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Abstract
The lateral distortion of a surface measuring Fizeau interferometer may cause distorted image features in the lateral direction, as well as the surface form error in the axial direction (which is a source of the retrace error). Traditional method for lateral distortion measurement requires a high-accuracy calibration plate featuring a grid pattern. Such a calibration plate is not always available, especially when the required accuracy of the grid pattern comes to the order of sub-micrometer or even nanometer level. To remove the dependence on the plate accuracy, we propose a self-calibration method for the measurement and correction of lateral distortion in Fizeau interferometer. The self-calibration technique may separate the lateral distortion and the geometric error of the calibration plate. This method is verified using a 108-mm-aperture Fizeau interferometer. The experiments show that the form measurement error of a surface tilted at approximately 5° and 16° can be reduced from 92 nm to 43 nm and from 251 nm to 144 nm (peak-to-valley value), respectively, after the distortion correction.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fizeau interferometers are widely used to measure the surface topography of optical components including flat, spherical, aspherical, and free-form surfaces [1–3]. With effective control of environmental influences, such as air disturbance, vibration, temperature, and humidity, the Fizeau interferometer can measure with a repeatability of tens of pm. Systematic error is an important component of the overall measurement uncertainty [4,5]. The systematic error of the Fizeau interferometer, an interferometric imaging system, is derived from optical aberrations. The presence of aberrations may lead to the so-called retrace error [6] when the common path condition is not satisfied. For some special measurement tasks, such as lenses and mirrors from a lithography projection lens, precise compensation of the system error is required to achieve sub-nanometer accuracy.
Lateral distortion is a common source of systematic error, which is mainly caused by the optical distortion from the Seidel aberration. The lateral distortion of the imaging system not only distorts the surface features in the image by varying the magnification within the image plane with the position in the field of view, but also results in retrace error in Fizeau interferometry as illustrated in Fig. 1, which shows how lateral distortion can lead to error in topography measurements of tilted and spherical surfaces. When subaperture stitching is conducted using a Fizeau interferometer, field-dependent lateral distortion in a single field of view causes significant stitch marks at the edges, and the accumulated stitching errors reduce the overall measurement accuracy of the surface topography. In addition, accurately measuring lateral distortion of an imaging system is important for securing the shift-invariant condition that is vital for the transfer function evaluation [7]. Instrument-specific distortion information is also a useful input to the virtual instrument [8] – a new way to evaluate the task-specific uncertainty.
Fig. 1. Influence of lateral distortion on surface topography measurement of (a) an oblique surface and (b) a spherical surface. Taking the optical axis of the Fizeau interferometer as the Z-axis, the actual measured coordinate of a point (x, y, z) in the nominal surface (red) becomes (x+Δx, y+Δy, z) in the measured surface (blue) owing to the lateral distortion.
The traditional method for measuring lateral distortion involves imaging a calibration plate containing a grid pattern and obtaining the distortion by comparing the measured image with the nominal image [9]. The accuracy of this method is limited by the accuracy of the calibration plate. Such a calibration plate is not always available, especially when the required accuracy of the grid pattern comes to the order of sub-micrometer or even nanometer level. For medium- and large-aperture (>100 mm aperture) Fizeau interferometers, it would be even more difficult to produce a large-area calibration plate with sufficient accuracy. To remove the dependence on the plate accuracy, we propose a self-calibration method for the measurement and correction of lateral distortion in Fizeau interferometer.
Self-calibration technique can be summarized into four categories: reversal, redundancy, error separation, and ‘absolute testing’ [10]. The method introduced in this paper is also known as the two-dimensional (2D) self-calibration originated in 1985 [11] and was used to quantify the relative accuracy of the image of an integrated circuit to specification constructed by the electron beam lithography in the semiconductor manufacturing industry. In this method, a calibration plate is measured in several different positions (usually translated and rotated positions). The accuracy of the plate is usually equal to or lower than the accuracy of instrument. The 2D self-calibration has been implemented by using the Fourier analysis method [12,13], iterative optimization [14], least squares [15–17], and learning-based method [18]. Ekberg et al. have used self-calibration techniques to measure the lateral distortion of surface measuring interference microscope with sub-nanometer precision [19,20]. However, the self-calibration technique has not been used to measure the distortion of Fizeau interferometer.
In our proposed method, the lateral distortion of a Fizeau interferometer and the geometric error of the calibration plate was first separated using a self-calibration method, and then the error of the surface form measurement was compensated by the distortion correction.
2. Methods
The lateral distortion measurement method for Fizeau interferometer based on the self-calibration technique comprises three main steps, as shown in Fig. 2(a).
- 1. Measure the surface topography of the calibration plate in three positions, i.e. the initial, relatively rotated and relatively translated positions, as detailed in Section 2.1.
- 2. Extract the location coordinates of marker points from the surface topography by image processing, as detailed in Section 2.2.
- 3. Substitute the location coordinates of the marker points of the calibration plate at the three positions into the self-calibration model based on least squares, to separate the lateral distortion of the Fizeau interferometer and the arrangement error of the grid points on the calibration plate itself, obtaining the lateral distortion of the Fizeau interferometer, as detailed in Section 2.3.
Fig. 2. Flowchart of (a) the measurement and (b) correction of lateral distortion in Fizeau interferometer based on the self-calibration technique.
The lateral distortion correction method consists of two main steps, as shown in Fig. 2(b). First, we predict the surface form error using the obtained distortion information. Later, the predicted form error is subtracted from the actual surface measurement to compensate the retrace error.
It should be noted that the self-calibration only finds the shape of the distortion. The absolute linear scaling problem can be resolved by measuring a traceable linear scale. Manufacturing a linear scale is much easier and less costly compared to a large, high-accuracy 2D calibration plate. In this paper, we focus on the correction of nonlinear lateral distortion, while the linear scaling process is out of the scope.
2.1 Material and instrument
A commercial Fizeau interferometer with an operating wavelength of 1053 nm and an aperture size of 108 mm was used. The camera has 2304 × 2304 pixels. The base of the calibration plate was a quartz glass with a uniform arrangement of solid chromium-plated circles. All circles were arranged in a square grid pattern within the interferometer aperture, with a height of approximately 100 nm from the substrate. The circle has a nominal diameter of 2 mm and a pitch of 4 mm. As shown in Fig. 3, the calibration plate was fixed on a five-dimensional adjustment mechanism and its front surface was measured using the Fizeau interferometer.
Fig. 3. Experimental setup of the measurement and correction of lateral distortion in a commercial Fizeau interferometer.
2.2 Image processing from Z to X-Y
The measured surface topography is shown in Fig. 4(a). The PV of the overall surface form of the calibration plate is 4.266 µm. Increasing the flatness of the substrate will significantly rise the cost of the calibration plate.
Fig. 4. Image processing to find the centers of the circles in the calibration plate. (a) Surface topography of the calibration plate; (b) the surface after form removal; (c) find the circle centers using an edge detection method.
The measured surface form was first removed by subtracting the best-fit 4th-order polynomials. Then, the edges of the marker points were extracted using the edge detection method with a Sobel operator [21]. The circle centers were obtained by least-squares fitting of the edges of the circles, as shown in Fig. 4(c). The grid composed of all circle centers was input to the self-calibration algorithm.
2.3 Self-calibration
The X-Y coordinates of the marker points obtained in the previous section contain both the lateral distortion of the Fizeau interferometer and the arrangement error of the marker points of the calibration plate. To separate these two, the coordinates of each marker point on the calibration plate in the initial, relative rotation (90° counterclockwise rotation around the Z-axis), and relative translation (one pitch forward translation along the X-axis) positions were substituted into the self-calibration model based on least squares.
The X-Y coordinates of the marker points at location (m, n) on the calibration plate (with the center marker point of the plate as the origin) corresponding to the three positions are expressed as Eqs. (1), (2), and (3), respectively.
Equations (1), (2), and (3) for all marker points are combined to form the self-calibration model of an over-determined equation set. The known parameters of X-Y coordinates measurements (x, y) and the nominal coordinates (Ix, Iy) for the three positions are substituted into the self-calibration model to calculate the unknown parameters using least squares, obtaining the lateral distortion (Δx, Δy).
3. Results
3.1 Measurement of the lateral distortion
The measured grid pattern with 17 × 17 marker points is shown in Fig. 5. A nominal grid with a pitch of 88.762 pixels was derived from the X-Y location coordinates of all marker points on the calibration plate in the initial position. Based on this nominal grid, the error vector between each marker point and the corresponding nominal point was calculated. The mean value of the error vector lengths of all the marker points was 0.174 pixels, which was very small relative to the pitch of the grid. Therefore, for a better observation of the experimental results, Fig. 5(a), (b) and (c) show the error between the X-Y location coordinates of each marker point and the ideal coordinates after a 150× magnification. The measurement of the marker points was repeated ten times. The repeatability (the mean value for the 17 × 17 marker points) is 0.008 pixels (corresponding to 0.36 µm). This result confirms the measurement repeatability of the Fizeau interferometer and the robustness of the image processing algorithm.
Fig. 5. The measured grid pattern at (a) the initial position, (b) the 90°-rotated position, and (c) the translated position. (d) The lateral distortion information of the Fizeau interferometer obtained using the self-calibration method, represented by the image of a perfect grid in the presence of the distortion. In all plots, the deviation from the nominal grid is magnified 150× for visualization purpose.
The results in Fig. 5(a), (b) and (c) were substituted into the self-calibration model to determine the least-squares solution of the lateral distortion. The lateral distortion of the Fizeau interferometer and the arrangement error of the grid points on the calibration plate are separated. Figure 5(d) shows the image of a perfect grid in the presence of lateral distortion of the Fizeau interferometer. The mean value of the lateral distortion measured by the self-calibration technique is 0.200 pixels and the maximum value is 0.943 pixels, corresponding to 9.01 µm and 42.49 µm, respectively.
The distortion measurement process was repeated five times, and the repeatability (averaged for all points) was 0.031 pixels and 0.027 pixels for Fig. 6(a) and (b), respectively. Figure 6(b) shows the lateral distortion obtained by the self-calibration method using a different calibration plate with 9 × 9 marker points and a 7-mm pitch. The two calibration plates have different surface forms and cover the similar areas in the field of view. The similar grid shapes shown in Fig. 6(a) and (b) indicate that the self-calibration method is robust in the presence of surface form error of the calibration plate.
Fig. 6. Repeatability of the lateral distortion measurement using the self-calibration technique for two calibration plates, one with (a) 17 × 17 marker points, 4-mm pitch and the other has (b) 9 × 9 marker points, 7-mm pitch.
It is worth mentioning that the result shown in Fig. 5(a) provides eventually the distortion information of the interferometer using a traditional calibration method (which has been mentioned in the introduction section). The discrepancy between Fig. 5(a) and (d) will be further used to verify the advantage of the proposed self-calibration method.
3.2 Verification through overlay assessment
Overlay of the distortion-corrected grid patterns that are measured at different positions is a typical and important parameter for assessing the performance of the self-calibration method. Here, the overlay is calculated as the mean value of the estimated standard deviations of the repeatedly measured X-Y coordinates for all marker points. The overlay evaluations before and after the distortion correction are shown in Fig. 7. The blue, green and magenta grids correspond to the calibration plate’s initial, rotated and translated positions, respectively (same color code as that in Fig. 5). We can see that the three grids in Fig. 7(b) converge very well after the distortion correction, and the overlay is significantly improved as shown in Table 1. In principle, the shape of the grid in Fig. 7(b) should represent the real shape of the grid in the calibration plate. The slight barrel shape is probably due to the curved surface form as shown in Fig. 4(a).
Fig. 7. Overlay (a) before the distortion correction, (b) after the distortion correction using the self-calibration, and (c) after the distortion correction using the traditional calibration method. In all plots, the deviation from the nominal grid is magnified 150× for visualization purpose.
Table 1. Overlay result
Figure 7(c) shows the overlay after the distortion correction using the traditional calibration method, where the distortion information is obtained from Fig. 5(a) which is a direct measurement of the calibration plate in its initial position. The blue grid in Fig. 7(c) is meaningless and therefore omitted.
The overlay after distortion correction using our new method is compared with that using the traditional calibration. Overall, the grids in Fig. 7(b) converge to the real shape of the calibration plate and are more consistent than those in Fig. 7(c). The result of five repeated experiments is shown in Table 1. It is clear that both methods improve the overlay through reducing the distortion effect. The self-calibration method reduces the overlay by 80%, and outperforms the traditional method which reduces the overlay by 67%. The maximum residual error after the distortion correction is 0.064 pixels achieved by the self-calibration method, while the value is 0.208 pixels, more than three times larger, by using the traditional method.
This result is not surprising because the traditional method relies on the accuracy of the calibration plate. However, the systematic and random errors of the locations of marker points in the calibration plate are inevitable, limiting the accuracy of traditional distortion measurement.
3.3 Verification through retrace error reduction
In a Fizeau interferometer, retrace error may occur under the non-null testing condition, e.g. when an oblique surface is measured [6]. Lateral distortion is one of the main sources of the retrace error (see Fig. 1(a)). Therefore, it should be able to reduce the retrace error by distortion correction.
A reference optical flat with a PV of 1/40 λ (the wavelength λ = 1053 nm) was tilted at approximately 5° and 16° with regards to the X-Y plane, and measured respectively. The field-of-view of this measurement is the same as the size of the grid pattern in Fig. 5(d). The measured height maps of the two surfaces with different tilt angles are shown in Fig. 8(a) and (d). The surface was fitted using the function z = ax + by + c. The best-fit plane was subtracted from the measured surface topography to obtain the residual error.
Fig. 8. Reduction of the retrace error in a commercial Fizeau interferometer through distortion correction. (a,d) The measured surface height maps, (b,e) the retrace error after subtracting the best-fit plane, and (d,f) the residual errors after distortion correction for the 5°- and 16°-tilted surfaces, respectively.
Figure 8(b) and (e) show the residual error after subtracting the best-fit plane for the oblique surface of 5° and 16°. The distributions of the errors are similar, but the magnitude of the error increases with the tilt angle. The increasing tilt-dependent error cannot be the form error of the rigid optical flat but is associated with the retrace error. When the lateral distortion is only caused by the 3rd-order optical distortion, the retrace error for a tilted flat would have a comma shape like that shown in Fig. 1(a). However, in this case, the retrace error is not in a perfect comma shape. It is likely that the lateral distortion also contains the effects of other optical misalignments.
The retrace error caused by the lateral distortion can be predicted if one measures the distortion accurately. The relationship between the lateral distortion Δx(x, y) and Δy(x, y) and the induced surface height error Δz at (x, y) is given by
As we know the slopes a and b from the best-fit plane, the surface height error map Δz(x, y) can be calculated for the 5°- and 16°-tilted surfaces. Then, the retrace error can be effectively reduced, as shown in Fig. 8(c) and (f), by subtracting Δz(x, y) from the surface measurement in Fig. 8(a) and (d). The quantitative comparison of the effect before and after the distortion correction is summarized in Table 2.
Table 2. Results of correction for lateral distortion
Since the lateral distortion is just one of the contributors to the retrace error. Reducing the effect of lateral distortion will not completely eliminate the retrace error because other optical aberrations are not compensated. The residual error is relatively high in the left part of the field of view because the calibration plate was translated in the positive direction along the X-axis for one pitch in the self-calibration process. The translation makes the measurements of the marker points in the leftmost column of the calibration plate not involved in the self-calibration distortion calculation. The uncertainties of the distortion information at these marker points are higher than that of the other points [22].
The PV and RMS of the retrace error decrease after the lateral distortion was corrected (see Table 2). At λ = 1053 nm, the PV values of the retrace errors for the 5°-tilted and 16°-tilted surfaces are reduced from 92 nm to 43 nm and from 251 nm to 144 nm, respectively.
4. Conclusion
In this study, we propose a self-calibration method for the measurement and correction of lateral distortion in Fizeau interferometer. Different from the traditional method, in which a high-accuracy calibration plate is required, the proposed method can remove the dependence on the plate accuracy. For a commercial 108-mm-aperture Fizeau interferometer, we demonstrate that the overlay of the grid pattern measurement can be improved by 80% after the distortion correction using the new method, while the overlay is improved by 67% using the traditional method. The self-calibration method is also proved effective to reduce the retrace error in the Fizeau interferometer. For a reference optical flat tilted at approximately 5° and 16°, the PV values of the retrace error are reduced from 92 nm to 43 nm and from 251 nm to 144 nm after the distortion correction. In the future, this method can be extended to the lateral distortion correction of Fizeau interferometer with a much larger aperture, e.g. 800 mm.
Funding
National Key Research and Development Program of China (2019YFF0216401); International Partnership Program of Chinese Academy of Sciences (181231KYSB20200040).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. J. H. Burge, “Fizeau interferometry for large convex surfaces,” Proc. SPIE 2536, 127–138 (1995). [CrossRef]
2. J.-M. Asfour and A. G. Poleshchuk, “Asphere testing with a Fizeau interferometer based on a combined computer-generated hologram,” J. Opt. Soc. Am. A 23(1), 172–178 (2006). [CrossRef]
3. P. de Groot, L. L. Deck, R. Su, and W. Osten, “Contributions of holography to the advancement of interferometric measurements of surface topography,” gxjzz 3(1), 1–20 (2022). [CrossRef]
4. P. de Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34(16), 2856–2863 (1995). [CrossRef]
5. P. de Groot, “Correlated errors in phase-shifting laser Fizeau interferometry,” Appl. Opt. 53(19), 4334–4342 (2014). [CrossRef]
6. C. B. Kreischer, “Retrace error: interferometry’s dark little secret,” Proc. SPIE 8884, 88840X (2013). [CrossRef]
7. R. Su, M. Thomas, M. Liu, J. Drs, Y. Bellouard, C. Pruss, J. Coupland, and R. K. Leach, “Lens aberration compensation in interference microscopy,” Opt. Lasers Eng. 128, 106015 (2020). [CrossRef]
8. R. Su and R. K. Leach, “Physics-based virtual coherence scanning interferometer for surface measurement,” gxjzz 2(2), 120–135 (2021). [CrossRef]
9. A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Annals 62(1), 547–550 (2013). [CrossRef]
10. C. Evans, R. Hocken, and W. Estler, “Self-calibration: reversal, redundancy, error separation, and ‘absolute testing’,” CIRP Annals 45(2), 617–634 (1996). [CrossRef]
11. M. R. Raugh, “Absolute two-dimensional sub-micron metrology for electron beam lithography: A calibration theory with applications,” Precis. Eng. 7(1), 3–13 (1985). [CrossRef]
12. J. Ye, M. T. Takac, C. N. Berglund, G. Owen, and R. F. Pease, “An exact algorithm for self-calibration of two-dimensional precision metrology stages,” Precis. Eng. 20(1), 16–32 (1997). [CrossRef]
13. H.-G. Rhee, J. Chu, and Y.-W. Lee, “Absolute three-dimensional coordinate measurement by the two-point diffraction interferometry,” Opt. Express 15(8), 4435–4444 (2007). [CrossRef]
14. P. Ekberg, L. Stiblert, and L. Mattsson, “A new general approach for solving the self-calibration problem on large area 2D ultra-precision coordinate measurement machines,” Meas. Sci. Technol. 25(5), 055001 (2014). [CrossRef]
15. X. Qiao, X. Chen, P. Ekberg, G. Ding, X. Cai, J. Wei, and J. Wu, “Self-calibration for the 2D stage based on weighted least squares,” Meas. Sci. Technol. 30(12), 125015 (2019). [CrossRef]
16. X. Qiao, G. Ding, X. Chen, and P. Cai, “Three-dimensional Self-Calibration for High-Precision Measurement Instruments with Hybrid Positions,” IEEE Trans. Instrum. Meas. 70, 1–8 (2021). [CrossRef]
17. X. Qiao, G. Ding, X. Chen, P. Cai, and L. Shao, “Comparison of 3D Self-calibration Methods for High-precision Measurement Instruments,” IEEE Trans. Instrum. Meas. 71, 1–9 (2022). [CrossRef]
18. X. You, Y. Wang, Y. Li, J. Liu, and K. Gu, “Learning-based self-calibration for correcting lateral and axial field distortions in 3D surface topography measurement,” Opt. Lett. 46(13), 3263–3266 (2021). [CrossRef]
19. P. Ekberg, R. Su, and R. K. Leach, “High-precision lateral distortion measurement and correction in coherence scanning interferometry using an arbitrary surface,” Opt. Express 25(16), 18703–18712 (2017). [CrossRef]
20. P. Ekberg and L. Mattsson, “Traceable X,Y self-calibration at single nm level of an optical microscope used for coherence scanning interferometry,” Meas. Sci. Technol. 29(3), 035005 (2018). [CrossRef]
21. S. Gupta and S. G. Mazumdar, “Sobel edge detection algorithm,” Int. J. Comput. Sci. Manage. Res. 2, 1578–1583 (2013).
22. X. Qiao, C. Fan, X. Chen, G. Ding, P. Cai, and L. Shao, “Uncertainty analysis of two-dimensional self-calibration with hybrid position using the GUM and MCM methods,” Meas. Sci. Technol. 32(12), 125012 (2021). [CrossRef]
