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Testing convex hyperbolic mirrors with two or more annuluses by Hindle and stitching methods
Optics and Lasers in Engineering 61 (2014) 52–56
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Testing convex hyperbolic mirrors with two or more annuluses by Hindle and stitching methods Fengtao Yan a,b,n, Bin Fan a, Xi Hou a, Fan Wu a a b
Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China University of Chinese Academy of Sciences, Beijing 100039, China
art ic l e i nf o
a b s t r a c t
Article history: Received 13 October 2013 Received in revised form 10 April 2014 Accepted 26 April 2014 Available online 20 May 2014
Based on our previously proposed convex hyperbolic mirrors testing method, we extend the application of this method to test the convex hyperbolic mirrors with two or more annuluses. First, the data deduction for this method is developed. The correspondence of overlapping point pairs is determined by the vertex of the hyperbolic mirror coordinate in image coordinate. Then the misalignment errors in different subapertures are compensated and the subapertures are stitched efficiently. Second, the experimental setup for testing of convex hyperbolic mirror with two annuluses is developed. We present an experiment to verify the validity of this testing method. A convex hyperbolic mirror is tested and stitched with two annuluses and 20 subapertures. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Surface measurement Convex hyperbolic Two or more annuluses Hindle and stitching methods
1. Introduction For giant telescope, the convex aspheric secondary mirror could be several meters in diameter. The form error of optical surface is primarily determined by the quality of the measurement system. The concave shape mirrors allow the measurement system to be small compared to the mirrors being measured. However, the convex shape mirror always needs the measurement system at least as large as the mirror dimension. Consequently, to test the convex aspheric mirrors is very difficult. According to the report, there are some measurement methods that have been developed to test the large convex hyperbolic mirrors. The classic method is the Hindle test. This method requires an auxiliary sphere which would be much larger than the mirror to be measured. For instance, the VISTA Telescope's M2 mirror was measured by this method with a Φ2.4 m sphere [1]. A variation of the auxiliary sphere is to utilize a Hindle shell or an optical gauge, which is only as large as the mirror to be measured. For example, VLT's and Gemini Telescope's M2 mirrors [2] were measured by the large optical gauge test method. However, the measurement accuracy is limited by the transmission quality of the glass. The Φ1.7 m secondary mirror for the MMT was tested by the holographic test plate method [3,4]. A full-aperture spherical test plate with computer-generated hologram (CGH) on the spherical reference surface is utilized in this method. And the accuracy of this method
n Corresponding author at: Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China. E-mail address: [email protected] (F. Yan).
http://dx.doi.org/10.1016/j.optlaseng.2014.04.014 0143-8166/& 2014 Elsevier Ltd. All rights reserved.
is limited by the quality of the spherical surface and the accuracy of the CGH. These test methods have been successful for large secondary mirrors. However, today's new giant telescopes are considering secondary mirrors up to 4 m in diameter. Two of these are listed in Table 1. It would be a very difficult task to test such large convex hyperbolic mirrors with the above measurement methods. The main technology to solve this problem is to use the subaperture stitching method [5,8–23]. The primary goal of this method is to obtain a full aperture mirror shape from several subaperture measurements without measuring the entire mirror one time. Most efforts were made to test large plane optics [12,13] and large concave parabolic mirrors [19,20]. Despite testing that aspheric mirrors with the subaperture stitching method would face more challenges. the QED Technologies has developed a series of stitching interferometer workstation [14–17] that can automatically carry out high-quality subaperture stitching of plane, spherical, aspheric and even the steep aspheric mirrors. However, the largest aperture that the workstation can test is only 300 mm in diameter. The University of Arizona developed the large aperture sub-aperture Fizeau interferometer with stitching software to test the large convex aspheric mirrors [5]. Though this method could obtain the large secondary mirror shape errors, the low order shape errors should be corroborated by other methods. In our previous work, the theoretical and experimental results show that the hyperbolic mirror could be measured with one annulus accurately and efficiently [23]. However, when the giant convex hyperbolic mirrors are measured with one annulus, for example in Table 1, the aperture of the auxiliary sphere would be also too large to fabricate as shown in Table 2. As we proposed in
F. Yan et al. / Optics and Lasers in Engineering 61 (2014) 52–56
Table 1 Giant telescopes that have proposed with very large convex secondary mirrors. Telescope
TMT (m) [6]
European-ELT (m) [7]
Primary mirror diameter Secondary mirror diameter Secondary mirror inner diameter Secondary mirror radius of curvature
30 3.11 0.18 6.2277
39 4.2 1.091 9.313
53
optical axis with a special angle to different positions. When the subapertures in different annuluses cover the full aperture, the full aperture surface shape could be obtained with stitching arithmetic. The data deduction in this measurement method is introduced. It mainly includes two important points. The first is the correspondence of overlapping point pairs. The secondary is the compensation of misalignment errors in different subapertures. 2.1. Correspondence of overlapping point pairs
Table 2 Parameters of the auxiliary sphere with different annuluses. Telescope TMT Annulus
1 2
Tested annulus (m)
European-ELT Sphere diameter (m)
0.18–3.11 12.766 0.18–1.143 2.992 1.143–3.11 2.992
Sphere radius (m)
Tested Sphere annulus (m) diameter (m)
Sphere radius (m)
25.249 15.767 8.036
1.091–4.2 4.414 1.091–2.379 2.150 2.379–4.2 2.150
10.238 8.512 6.871
The important process to keep the measurement accuracy of this method is the correspondence of overlapping point pairs. In other words, the relationship between the subaperture local coordinate and the global coordinate should be obtained accurately. This problem could be described as N1
s1 ¼ ∑
N
∑ ððxi xj Þ2 þ ðyi yj Þ2 Þ
ð1Þ
i ¼ 1 j ¼ iþ1
where ðxi ; yi Þ and ðxj ; yj Þ are the object global coordinates which are uniformed by the adjacent subapertures i and j local coordinates, respectively. s1 represents the coordinate deviation of the overlapping region in the object coordinate. According to the camera projection model, the subaperture local image coordinate ðx0 ; y0 Þ and local object coordinate ðx; y; zÞ are related as follows: ½x0 ; y0 ; 1T ¼ M U ½x; y; z; 1T
ð2Þ
where M is the projection matrix. It does not change in every subaperture during the experiment. The real projection matrix is deeper and harder to define precisely. Fortunately, in our method, it is just an intermediate variable which does not need to know. Eq. 1 could be replaced by the following equation: N1
Fig. 1. Hindle and subaperture testing system with two annuluses.
s2 ¼ ∑
N
∑
i ¼ 1 j ¼ iþ1
previous work, the convex mirror should be measured with two or more annulus. The auxiliary spheres which are utilized in two annulus measurement are also listed in Table 2. The diameters of the auxiliary spheres are much smaller in two annulus measurement. The more the annuluses, the smaller the diameters of the auxiliary spheres. The two or more annuluses measurements are discussed in our present work. This paper is organized as follows. In Section 2, the data deduction for this method is introduced. The correspondence of overlapping point pairs and the compensation of misalignment errors are mainly discussed. In Section 3, we describe the experimental setup and stitching results. The convex hyperbolic mirror is measured with two annuluses in the experiment. Good consistency is observed between the two annulus stitching results and one annulus stitching results. The conclusion is given in Section 4.
2. The theory of Hindle and stitching measurement method The testing system with two annuluses is shown in Fig. 1. The parameters of the auxiliary sphere could be obtained from our previous work [22,23]. However, any sphere in the optical shop can be the auxiliary sphere if their specifications meet the requirements of testing system. In this work, there are two auxiliary spheres in our testing system. The inner annulus subapertures are obtained by the auxiliary sphere 1, and the outer annulus subapertures are obtained by the auxiliary sphere 2. In our measurement, one subaperture surface phase data is obtained by one measurement. The whole subapertures phase data are obtained by rotating the convex hyperbolic mirror around the
ðxi 0 xj 0 Þ2 þ ðyi 0 yj 0 Þ2
ð3Þ
where ðxi 0 ; yi 0 Þ and ðxj 0 ; yj 0 Þ are the image global coordinates which are uniformed by subaperture i and subaperture j local coordinates, respectively. s2 represents the coordinate deviation of the overlapping region in the image coordinate. The projection coordinate in image coordinate of the vertex of the hyperbolic mirror is ðxc 0 ; yc 0 Þ, which would not be changed in every subaperture measurement. As ðxc 0 ; yc 0 Þ is the origin point, the kth subaperture coordinate in image could be described by " # cos ðθk Þ sin ðθk Þ ð4Þ ðxkc 0 ; ykc 0 Þ ¼ ðxk 0 xc 0 ; yk 0 yc 0 Þ cos ðθk Þ sin ðθk Þ where ðxkc 0 ; ykc 0 Þ is the normalized global cordinate system for fullaperture, ðxk 0 ; yk 0 Þ is the normalized local image coordinate system for kth subaperture, and θk is the angle of the kth subaperture rotated. The deviation of the coordinate by the angle θk could be below 0.01 pixel as the precision of the angle θk is 0.001251. This error would be neglected here as the translation accuracy would be 0.1 pixel. ðxc 0 ; yc 0 Þ is neither the CCD center nor image center, which is influenced by the globle image coordinate. It could be obtained by the relationship of ith and jth subaperture phase data. The difference of the subaperture overlapping phase data which are removed from the first order aberration should be the global minimum as s3 ¼ Rf W i 0 ðxic ; yic Þ W i þ 1 0 ðxi þ 1c ; yi þ 1c Þ 2 ð5Þ where Rf stands for ‘removed first order aberration’, W i 0 ðxic ; yic Þ and W i þ 1 0 ðxi þ 1c ; yi þ 1c Þ are the ith and jth subaperture phase data and ðxic ; yic Þ and ðxi þ 1c ; yi þ 1c Þ are the computed vertex of the hyperbolic mirror's in each image coordinate. s3 represents the
54
F. Yan et al. / Optics and Lasers in Engineering 61 (2014) 52–56
phase deviation of the overlapping region between ith and jth subaperture. After ðxic ; yic Þ and ðxi þ 1c ; yi þ 1c Þ are obtained, the projection coordinate ðxc 0 ; yc 0 Þ in image coordinate of the vertex of the hyperbolic mirror could be obtained by " ð x0c ; y0c Þ
cos ðθi Þ cos ðθi þ 1 Þ sin ðθi Þ sin ðθi þ 1 Þ
# sin ðθ i Þ þ sin ðθi þ 1 Þ ¼ cos ðθi Þ cos ðθi þ 1 Þ
xic U cos ðθ i Þ yic U sin ðθ i Þ þ xi þ 1c U cos ðθi þ 1 Þ þ yi þ 1c U sin ðθi þ 1 Þ; xic U sin ðθi Þ yic U cos ðθ i Þ xi þ 1c U sin ðθ i þ 1 Þ þ yi þ 1c U cos ðθ i þ 1 Þ
!
ð6Þ According to Eq. (4), the subaperture local coordinate can be uniformed to the global coordinate after ðx0c ; y0c Þ is obtained from Eq. (6). With the fixed overlapping correspondence, the next step is to compensate the misalignment errors among the subapertures phase data.
2.2. Compensation of misalignment errors As the interferometer, sphere and convex hyperbolic mirror could not be in the ideal position, there would be different misalignment errors in each subaperture phase data. Consequently, the subaperture phase data obtained by the interferometer should be calibrated by compensating the misalignment errors. The subapertures phase data compensated by Eq. (7) could be in the same criterion W i ðx; yÞ ¼ W i ðx; yÞ þ Ai Z 1 ðx; yÞ þ Bi Z 2 ðx; yÞ þ C i Z 3 ðx; yÞ þ Di Z 4 ðx; yÞ þ Ei Z 7 ðx; yÞ þF i Z 8 ðx; yÞ
ð7Þ
where W i is the compensation result of W i ; Ai , …, F i are the stitching coefficients to compensate the misalignment errors. Z 1 , …, Z 8 are the Zernike polynomial. In our previous work, we only considered the overlapping region data between the adjacent subapertures [23]. However, in this work, all the overlapping region data are taken to computer the stitching coefficients. The stitching coefficients could be obtained by making the Eq.(8) to be the global minimum N1
s4 ¼ ∑
N
∑ ðW i ðx; yÞ W j ðx; yÞÞ
i ¼ 1 j ¼ iþ1
2
ð8Þ
here s4 represents the phase deviation of all overlapping region data. In order to improve the computational efficiency, Eq. 8 could be expanded in terms of a complete set of Zernike polynomials in
matrix. The matrix form can be described as follows: 2 2 3 H 1;2 6 6 7 6 6 H 1;3 7 6 6 7 6 6 ⋮ 7 6 4 5 6 6 H 1;N 6 6 6 s4 ¼ 6 6 6 6 0 6 6 6 6 6 6 ⋮ 4 0
3 H 1;2 6 0 7 7 6 6 7 4 ⋮ 5 2
2
0 H 2;3
3
7 6 6 H 2;4 7 7 6 6 ⋮ 7 5 4 H 2;N
3 0 6H 7 6 1;3 7 6 7 4 ⋮ 5 2
2 ⋯
0 3 H 2;3 6 0 7 7 6 6 7 4 ⋮ 5
7 6 7 6 6 7 4 H 1;N 1 5 0 3 0 6 ⋮ 7 7 6 6 7 4 H 2;N 1 5 2
2
⋮
0 ⋮
⋱
0
0
⋯
3 32 W1 W2 6 6 W W 7 7 66 1 3 7 7 66 77 ⋮ 57 4 3 6 2 7 6 ½M 1 7 6 W W 7 N 1 6 ½M 7 6 2 7 6 6 3 7 2 7 6 6 W2 W3 7 7 6 7 6 ⋮ U6 7þ6 6 W2 W4 7 7 7 6 6 ½M 77 6 6 4 N1 5 6 6 77 7 ⋮ 5 7 64 ½M N 7 6 6 W 2 W N 7 7 6 7 6 ⋮ 5 4 WN 1 WN
3
0 ⋮
0 ⋮ H N 1;N
33 0 77 6 6 ⋮ 77 77 6 6 0 77 57 4 7 H 1;N 7 37 2 7 0 7 77 6 ⋮ 77 6 77 6 6 0 77 57 4 7 H 2;N 7 7 7 7 ⋮ 5 H N 1;N 2
22
ð9Þ
where M i (i¼1,2,…,N) is the stitching coefficients for compensating W i misalignment error, H i;j is the coefficient matrix in the overlapping region of ith and jth subapertures. It would be zero if there is no overlapping region between the subapertures. The stitching parameters M i are obtained when s4 is the global minimum to Eq. (9) which was solved by the optimization algorithm and the least-squares method. And then the adjusted results W 1 , W 2 ,…, W N could be worked out by putting M i into Eq. (7). The full aperture shapes of the tested mirror could be obtained by stitching the adjusted results. However, the piston, tilts, defocus, coma and primary spherical aberration should be removed in the stitched results as the interferometer, sphere and convex hyperbolic mirror could not be in ideal position [24].
3. Experimental verification A convex hyperbolic mirror is tested to verify the two annulus metrology capability of the Hindle and stitching method. In this experiment, the clear aperture of the mirror is about 150 mm, the radius of the vertex is 350 mm, and the conic constant is 1.9732. The convex hyperboloid mirror is mounted with a 6-axis adjustor, which could rotate via computer control to an accuracy of 0.001251. A 6ʺ Zygo interferometer with an f/5.3 transmission sphere is utilized in this
Fig. 2. Two annuluses test: (a) experimental setup and (b) lattice scheme.
F. Yan et al. / Optics and Lasers in Engineering 61 (2014) 52–56
55
experiment. Two auxiliary spheres with 180 mm in diameter are needed to obtain the inner and outer subaperture phase data. The radiuses of the spheres are about 629.81 mm. The experiment setup is shown in Fig. 2(a). In this experiment, the mirror is tested with two annuluses. In total 20 subapertures, 8 subapertures in the inner annulus and 12 subapertures in the outer annulus, are required to cover the full-aperture surface as shown in Fig. 2(b). The subapertures are tested one by one. One of the inner subaperture surface phase data is shown in Fig. 3(a). And Fig. 3(b) shows one of the outer annulus subaperture surface phase data. From Fig. 3, we can clearly see that the misalignment errors are different in the two raw subaperture surface phase data. Theoretically, if the sphere, convex hyperbolic and interferometer are in the ideal position when the measurement was taken, the phase data that we obtained from the interferometer is the mirror surface shape. However, there are always some tilt and defocus misalignments among the interferometer, mirror and sphere. These misalignments would introduce some piston, tilts, defocus, coma and primary spherical error [24] in the raw subaperture phase data. Generally, the piston, tilts, defocus, coma and primary spherical aberration should be removed from the stitching results in this measurement method. The mirror surface shape is successfully retrieved with our measurement method. And the stitching results which removed piston, tilts, defocus, coma and primary spherical are shown in Fig. 4 (a), and the PV and RMS values are approximately 340.34 nm and 43.17 nm, respectively. In order to estimate the accuracy of this method, the mismatch error in the overlapping region among the 20 subapertures is calculated as shown in Fig. 4(b), and the PV¼ 151.69 nm, RMS¼4.23 nm. From Fig. 4, we can see that the RMS in mismatch error is only about ten percent of the tested mirror.
Fig. 4. Experiment results: (a) stitching phase data with two annuluses (PV¼ 340.34 nm, RMS ¼ 43.17 nm) and (b) overlapping mismatch data (PV¼ 151.69 nm, RMS¼ 4.23 nm).
Fig. 3. Measured subapertures: (a) subaperture 1 and (b) subaperture 9.
This accuracy is enough to testify the convex hyperboloid mirror measurement precision. However, the center of the mirror cannot be tested by this measurement method. In order to check the theory exactitude of the measurement and the algorithm, this convex hyperbolic mirror was measured with one annulus as introduced in our previous work [23]. The results removed piston, tilts, defocus, coma and primary spherical are shown in Fig. 5. We can see that the results are in agreement with those in Fig. 4(a), and the PV and RMS values are approximately 360.81 nm and 43.33 nm, respectively. Comparing Fig. 4(a) with Fig. 5, the surface shape is highly similar, and the difference of PV and RMS value is approximate 20.47 nm and 0.16 nm, respectively. The first 36 terms Zernike polynomial coefficients which are calculated for the above results of Figs. 4(a) and 5 are shown in Fig. 6. The coincidence of the two coefficients variation can testify the coincidence of the low order surface that is tested by the two methods. Fig. 7 shows the measurement results’ residual figure between the two methods, specifically the difference between Fig. 4(a) and Fig. 5, where the PV difference is 75.26 nm and the RMS is 5.62 nm. This difference is mainly caused by auxiliary sphere surface deviation. These results show that the Hindle and stitching method with two annuluses is not inducing significant
56
F. Yan et al. / Optics and Lasers in Engineering 61 (2014) 52–56
4. Conclusion
Fig. 5. Full aperture test with one annulus (PV ¼ 360.81 nm, RMS¼ 43.33 nm).
In summary, we extended the measurement method of measuring the large convex hyperbolic mirror and discussed the testing method with two or more annulus. The correspondence of overlapping point pairs and the compensation of misalignment errors in different subapertures has been studied. Through the experiment of two annuluses, it has demonstrated that the convex hyperbolic mirrors could be tested with two annuluses by two small auxiliary spheres and the full aperture surface shape could be obtained with satisfactory accuracy. However, the obscuration area surface shape in the mirror center could not be obtained by this method. Though the convex hyperbolic mirror is tested with two annuluses in the experiments, it can be extended in more annuluses theoretically. At the same time, we are aware that the experiment does prove something, but not everything. We will focus our future work on the following issues. The systematic errors induced by the auxiliary sphere and the interferometer must be separated from the measurements; otherwise the accuracy is mainly limited by the auxiliary sphere.
Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant no. 60908042. References
Fig. 6. Comparison of the first 36 terms Zernike decomposition of results shown in Fig. 4(a) and Fig. 5.
Fig. 7. Residual figure between Fig. 4(a) and Fig. 5 (PV¼ 75.26 nm, RMS¼ 5.62 nm).
inaccuracy. In summary, the stitching measurements of the convex hyperbolic with two annuluses have good agreement with the one annulus stitching measurement results.
[1] Abdulkadyrov MA, Ignatov AN, Patrikeev VE, Pridnya VV, Polyanchikov AV, Semenov AP, et al. M1 and M2 mirrors manufacturing for VISTA telescope. Proc SPIE 2004;5494:374–81. [2] Geyl R, Ruch E, Primet S, Chaussat G. Progress toward a third Gemini M2 mirror. Proc SPIE 2005;5869:586901. [3] Burge JH, Anderson DS. Full-aperture interferometric test of convex secondary mirrors using holographic test plates. Proc SPIE 1994;2199:181–92. [4] Burge JH. Measurement of large convex aspheres. Proc SPIE 1997;2871:362–73. [5] Burge JH, Su P, Zhao C. Optical metrology for very large convex aspheres. Proc SPIE 2008;7018:701818. [6] TMT statement of work, “Secondary mirror cell assembly,” 〈http://www.tmt. org/documents/statement-work-secondary-mirror-cell-assembly〉. [7] Cayrel M. E-ELT optomechanics: overview. Proc SPIE 2012;8444:84441X. [8] Kim CJ. Polynomial fit of interferograms. Appl Opt 1982;21:4521–5. [9] Chow WW, Lawrence GN. Method for subaperture testing interferogram reduction. Opt Lett 1983;8:468–70. [10] Stuhlinger TW. Subaperture optical testing: experimental verification. Proc SPIE 1986;656:118–27. [11] Cheng W, Chen M. Transformation and connection of subapertures in the multiaperture overlap-scanning technique for large optics tests. Opt Eng 1993;32:1947–50. [12] Otsubo M, Okada K, Tsujiuchi J. Measurement of large plane surface shapes by connecting small-aperture interferograms. Opt Eng 1994;33:608–13. [13] Bray M. Stitching interferometer for large piano optics using a standard interferometer. Proc SPIE 1997;3134:39–50. [14] Murphy P, Forbes G, Fleig J, Dumas P, Tricard M. Stitching interferometry: a flexible solution for surface metrology. Opt Photonics News 2003;14:38–43. [15] Fleig J, Dumas P, Murphy PE, Forbes G. An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces. Proc SPIE, 5188; 2003; 296–307. [16] Kulawiec A, Murphy P, Demarco M. Measurement of high-departure aspheres using subaperture stitching with the Variable Optical Null (VON™). Proc SPIE 2010;7655:765512. [17] Aspheric Stitching Interferometer with QIS™, 〈http://www.qedmrf.com/qis〉. [18] Li B, Chen L, Bian B, Li Y. A demodulation method for the circular carrier interferogram using phase stitching. Opt Lasers Eng 2011;49(9–10):1118–23. [19] Chen S, Li S, Dai Y, Zheng Z. Testing of large optical surfaces with subaperture stitching. Appl Opt 2007;46:3054–9. [20] Hou X, Wu F, Yang L, Chen Q. Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method. Opt Express 2007;15:4504–13. [21] Zhang P, Zhao H, Zhou X, Li J. Sub-aperture stitching interferometry using stereovision positioning technique. Opt Express 2010;18:15216–22. [22] Yan F, Fan B, Hou X, Wu F. Hindle-sphere method based on sub-aperture stitching. High Power Laser and Particle Beams 2012;24:2555–9 (in Chinese). [23] Yan F, Fan B, Hou X, Wu F. Measurement of large convex hyperbolic mirrors using hindle and stitching methods. Opt Lasers Eng 2013;51(7):856–60. [24] Born M, Wolf E. Principles of optics. 6th ed. Great Britain, 1980.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Testing convex hyperbolic mirrors with two or more annuluses by Hindle and stitching methods Fengtao Yan a,b,n, Bin Fan a, Xi Hou a, Fan Wu a a b
Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China University of Chinese Academy of Sciences, Beijing 100039, China
art ic l e i nf o
a b s t r a c t
Article history: Received 13 October 2013 Received in revised form 10 April 2014 Accepted 26 April 2014 Available online 20 May 2014
Based on our previously proposed convex hyperbolic mirrors testing method, we extend the application of this method to test the convex hyperbolic mirrors with two or more annuluses. First, the data deduction for this method is developed. The correspondence of overlapping point pairs is determined by the vertex of the hyperbolic mirror coordinate in image coordinate. Then the misalignment errors in different subapertures are compensated and the subapertures are stitched efficiently. Second, the experimental setup for testing of convex hyperbolic mirror with two annuluses is developed. We present an experiment to verify the validity of this testing method. A convex hyperbolic mirror is tested and stitched with two annuluses and 20 subapertures. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Surface measurement Convex hyperbolic Two or more annuluses Hindle and stitching methods
1. Introduction For giant telescope, the convex aspheric secondary mirror could be several meters in diameter. The form error of optical surface is primarily determined by the quality of the measurement system. The concave shape mirrors allow the measurement system to be small compared to the mirrors being measured. However, the convex shape mirror always needs the measurement system at least as large as the mirror dimension. Consequently, to test the convex aspheric mirrors is very difficult. According to the report, there are some measurement methods that have been developed to test the large convex hyperbolic mirrors. The classic method is the Hindle test. This method requires an auxiliary sphere which would be much larger than the mirror to be measured. For instance, the VISTA Telescope's M2 mirror was measured by this method with a Φ2.4 m sphere [1]. A variation of the auxiliary sphere is to utilize a Hindle shell or an optical gauge, which is only as large as the mirror to be measured. For example, VLT's and Gemini Telescope's M2 mirrors [2] were measured by the large optical gauge test method. However, the measurement accuracy is limited by the transmission quality of the glass. The Φ1.7 m secondary mirror for the MMT was tested by the holographic test plate method [3,4]. A full-aperture spherical test plate with computer-generated hologram (CGH) on the spherical reference surface is utilized in this method. And the accuracy of this method
n Corresponding author at: Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China. E-mail address: [email protected] (F. Yan).
http://dx.doi.org/10.1016/j.optlaseng.2014.04.014 0143-8166/& 2014 Elsevier Ltd. All rights reserved.
is limited by the quality of the spherical surface and the accuracy of the CGH. These test methods have been successful for large secondary mirrors. However, today's new giant telescopes are considering secondary mirrors up to 4 m in diameter. Two of these are listed in Table 1. It would be a very difficult task to test such large convex hyperbolic mirrors with the above measurement methods. The main technology to solve this problem is to use the subaperture stitching method [5,8–23]. The primary goal of this method is to obtain a full aperture mirror shape from several subaperture measurements without measuring the entire mirror one time. Most efforts were made to test large plane optics [12,13] and large concave parabolic mirrors [19,20]. Despite testing that aspheric mirrors with the subaperture stitching method would face more challenges. the QED Technologies has developed a series of stitching interferometer workstation [14–17] that can automatically carry out high-quality subaperture stitching of plane, spherical, aspheric and even the steep aspheric mirrors. However, the largest aperture that the workstation can test is only 300 mm in diameter. The University of Arizona developed the large aperture sub-aperture Fizeau interferometer with stitching software to test the large convex aspheric mirrors [5]. Though this method could obtain the large secondary mirror shape errors, the low order shape errors should be corroborated by other methods. In our previous work, the theoretical and experimental results show that the hyperbolic mirror could be measured with one annulus accurately and efficiently [23]. However, when the giant convex hyperbolic mirrors are measured with one annulus, for example in Table 1, the aperture of the auxiliary sphere would be also too large to fabricate as shown in Table 2. As we proposed in
F. Yan et al. / Optics and Lasers in Engineering 61 (2014) 52–56
Table 1 Giant telescopes that have proposed with very large convex secondary mirrors. Telescope
TMT (m) [6]
European-ELT (m) [7]
Primary mirror diameter Secondary mirror diameter Secondary mirror inner diameter Secondary mirror radius of curvature
30 3.11 0.18 6.2277
39 4.2 1.091 9.313
53
optical axis with a special angle to different positions. When the subapertures in different annuluses cover the full aperture, the full aperture surface shape could be obtained with stitching arithmetic. The data deduction in this measurement method is introduced. It mainly includes two important points. The first is the correspondence of overlapping point pairs. The secondary is the compensation of misalignment errors in different subapertures. 2.1. Correspondence of overlapping point pairs
Table 2 Parameters of the auxiliary sphere with different annuluses. Telescope TMT Annulus
1 2
Tested annulus (m)
European-ELT Sphere diameter (m)
0.18–3.11 12.766 0.18–1.143 2.992 1.143–3.11 2.992
Sphere radius (m)
Tested Sphere annulus (m) diameter (m)
Sphere radius (m)
25.249 15.767 8.036
1.091–4.2 4.414 1.091–2.379 2.150 2.379–4.2 2.150
10.238 8.512 6.871
The important process to keep the measurement accuracy of this method is the correspondence of overlapping point pairs. In other words, the relationship between the subaperture local coordinate and the global coordinate should be obtained accurately. This problem could be described as N1
s1 ¼ ∑
N
∑ ððxi xj Þ2 þ ðyi yj Þ2 Þ
ð1Þ
i ¼ 1 j ¼ iþ1
where ðxi ; yi Þ and ðxj ; yj Þ are the object global coordinates which are uniformed by the adjacent subapertures i and j local coordinates, respectively. s1 represents the coordinate deviation of the overlapping region in the object coordinate. According to the camera projection model, the subaperture local image coordinate ðx0 ; y0 Þ and local object coordinate ðx; y; zÞ are related as follows: ½x0 ; y0 ; 1T ¼ M U ½x; y; z; 1T
ð2Þ
where M is the projection matrix. It does not change in every subaperture during the experiment. The real projection matrix is deeper and harder to define precisely. Fortunately, in our method, it is just an intermediate variable which does not need to know. Eq. 1 could be replaced by the following equation: N1
Fig. 1. Hindle and subaperture testing system with two annuluses.
s2 ¼ ∑
N
∑
i ¼ 1 j ¼ iþ1
previous work, the convex mirror should be measured with two or more annulus. The auxiliary spheres which are utilized in two annulus measurement are also listed in Table 2. The diameters of the auxiliary spheres are much smaller in two annulus measurement. The more the annuluses, the smaller the diameters of the auxiliary spheres. The two or more annuluses measurements are discussed in our present work. This paper is organized as follows. In Section 2, the data deduction for this method is introduced. The correspondence of overlapping point pairs and the compensation of misalignment errors are mainly discussed. In Section 3, we describe the experimental setup and stitching results. The convex hyperbolic mirror is measured with two annuluses in the experiment. Good consistency is observed between the two annulus stitching results and one annulus stitching results. The conclusion is given in Section 4.
2. The theory of Hindle and stitching measurement method The testing system with two annuluses is shown in Fig. 1. The parameters of the auxiliary sphere could be obtained from our previous work [22,23]. However, any sphere in the optical shop can be the auxiliary sphere if their specifications meet the requirements of testing system. In this work, there are two auxiliary spheres in our testing system. The inner annulus subapertures are obtained by the auxiliary sphere 1, and the outer annulus subapertures are obtained by the auxiliary sphere 2. In our measurement, one subaperture surface phase data is obtained by one measurement. The whole subapertures phase data are obtained by rotating the convex hyperbolic mirror around the
ðxi 0 xj 0 Þ2 þ ðyi 0 yj 0 Þ2
ð3Þ
where ðxi 0 ; yi 0 Þ and ðxj 0 ; yj 0 Þ are the image global coordinates which are uniformed by subaperture i and subaperture j local coordinates, respectively. s2 represents the coordinate deviation of the overlapping region in the image coordinate. The projection coordinate in image coordinate of the vertex of the hyperbolic mirror is ðxc 0 ; yc 0 Þ, which would not be changed in every subaperture measurement. As ðxc 0 ; yc 0 Þ is the origin point, the kth subaperture coordinate in image could be described by " # cos ðθk Þ sin ðθk Þ ð4Þ ðxkc 0 ; ykc 0 Þ ¼ ðxk 0 xc 0 ; yk 0 yc 0 Þ cos ðθk Þ sin ðθk Þ where ðxkc 0 ; ykc 0 Þ is the normalized global cordinate system for fullaperture, ðxk 0 ; yk 0 Þ is the normalized local image coordinate system for kth subaperture, and θk is the angle of the kth subaperture rotated. The deviation of the coordinate by the angle θk could be below 0.01 pixel as the precision of the angle θk is 0.001251. This error would be neglected here as the translation accuracy would be 0.1 pixel. ðxc 0 ; yc 0 Þ is neither the CCD center nor image center, which is influenced by the globle image coordinate. It could be obtained by the relationship of ith and jth subaperture phase data. The difference of the subaperture overlapping phase data which are removed from the first order aberration should be the global minimum as s3 ¼ Rf W i 0 ðxic ; yic Þ W i þ 1 0 ðxi þ 1c ; yi þ 1c Þ 2 ð5Þ where Rf stands for ‘removed first order aberration’, W i 0 ðxic ; yic Þ and W i þ 1 0 ðxi þ 1c ; yi þ 1c Þ are the ith and jth subaperture phase data and ðxic ; yic Þ and ðxi þ 1c ; yi þ 1c Þ are the computed vertex of the hyperbolic mirror's in each image coordinate. s3 represents the
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F. Yan et al. / Optics and Lasers in Engineering 61 (2014) 52–56
phase deviation of the overlapping region between ith and jth subaperture. After ðxic ; yic Þ and ðxi þ 1c ; yi þ 1c Þ are obtained, the projection coordinate ðxc 0 ; yc 0 Þ in image coordinate of the vertex of the hyperbolic mirror could be obtained by " ð x0c ; y0c Þ
cos ðθi Þ cos ðθi þ 1 Þ sin ðθi Þ sin ðθi þ 1 Þ
# sin ðθ i Þ þ sin ðθi þ 1 Þ ¼ cos ðθi Þ cos ðθi þ 1 Þ
xic U cos ðθ i Þ yic U sin ðθ i Þ þ xi þ 1c U cos ðθi þ 1 Þ þ yi þ 1c U sin ðθi þ 1 Þ; xic U sin ðθi Þ yic U cos ðθ i Þ xi þ 1c U sin ðθ i þ 1 Þ þ yi þ 1c U cos ðθ i þ 1 Þ
!
ð6Þ According to Eq. (4), the subaperture local coordinate can be uniformed to the global coordinate after ðx0c ; y0c Þ is obtained from Eq. (6). With the fixed overlapping correspondence, the next step is to compensate the misalignment errors among the subapertures phase data.
2.2. Compensation of misalignment errors As the interferometer, sphere and convex hyperbolic mirror could not be in the ideal position, there would be different misalignment errors in each subaperture phase data. Consequently, the subaperture phase data obtained by the interferometer should be calibrated by compensating the misalignment errors. The subapertures phase data compensated by Eq. (7) could be in the same criterion W i ðx; yÞ ¼ W i ðx; yÞ þ Ai Z 1 ðx; yÞ þ Bi Z 2 ðx; yÞ þ C i Z 3 ðx; yÞ þ Di Z 4 ðx; yÞ þ Ei Z 7 ðx; yÞ þF i Z 8 ðx; yÞ
ð7Þ
where W i is the compensation result of W i ; Ai , …, F i are the stitching coefficients to compensate the misalignment errors. Z 1 , …, Z 8 are the Zernike polynomial. In our previous work, we only considered the overlapping region data between the adjacent subapertures [23]. However, in this work, all the overlapping region data are taken to computer the stitching coefficients. The stitching coefficients could be obtained by making the Eq.(8) to be the global minimum N1
s4 ¼ ∑
N
∑ ðW i ðx; yÞ W j ðx; yÞÞ
i ¼ 1 j ¼ iþ1
2
ð8Þ
here s4 represents the phase deviation of all overlapping region data. In order to improve the computational efficiency, Eq. 8 could be expanded in terms of a complete set of Zernike polynomials in
matrix. The matrix form can be described as follows: 2 2 3 H 1;2 6 6 7 6 6 H 1;3 7 6 6 7 6 6 ⋮ 7 6 4 5 6 6 H 1;N 6 6 6 s4 ¼ 6 6 6 6 0 6 6 6 6 6 6 ⋮ 4 0
3 H 1;2 6 0 7 7 6 6 7 4 ⋮ 5 2
2
0 H 2;3
3
7 6 6 H 2;4 7 7 6 6 ⋮ 7 5 4 H 2;N
3 0 6H 7 6 1;3 7 6 7 4 ⋮ 5 2
2 ⋯
0 3 H 2;3 6 0 7 7 6 6 7 4 ⋮ 5
7 6 7 6 6 7 4 H 1;N 1 5 0 3 0 6 ⋮ 7 7 6 6 7 4 H 2;N 1 5 2
2
⋮
0 ⋮
⋱
0
0
⋯
3 32 W1 W2 6 6 W W 7 7 66 1 3 7 7 66 77 ⋮ 57 4 3 6 2 7 6 ½M 1 7 6 W W 7 N 1 6 ½M 7 6 2 7 6 6 3 7 2 7 6 6 W2 W3 7 7 6 7 6 ⋮ U6 7þ6 6 W2 W4 7 7 7 6 6 ½M 77 6 6 4 N1 5 6 6 77 7 ⋮ 5 7 64 ½M N 7 6 6 W 2 W N 7 7 6 7 6 ⋮ 5 4 WN 1 WN
3
0 ⋮
0 ⋮ H N 1;N
33 0 77 6 6 ⋮ 77 77 6 6 0 77 57 4 7 H 1;N 7 37 2 7 0 7 77 6 ⋮ 77 6 77 6 6 0 77 57 4 7 H 2;N 7 7 7 7 ⋮ 5 H N 1;N 2
22
ð9Þ
where M i (i¼1,2,…,N) is the stitching coefficients for compensating W i misalignment error, H i;j is the coefficient matrix in the overlapping region of ith and jth subapertures. It would be zero if there is no overlapping region between the subapertures. The stitching parameters M i are obtained when s4 is the global minimum to Eq. (9) which was solved by the optimization algorithm and the least-squares method. And then the adjusted results W 1 , W 2 ,…, W N could be worked out by putting M i into Eq. (7). The full aperture shapes of the tested mirror could be obtained by stitching the adjusted results. However, the piston, tilts, defocus, coma and primary spherical aberration should be removed in the stitched results as the interferometer, sphere and convex hyperbolic mirror could not be in ideal position [24].
3. Experimental verification A convex hyperbolic mirror is tested to verify the two annulus metrology capability of the Hindle and stitching method. In this experiment, the clear aperture of the mirror is about 150 mm, the radius of the vertex is 350 mm, and the conic constant is 1.9732. The convex hyperboloid mirror is mounted with a 6-axis adjustor, which could rotate via computer control to an accuracy of 0.001251. A 6ʺ Zygo interferometer with an f/5.3 transmission sphere is utilized in this
Fig. 2. Two annuluses test: (a) experimental setup and (b) lattice scheme.
F. Yan et al. / Optics and Lasers in Engineering 61 (2014) 52–56
55
experiment. Two auxiliary spheres with 180 mm in diameter are needed to obtain the inner and outer subaperture phase data. The radiuses of the spheres are about 629.81 mm. The experiment setup is shown in Fig. 2(a). In this experiment, the mirror is tested with two annuluses. In total 20 subapertures, 8 subapertures in the inner annulus and 12 subapertures in the outer annulus, are required to cover the full-aperture surface as shown in Fig. 2(b). The subapertures are tested one by one. One of the inner subaperture surface phase data is shown in Fig. 3(a). And Fig. 3(b) shows one of the outer annulus subaperture surface phase data. From Fig. 3, we can clearly see that the misalignment errors are different in the two raw subaperture surface phase data. Theoretically, if the sphere, convex hyperbolic and interferometer are in the ideal position when the measurement was taken, the phase data that we obtained from the interferometer is the mirror surface shape. However, there are always some tilt and defocus misalignments among the interferometer, mirror and sphere. These misalignments would introduce some piston, tilts, defocus, coma and primary spherical error [24] in the raw subaperture phase data. Generally, the piston, tilts, defocus, coma and primary spherical aberration should be removed from the stitching results in this measurement method. The mirror surface shape is successfully retrieved with our measurement method. And the stitching results which removed piston, tilts, defocus, coma and primary spherical are shown in Fig. 4 (a), and the PV and RMS values are approximately 340.34 nm and 43.17 nm, respectively. In order to estimate the accuracy of this method, the mismatch error in the overlapping region among the 20 subapertures is calculated as shown in Fig. 4(b), and the PV¼ 151.69 nm, RMS¼4.23 nm. From Fig. 4, we can see that the RMS in mismatch error is only about ten percent of the tested mirror.
Fig. 4. Experiment results: (a) stitching phase data with two annuluses (PV¼ 340.34 nm, RMS ¼ 43.17 nm) and (b) overlapping mismatch data (PV¼ 151.69 nm, RMS¼ 4.23 nm).
Fig. 3. Measured subapertures: (a) subaperture 1 and (b) subaperture 9.
This accuracy is enough to testify the convex hyperboloid mirror measurement precision. However, the center of the mirror cannot be tested by this measurement method. In order to check the theory exactitude of the measurement and the algorithm, this convex hyperbolic mirror was measured with one annulus as introduced in our previous work [23]. The results removed piston, tilts, defocus, coma and primary spherical are shown in Fig. 5. We can see that the results are in agreement with those in Fig. 4(a), and the PV and RMS values are approximately 360.81 nm and 43.33 nm, respectively. Comparing Fig. 4(a) with Fig. 5, the surface shape is highly similar, and the difference of PV and RMS value is approximate 20.47 nm and 0.16 nm, respectively. The first 36 terms Zernike polynomial coefficients which are calculated for the above results of Figs. 4(a) and 5 are shown in Fig. 6. The coincidence of the two coefficients variation can testify the coincidence of the low order surface that is tested by the two methods. Fig. 7 shows the measurement results’ residual figure between the two methods, specifically the difference between Fig. 4(a) and Fig. 5, where the PV difference is 75.26 nm and the RMS is 5.62 nm. This difference is mainly caused by auxiliary sphere surface deviation. These results show that the Hindle and stitching method with two annuluses is not inducing significant
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F. Yan et al. / Optics and Lasers in Engineering 61 (2014) 52–56
4. Conclusion
Fig. 5. Full aperture test with one annulus (PV ¼ 360.81 nm, RMS¼ 43.33 nm).
In summary, we extended the measurement method of measuring the large convex hyperbolic mirror and discussed the testing method with two or more annulus. The correspondence of overlapping point pairs and the compensation of misalignment errors in different subapertures has been studied. Through the experiment of two annuluses, it has demonstrated that the convex hyperbolic mirrors could be tested with two annuluses by two small auxiliary spheres and the full aperture surface shape could be obtained with satisfactory accuracy. However, the obscuration area surface shape in the mirror center could not be obtained by this method. Though the convex hyperbolic mirror is tested with two annuluses in the experiments, it can be extended in more annuluses theoretically. At the same time, we are aware that the experiment does prove something, but not everything. We will focus our future work on the following issues. The systematic errors induced by the auxiliary sphere and the interferometer must be separated from the measurements; otherwise the accuracy is mainly limited by the auxiliary sphere.
Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant no. 60908042. References
Fig. 6. Comparison of the first 36 terms Zernike decomposition of results shown in Fig. 4(a) and Fig. 5.
Fig. 7. Residual figure between Fig. 4(a) and Fig. 5 (PV¼ 75.26 nm, RMS¼ 5.62 nm).
inaccuracy. In summary, the stitching measurements of the convex hyperbolic with two annuluses have good agreement with the one annulus stitching measurement results.
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