching algorithm can resolve the problem caused by stage error parasitic motion not easy to solve in the classical stitching methods where a lot of sub aperture are stitched together. The stitching accuracy mainly depends on the accuracy of the tilt measurement. We will address the performance of the proposed Sub-aperture Stitching Interferometry with the TBI (SSI-TBI) in both simulation and experiment.
2. The 1D sub-aperture stitching system with the TBI
The schematic experimental setup is shown in Fig. 1. It is designed to measure long mirrors with sub-aperture stitching technique. The whole system contains a microscope interferometer for sub-aperture measurement, a linear stage for 1D scanning, a TBI to measure tilt and position simultaneously for each sub-aperture which is rigidly clamped to the system platform, a reflector (mirror) for laser reflection which is mechanically fixed on the stage, and a vibration isolation table. Obviously, the flatness of the reflecting mirror should be considered because it will influence the angular measurement.
Fig. 1 Scheme of the 1D sub-aperture stitching system with the TBI. The SUT is fixed on the linear stage.
During the SUT motion, each sub-aperture height map can be measured by the interferometer. However, the imperfection of the stage will introduce a small tilt between two neighboring sub-aperture maps, which will affect the stitching process. Fortunately, the TBI can measure the tilt angle and distance at each position. Let
where N is the total number of sub-aperture, is the local x-coordinate in each sub-aperture, dn is the nth scanning distance, is the global x-coordinate, is the nth sub-aperture original height map which was measured by the interferometer, is the height map after stitching in each stage position, tn is the nth stage tilt. Both of t1 and d1 are 0. pn is the relative piston between adjacent sub-apertures and can be calculated with the overlapping data by Eq. (3). is the overlapping height map at the nth position. Combining Eqs. (1)-(3), the whole mirror profile can be stitched out with these measured local height maps and parameters.3. Estimation of the accuracy and repeatability evaluation
To verify the feasibility of the proposed method, we simulate a set of sub-aperture height maps, numerically, using MATLAB®. The retrace error of the interferometer is incorporated and changed with the mirror curvatures and tilt angles. The retrace error is simulated by a sinusoidal waveform with different amplitudes, periods, and phases. In addition, normally distributed random noise is added as measurement noise.
An interferometer with a field view of about 2.8 mm × 2.1 mm and a CCD with 640 × 480 pixels are used in real measurements in Section 4. For our simulations, an interferometer with a 2 mm aperture and a 10 µm pixel size has been used to keep the simulation time manageable. The sub aperture overlap is set to 30%.
A 100-mm-long and 2-mm-wide mirror with a small curvature in x-dimension is simulated as shown in Fig. 2. The peak-to-valley (PV) of the mirror profile is less than 100 nm. To generate this mirror profile we used Eq. (4).
where coefficients a = 10−11 and b = 10−8.Fig. 2 The simulated mirror under test. (a) height mirror map, (b) height profile along the center line.
The theoretical sub-aperture measurement surfaces are then simulated for each sub-aperture position. On top of these simulated ideal sub-aperture height maps, a Gaussian noise with a standard deviation (STD) of 0.2 nm and a random retrace error with 0.5 nm PV are added. A set of typical simulated measurement noise and retrace error is shown in Fig. 3.
Fig. 3 An example of simulated measurement error for one sub-aperture. (a) measurement noise, (b) center line profile along x-direction in (a), (c) retrace error, (d) center line profile along x-direction in (c).
A series of simulations are carried out in order to compare the influences of the interferometer retrace error and measurement noise to the proposed SSI-TBI method and the classical stitching method. We used the Sub-aperture Stitching Interferometry with Least Squares (SSI-LS) method [17, 20] as the representative of the classical stitching methods. The effect of the STD of the angular measurement errors for the TBI method is calculated from 0 to 0.6 µrad, and for the SSI-LS method, the STD of the stage errors is also from 0 to 0.6 µrad. The simulated scans are repeated 10 times for each case. The accuracy is obtained by subtracting the average of the 10 scans to the true profile and the repeatability is calculated by using the 10 stitched profiles subtracted by their average value.
First, the stitching accuracy of SSI-TBI and SSI-LS are studied as illustrated in Fig. 4. The simulation includes the retrace error and system random noise, but with no angular measurement error for SSI-TBI and stage error for SSI-LS. Figure 4(a) is the result of the SSI-TBI. The upper image is the profile of the average of 10 scans, the center plot shows the stitched and true profile, and the lower one is the stitching error. Figure 4(b) is the result of SSI-LS.
Fig. 4 Stitching results and errors. The upper image is the stitched profile, the center plot shows the stitched and true profile, and the lower image is the stitching error. (a) SSI-TBI, (b) SSI-LS.
Next, the accuracy and repeatability of both SSI-TBI and SSI-LS are investigated in different angular measurement error or stage errors with and without the retrace errors to highlight the impact of the each error source when employing these two stitching methods.
In Fig. 5, the simulation results of SSI-TBI show a linear variation of the Root Mean Square (RMS) of stitching errors with the angular measurement errors. Comparing the results in Figs. 5(a) and 5(b), it is not difficult to notice that the stitching error of SSI-TBI mainly depends on the accuracy of the angular measurement and is not sensitive to the retrace errors with a retrace errors under 0.3 nm PV.
Fig. 5 The accuracy and repeatability of SSI-TBI with 10 scans were simulated in different STD of angular measurement errors. (a) without retrace errors, (b) with retrace errors.
The stitching repeatability is presented in detailed images when the STD of the angular measurement errors equals 0 µrad, 0.3 µrad, and 0.6 µrad. The vertical axes are kept identical to highlight their differences. This simulation highlights the needed accuracy of the angular measurement device for SSI-TBI for a stitching accuracy requirement. The ability to estimate the stitching performance enables the designers to take this into account for the overall system design during the conceptual phase of the metrology system.
Following the same evaluation method, similar simulations were done to evaluate the SSI-LS (Fig. 6).
Fig. 6 The accuracy and repeatability of SSI-LS with 10 scans were simulated in different STD of stage errors. (a) without retrace errors, (b) with retrace errors.
Comparing Figs. 6(a) and 6(b), the first conclusion is that the stitching RMS errors from SSI-LS increases dramatically from about 1 nm to 50 nm after the retrace errors are added to each individual sub-apertures map. The retrace error can affect the height profile stitched by SSI-LS with additional curvatures (either convex or concave) as shown in Fig. 4(b). When different retrace errors are produced at each position in each scan, the repeatability of SSI-LS shown in Fig. 6(b) is low and commonly cannot be tolerated for high accuracy applications. In most of the cases, the retrace error is the major failing factor of the SSI-LS method. Since the SSI-LS system is so sensitive to the retrace errors, fine adjustments to “null fringe” condition for individual measurements to reduce the retrace errors are critical to the stitching accuracy of SSI-LS. In addition, the measurement noise is another factor which limits the stitching accuracy in SSI-LS as shown in Fig. 6(a). The lower noise will enable a better and more robust stitching result, if the retrace error has been reduced to a negligible level. The tilt errors of the stage during translation have limited influence to SSI-LS with nearly the same level of repeatability and accuracy shown in Figs. 6(a) and 6(b).
4. Experiments
4.1 Measurement setup
Figure 7 shows the fundamental parts and their arrangement in our measurement system. The SUT is scanned under a Zygo© NewView® white light interferometer by a linear translation stage. The CCD camera used in the measurements has 640 × 480 pixels at a maximum speed rate of 60 Hz and we used a 2.5X microscope objective. The field of view for single measurement is 2.8 mm × 2.1 mm. The TBI is fixed on the platform and a reflecting mirror used as the reflector is attached to the sample stage.
The TBI device used in our experiment is a SIOS© SP-TR Series TBI with a specified angle resolution of 0.002 arcsec (≈9.7 nrad). The performance of the TBI in our system was tested in a fixed position. The stability of distance measurement is about 0.01 µm RMS as shown in Fig. 8(a) and the angular measurement is around 0.33 µrad RMS as shown in Fig. 8(b).
The linear alignment between the TBI and the translation direction of the stage should be carefully calibrated to reduce the error introduced by the surface imperfection of the reflector. When the linear stage translation is not perfect, the laser beam from TBI will have a small shift on the surface of the reflecting mirror, and the reflecting angle will then contain an additional error from the reflecting surface. Therefore, in addition to the alignment, the flatness of the reflecting mirror is important as well. Prior to the experiment, the mirror flatness was measured with a Zygo© Fizeau interferometer. The mirror surface profile in the center line was 0.82 nm RMS and 3.8 nm PV over 30 mm.
In order to investigate the performance of the proposed SSI-TBI, a flat mirror and a spherical mirror have been tested. By using the same raw data from the interferometer, the SSI-LS method is also implemented for comparison purpose.
4.2 Flat mirror test
To evaluate the measurement repeatability, a 90-mm-long flat mirror surface is scanned 10 times with 30% overlap, and 47 pieces of 2.8 mm × 2.1 mm sub-aperture maps are captured for each scan. Three typical measured profiles taken on the left side, center and right side of the mirror are shown in Fig. 9. The flat mirror was also measured using a 4-inch Fizeau interferometer. The Fizeau data is used as a benchmark for error evaluation.
Fig. 9 Sub-aperture profiles with two scans in three mirror positions. (a) left side, (b) center, (c) right side.
In Fig. 9, the random noise in a single sub-aperture is about 0.24 nm RMS. Meanwhile, small retrace errors are illustrated by the difference in two scans at the same position.
The stitching profile of SSI-TBI and SSI-LS are the average of 10 scans in Fig. 10(a) and 10(b). Figure 10(a) shows the mirror profile stitched by using the proposed SSI-TBI. The stitched result with SSI-LS method is shown in Fig. 10(b). Figure 10(c) shows the result measured using our 4-inch aperture Fizeau interferometer. From Fig. 10, it is obvious that the result of SSI-TBI is much closer to the Fizeau result compare to the SSI-LS result.
Fig. 10 A flat mirror was measured along the central line. (a) SSI-TBI, (b) SSI-LS, (c) Fizeau interferometer.
Figure 11 shows the difference of the measurement results. Figure 11 (a) compares the measured profiles with the three methods, and Fig. 11(b) shows the errors of SSI-TBI and SSI-LS with respect to the Fizeau result. We can notice that the SSI-TBI result agrees well with the Fizeau result, while the SSI-LS result generates an additional large curvature on top of the Fizeau result.
Figure 12(a) shows the repeatability of the SSI-TBI. The upper plot is the stitched profiles from 10 different scans. The differences of the 10 scans are within ± 2.5 nm from the average as shown in Fig. 12(a) lower image. The repeatability of the SSI-LS in Fig. 12(b) indicates the height difference of the 10 scans from the average is about ± 15 nm. In our measurement environment, SSI-TBI can offer a robust mirror profile, while SSI-LS method failed to provide any reliable result.
4.3 Spherical mirror test
A spherical mirror is tested to investigate the stitching performance on a curved surface. The experimental set up is similar to the one used to test a flat mirror. In order to compare the stitching results a 2nd order is used to fit the profile, and the residual is calculated by subtracting the best fit to the measured data. The stitching result of SSI-TBI is shown in Fig. 13(a) and the measured radius of curvature is 65.2 m. The upper image is the height map of the average of 10 scans, the center plot shows the profile and fitting line, and the lower one is the residual. Similarly, Fig. 13(b) displays the stitching result of SSI-LS, and the measured radius of curvature is 65.6 m.
Fig. 13 A spherical mirror was measured along the central line. (a) SSI-TBI, (b) SSI-LS, (c) Fizeau interferometer, (d) Comparison of the measured profiles from SSI-TBI and SSI-LS and Fizeau.
In order to evaluate the accuracy, the SUT was measured by the Fizeau interferometer with a flat reference as shown in Fig. 13(c), and the measured radius curvature is 65.0 m. The measured data from the Fizeau interferometer has a large number of fringes in the interference pattern due to the curvature difference between the SUT and the reference. Sinusoidal waviness from print-through effect can be observed at the edges of the residual in Fig. 13(c). Therefore, we only selected the data inside the green rectangle in Fig. 13(d) for our evaluation. The upper plot shows the profiles with the three methods of SSI-TBI, SSI-LS and Fizeau measurement, and the lower one is the difference with respect to the Fizeau result. The comparison indicates that SSI-TBI has much smaller residuals comparing to SSI-LS.
5. Discussion
Among all the error sources, one of the vital issues is the retrace error of the interferometer for SSI-LS. For a flat mirror test, we can adjust the interferometer to ensure a null-fringe status for each position, so that the retrace error can be reduced to a negligible level. For curved mirror, even though the interferometer can be adjusted for each sub-aperture, the retrace error caused by the curvature difference is still not negligible for stitching purpose. However, the “null-fringe adjustment” could be a time consuming process for a stitching system. Furthermore, the environment stability including temperature, humidity, etc. is more difficult to achieve for a long measurement time.
The simulation has a better stitching accuracy than illustrated by our experiment. The main reason is due to the precision of the TBI shown in Fig. 8 was measured in steady state. However, in fact, it usually takes the TBI system about 1 hour to enter in a steady state mode in our experimental environment as shown in Fig. 14. The error during the transition time is therefore included in the angular measurement during our stitching experiment. As a result, the equivalent precision of TBI in our system should be worse than that shown in Fig. 8. The second reason is linked to the fact that the measurement result of the Fizeau interferometer which is used as our benchmark in our comparison is also not perfect. The error of the Fizeau interferometer is therefore also included into the final stitching error.
6. Conclusion
Compared to Fizeau interferometer results, our stitching results indicate that the proposed SSI-TBI method has better performance in terms of accuracy (about 2.5 nm RMS) and repeatability (about 5 nm PV) than the SSI-LS method under the conditions describe in our simulation and experiment. The experiments demonstrate how mirror profile measurement could be performed using an additional angular measuring device like a triple-beam interferometer. According to our simulation illustrated in Fig. 5, better angular measurement should be considered if we want to improve the measurement accuracy. Currently, our effort is made on the 1D stitching, since the profile of interest (such as synchrotron mirrors) is usually in one dimension only. For our future work, a two-dimensional stitching system using a two-dimensional angular measurement solution will be investigated.
Funding
This work was supported by the US Department of Energy, Office of Science, Office of Basic Energy sciences (DE-AC-02-98CH10886); the fundamental Research Funds for the China Central Universities (SCU2015D014).
Acknowledgments
The authors greatly acknowledge the support of the China Scholarship Council. The authors also would like to thank Dennis Kuhne and Weihe Xu in the Optics and Metrology group of NSLS-II for helpful mechanical machining and advices.
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