- Email: [email protected]
Measurement of large convex hyperbolic mirrors using hindle and stitching methods
Optics and Lasers in Engineering 51 (2013) 856–860
Contents lists available at SciVerse ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Measurement of large convex hyperbolic mirrors using 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 Science, Chengdu 610209, China Graduate School of Chinese Academy of Sciences, Beijing 100039, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 November 2012 Received in revised form 14 January 2013 Accepted 20 January 2013 Available online 13 March 2013
Based on the traditional Hindle-Sphere method and Subaperture stitching method, we introduce a measurement method for large convex hyperbolic mirrors. This method uses a small sphere to get the subaperture’s surface shape of the convex hyperbolic mirror, and then uses the stitching method to get the full-aperture surface shape. The experimental demonstrations were performed on a convex hyperbolic mirror with our method. The results show that the repeatability of the RMS is 0.002l (3s). Compared with the traditional optical compensation method, it is shown that our results are in good agreement with it. Our method can get high-precision full-aperture surface shapes; furthermore, the cost of maintaining it is relatively low and the operation is convenient. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Large convex hyperbolic Surface measurements Hindle-sphere Stitching interferometry Aspheric
1. Introduction In the past decades, extensive studies have focused on the convex hyperbolic mirrors. They have attracted much attention for a wide range of utility in modern telescope as a secondary mirror (M2 mirror). And with the development of modern telescopes, their apertures have been extended to several meters [1–3]. Large mirror fabrication needs the large mirror test technology. High-precision fabrication also needs the high precise mirror test technology. Reportedly, there have been some methods to test the large convex hyperbolic mirrors. For example, VISTA Telescope’s M2 mirror was measured by the Hindle Sphere method which needed a 2.4 m diameter sphere [4]. However, it was difficult to make such a huge sphere. The SAGEM Optronics and AirLand Systems—REOSC High Performance Optics had used the large optical gauge test method to test the VLT’s and Gemini Telescope’s M2 mirrors [5]. This method needed a large size high optical homogeneity materials and a concave aspheric surface which were very difficult to produce. The holographic test plate method was used to test the 1.7 m secondary mirror for the MMT [6,7]. However, this test used a full-aperture test plate with a computer-generated hologram (CGH) which was very difficult to be fabricated onto the spherical reference surface. The large aperture vibration insensitive sub-aperture Fizeau interferometer
n Corresponding author at:Chinese Academy of Science, Institute of Optics and Electronics, P.O.B.350, Chengdu 610209, China. Tel.: þ86 13408590346. E-mail address: [email protected] (F. Yan).
0143-8166/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlaseng.2013.01.020
combined with stitching software was also used in a 1.7-m secondary mirror test [8]. However, a 1-m sub-aperture aspheric Fizeau test plate was difficult to fabricate in this method. In 2003, QED Technologies had developed a general-purpose stitching interferometer workstation (SSI) [9] that can automatically carry out high-quality subaperture stitching of flat, spherical, and milddeparture aspheric surfaces. In 2006, the QED’s SSI-ATM added asphere metrology to the SSI’s capabilities [10]. In 2010, with the help of the Variable Optical Null (VON) technology, QED Technologies had developed the Aspheric Stitching Interferometer (ASI). It even could test a wider range of aspheres with up to 1000 waves of departure [11]. But the largest aperture that the workstation can test is only 200 mm in diameter. Chen [12,13] had used the iterative stitching algorithm to test the large optical surface by a small aperture interferometer. But it was only applied to the concave mirror. The stitching algorithm was very complex, and it cost a long time to get the full aperture. In this paper, we introduce a simple measurement method for the large convex hyperbolic mirror surfaces in high precision. It is based on the traditional Hindle-Sphere method and Subaperture stitching method. In this method the assistant element is only a small sphere and a 4 in. interferometer. This paper is organized as follows. In Section 2, the method configuration is described. The assistant sphere parameters that were used in the method and the subaperture stitching theory are particularly described in this section. In Section 3, the experimental setup and the results are described. Our measurement method has been used to test a convex hyperboloid mirror with 150-mm diameter and f/1.17; then we compare the results with those of the traditional optical
F. Yan et al. / Optics and Lasers in Engineering 51 (2013) 856–860
compensation method. In Section 4, a brief summary and conclusions are given. We also discuss some limitations of this measurement in this section.
2. Prototype for the large convex hyperbolic mirrors testing 2.1. Measurement configuration and the assistant sphere parameters Our measurement configuration is shown in Fig. 1. According to Fig. 1, in our method the light path to and from the convex hyperbolic mirror is the same as the Hindle-sphere method in the operational instrument. But our sphere’s diameter is much smaller than that used in the traditional Hindle-sphere method. A part of the convex hyperbolic mirror surface would be tested at one position, and then we get one subaperture surface shape. Rotating the convex hyperbolic mirror around the optical axis with a special angle to next position, we get another subaperture surface shape. When the subapertures cover the full aperture, we can get the full aperture surface shape by stitching the subapertures together. The sphere parameters are defined as a function, which not only comprises the convex mirrors, but also the overlap areas between two subapertures. The radius of curvature RH and the diameters of its aperture DH are calculated as follows. The diameter of the test mirror outer border is D and hole border is e. c and k are the radius of curvature and the conic constant of the convex mirror, respectively. Based on the theory in Ref. [14], we can obtain RH and DH respectively by 8 y2 ðf 1 f Þ2 ½y21 þ ðz1 f 2 Þ2 1=2 > > < RH ¼ y2 ðz1 f 2 Þy1 ðz2 f 1 Þ 1=2 ð1Þ ðf z2 Þðf 2 z1 Þ þ y1 y2 1=2 > > : DH ¼ 2 RH 1 ½y2 þ ðf 2z Þ2 1=2 ½y2 þ ðf z Þ2 1=2 2
2
2
1
2
1
where cy21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi, y1 ¼ e=2, y2 ¼ D=2, z1 ¼ 1þ 1ðk þ1Þc2 y21 cy2 2 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi, f1 ¼ , c 1ðkÞ1=2 1þ 1ðk þ1Þc2 y2 2 1 1 f2 ¼ c 1 þ ðkÞ1=2 z2 ¼
believable when the overlap area between two subapertures is more than 20% of one subaperture. In our method, the subapertures are like a circle. We suppose it as one circle and the test area is shown in Fig. 2. P is the overlap factor, which is obtained by dividing the overlap area by the area of a subaperture. It can be described by ! 0 00 AB O O UAB 2 2 P ¼ arctan 0 00 U Z0:2 ð2Þ p p O0 O00 2 þ AB2 OO We obtain 9AB9=9O0 O00 9 Z 1:0576. According to Fig. 2 and based on the geometry we can obtain ( AB ¼ ðDeÞ=2 0 00 ð3Þ O O ¼ ðD þ eÞ=2Utanð+BOO’Þ
y ¼ +O0 OO00 ¼ 2+BOO0 is the angle at which we rotate the test mirror to get another subaperture surface shape. We can obtain y by De y r2arctan 0:9456 ð4Þ Dþe The number of subapertuers is 2p/y. We take its integer plus 1 as the subapertuer number N. The rotation angle y is all equal to 2p/N. And the 9OC9 and 9OD9 should be used to confirm the sphere parameters; they could be described as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ðD þ eÞ2 Dþe De > < jOC j ¼ 4cosð y=2Þ 16cos2 ðy=2Þ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ ðD þ eÞ2 > Dþe De : jODj ¼ 4cosð y=2Þ þ 16cos2 ðy=2Þ 4 In Eq. (1), let y1 ¼ jOC j,y2 ¼ jODj, then we obtain the sphere parameters that our method needs. 2.2. Subaperture stitching theory The subaperture stitching theory has been developed for 30 years and has many stitching algorithms [12–13,15–22]. In order to get high precision result, our method is based on the Discrete Phase Method. Because the tested mirror is aspheric, we take the piston, tilt, power and coma as the alignment errors. All the subaperture surface shapes are compensated by the information of the overlap area to include all the subapertures into one
If we want to use a smaller sphere to test the hyperbolic mirror, we could divide the hyperbolic mirror in different zones. Then the spherical parameters are obtained by the zone parameters. The sphere parameters obtained from Eq. (1) only take into account the convex mirror parameters. The subaperture tested by this sphere could not cover the full convex mirror surface. Based on the subaperture stitching theory, the stitching results are
Fig. 1. Mainframe of our system configuration for large convex hyperbolic mirrors test.
857
Fig. 2. The position of the subapertures.
858
F. Yan et al. / Optics and Lasers in Engineering 51 (2013) 856–860
3. Experiment results
coordinate: W i ¼ W i þ Ai Z 1 ðr, yÞ þBi Z 2 ðr, yÞ þ C i Z 3 ðr, yÞ þ Di Z 4 ðr, yÞ þ Ei Z 7 ðr, yÞ þ F i Z 8 ðr, yÞ
ð6Þ
where W i is the compensation result of Wi, Ai, y, Fi are the stitching parameters for compensating the alignment error, Z1, y, Z8 are the Zernike polynomial. The least-squares method is used to calculate the stitching parameters. Ideally, the compensation results between the adjoining subapertures W i and W i þ 1 should be equal in the overlap area. We could get the matrix equation as " # ½Mi ð7Þ ½½Hi ,½Hi U þ ½W i W i þ 1 ¼ 0 ½M i þ 1 where Hi is the coefficient matrix in the overlap area, Mi and Mi þ 1 are the stitching parameters for compensating, Wi and Wi þ 1 are the alignment errors, respectively. If there are N subapertures when the mirror is tested by our measurement method, the matrix equations could be given by 2 6 6 6 6 6 6 4
½H 1 0 ^ 0 ½HN
½H1 ½H 2 ^ 0
0 ½H2 ^ 0
&
0 0 ^ ½HN1
0
0
0
3 3 2 3 2 ½M 1 0 W 1 W 2 7 6 ½M 7 6 W W 7 0 2 2 3 7 7 6 7 6 7 7 6 7 6 7¼0 7U6 ^ 7 þ 6 ^ ^ 7 7 6 7 6 7 7 6W 6 ½M ½HN1 7 W 5 4 N1 5 4 N1 N5 ½H N ½M N W N W 1
ð8Þ The least-squares method is used to obtain the stitching parameters M1, M2 y, MN. The compensation results (W 1 , W 2 ,y, W N ) can be worked out by Eq. (6). Then we stitch them to get the full tested mirrors surface shape.
As shown in Fig. 3, the experimental setup for testing the convex hyperbolic mirror is very simple, which consists of a Fizeau interferometer (ZYGO GPI XP 4’), a 6-axis adjustor and a small sphere. The f/1.17 and 150 mm convex hyperboloid mirror under test is mounted with a 6-axis adjustor. The subaperture surface shape can be accurately extracted by Fizeau interferometer. In this experiment, eight subapertures were required to cover the full-aperture surface. The experimental results of eight subapertures are shown in Fig. 4. In theory, if the interferometer, sphere and aspheric mirror are in the ideal position, the subaperture surface shape which we had obtained from interferometer is the convex aspheric mirror subaperture surface shape. Actually, we could not make them in the ideal position, so there are alignment errors of the piston, tilts, defocus and coma in each subaperture. Eq. (8) was used to compensate the subaperture alignment errors for the full-aperture shape. They would be removed in the stitched results. The stitched result without alignment errors and nominal aspheric prescription is shown in Fig. 5. The results of the traditional optical compensation method removed with the alignment errors of piston, tilts, defocus and coma are shown in Fig. 6. Compared with Fig. 5, we found that there was a big primary spherical error in the stitching result. We reanalyzed our measurement configuration and found that the primary spherical aberrations were mainly caused by the defocus between the interferometer and convex hyperbolic mirror. In recent years, the adaptive mirrors are widely used as secondary mirrors [23]. They can compensate the low order aberrations, so the tested low order aberrations could be flexible. If we use the laser tracker or computer-generated hologram (CGH) to make the
Fig. 3. Experimental setup for testing a convex hyperboloid mirror with our method.
Fig. 4. Experiment results: the eight subapertures in our measurement.
F. Yan et al. / Optics and Lasers in Engineering 51 (2013) 856–860
859
Fig. 6. Traditional optical compensation method measurement results removed aligment errors like piston, tilts, defocus and coma. (a) Phase data (PV¼ 0.530l, RMS¼ 0.054l). (b) First 36-term Zernike fitting (PV ¼0.252l, RMS ¼0.035l). Fig. 5. Stitched results removed aligment errors like piston, tilts, defocus and coma. (a) Stitched phase data (PV¼ 0.880l, RMS ¼0.138l). (b) First 36-term Zernike fitting (PV ¼ 0.590l, RMS ¼0.140l).
interferometer, sphere and aspheric mirror in the ideal position, more low order aberrations can be measured. We compared the results again with our results from which piston, tilts, defocus, coma, and primary spherical aberrations were removed. Our results are shown in Fig. 7. The mean RMS of 30 measurements is 0.052l. The repeatability of the RMS is 0.002l (3s). The size of Fig. 6(a) is only 435 435 pixels. Our results are shown in Fig. 7(a), which has the size of 635 635 pixels. It is clear that our method has a higher lateral resolution than the traditional method. Comparing Fig. 6 with Fig. 7, the surface shapes are highly similar, and the differences of PV and RMS values are approximately 0.051l and 0.001l, respectively. In order to clearly compare the results of the two methods, we make the size of the figures depicting both the results of the same size and obtain a direct subtraction of Fig. 6 (a) and Fig. 7(a). This was shown in Fig. 8. The differences of PV and RMS values are 0.125l and 0.015l, respectively. As shown in Fig. 8, the differences looked like ripple and ‘‘swirling’’; the reason may be affected by the environment, as vibration and air disturbance [24]. In summary, the stitching results of our measurement method have good agreement with those of the traditional optical
compensation method measurement without the alignment errors (piston, tilts, defocus, coma and primary spherical).
4. Conclusion and discussion In this paper, a new measurement of large convex hyperbolic mirrors using Hindle and stitching methods was introduced. The measurement configuration and the stitching theory have been clearly described. This method used a small sphere to obtain the subaperture’s surface of the convex hyperbolic mirror, and then used the stitching method to get the full surface shape. The experimental demonstrations were performed on a convex hyperbolic mirror with our method. The results show that the repeatability of the RMS is 0.002l (3s). The experimental results prove that the proposed method can obtain the stitched full-aperture surface shape with satisfactory accuracy. But there are some problems. Firstly, it will consume a little longer time to test the large convex aspheric surface shape as it would be susceptible to the environment. Secondly, the defocus between the interferometer and the convex aspheric mirror is not easy to confirm that would cause primary spherical error. So we used the laser tracker or CGH to obtain the interferometer, sphere and aspheric mirror in
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F. Yan et al. / Optics and Lasers in Engineering 51 (2013) 856–860
Fig. 8. Direct subtraction of Fig. 6 (a) and Fig. 7(a) maps (PV¼ 0.125l, RMS¼0.015l).
Fig. 7. Stitched results removed aligment errors like piston, tilts, defocus, coma and primary spherical. (a) Stitched phase data (PV¼ 0.479l, RMS ¼0.053l). (b) First 36-term Zernike fitting (PV ¼0.216l, RMS ¼0.029l).
the ideal position; then the convex hyperbolic mirror surfaces could be measured for more low-order aberrations. In summary, it is shown that our results are in good agreement compared with those of the traditional optical compensation method. Our method can yield high-precision full-aperture surface shapes; furthermore, the cost of maintaining is relatively low and the operation is convenient. Further theoretical and experimental development of the proposed method will allow the characterization of some interesting higher precision full-aperture surface shapes in the full convex hyperbolic aperture stitching.
Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant no. 60908042. References [1] Olivier S, Seppala L, Gilmore K, Hale L, Whistler W. LSST camera optics. Proc SPIE 2006;6273:62730Y.
[2] The telescope: optics: /http://www.tmt.org./observatory/telescope/opticsS. [3] The European Extremely Large Telescope: /http://www.eso.org/public/tele s-instr/e-elt.htmlS. [4] 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. [5] Geyl R, Ruch E, Primet S, Chaussat G. Progress toward a third Gemini M2 mirror. Proc SPIE 2005;5869:586901. [6] Burge JH, Anderson DS. Full-aperture interferometric test of convex secondary mirrors using holographic test plates. Proc SPIE 1994;2199:181–92. [7] Burge JH. Measurement of large convex aspheres. Proc SPIE 1997;2871: 362–73. [8] Burge JH, Su P, Zhao C. Optical metrology for very large convex aspheres. Proc SPIE 2008;7018:701818. [9] 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. [10] Tricard, M and Murphy, PE, Subaperture stitching for large aspheric surfaces, in Talk for NASA Tech Day 2006. [11] Kulawiec A, Murphy P, Demarco M. Measurement of high-departure aspheres using subaperture stitching with the Variable Optical Null (VONTM). Proc SPIE 2010;7655:765512. [12] Chen S, Li S, Dai Y, Zheng Z. Testing of large optical surfaces with subaperture stitching. Appl Opt 2007;46(17):3504–9. [13] Chen S, Li S, Dai Y, Ding L, Zeng S. Experimental study on subaperture testing with iterative stitching algorithm. Opt Express 2008;16(7):4760–5. [14] Percino-Zacarias ME, Cordero-Da´vila A. n-Hindle-sphere arrangement with an exact ray trace for testing hyperboloid convex mirrors. Appl Opt 1999;38(28):6050–4. [15] Kim CJ. Polynomial fit of interferograms. Appl Opt. 1982;21(24):4521–5. [16] Otsubo M, Okada K, Tsujiuchi J. Measurement of large plane surface shapes by connecting small-aperture interferograms. Opt Eng 1994;33(2):608–13. [17] Cheng Weiming, Chen Mingyi. Transformation and connection of subapertures in the multiaperture overlap-scanning technique for large optics tests. Opt Eng 1993;32(8):1947–50. [18] Hongwei Guo, Chen Mingyi. Multiview connection technique for 360-deg three-dimensional measurement. Opt Eng 2003;42(04):900–5. [19] Murphy PE, Fleig J, Forbes G, Tricard M. High precision metrology of domes and aspheric optics. Proc SPIE 2005;5786:112–21. [20] Hou X, Wu F, Yang L, Wu S, Chen Q. Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces. Appl Opt 2006;45(15):3442–55. [21] Su P, Burge J, Sprowl RA, Sasian J. Maximum likelihood estimation as a general method of combining sub-aperture data for interferometric testing. Proc SPIE OSA 2006; 6342, 63421X. [22] Xu H, Xian H, Zhang Y. Algorithm and experiment of whole-aperture wavefront reconstruction from annular subaperture Hartmann–Shack gradient data. Opt Express 2010;18(13):13431–43. [23] Esposito S, Riccardi A, Quiro´s-pacheco F, Pinna E, Puglisi A, Xompero M, et al. Laboratory characterization and performance of the high-order adaptive optics system for the Large Binocular Telescope. Appl Opt 2010;49(31): G174–89. [24] Lars A. Selberg, Interferometer accuracy and precision. Proc SPIE 1990;1400: 24–32.
Contents lists available at SciVerse ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Measurement of large convex hyperbolic mirrors using 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 Science, Chengdu 610209, China Graduate School of Chinese Academy of Sciences, Beijing 100039, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 November 2012 Received in revised form 14 January 2013 Accepted 20 January 2013 Available online 13 March 2013
Based on the traditional Hindle-Sphere method and Subaperture stitching method, we introduce a measurement method for large convex hyperbolic mirrors. This method uses a small sphere to get the subaperture’s surface shape of the convex hyperbolic mirror, and then uses the stitching method to get the full-aperture surface shape. The experimental demonstrations were performed on a convex hyperbolic mirror with our method. The results show that the repeatability of the RMS is 0.002l (3s). Compared with the traditional optical compensation method, it is shown that our results are in good agreement with it. Our method can get high-precision full-aperture surface shapes; furthermore, the cost of maintaining it is relatively low and the operation is convenient. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Large convex hyperbolic Surface measurements Hindle-sphere Stitching interferometry Aspheric
1. Introduction In the past decades, extensive studies have focused on the convex hyperbolic mirrors. They have attracted much attention for a wide range of utility in modern telescope as a secondary mirror (M2 mirror). And with the development of modern telescopes, their apertures have been extended to several meters [1–3]. Large mirror fabrication needs the large mirror test technology. High-precision fabrication also needs the high precise mirror test technology. Reportedly, there have been some methods to test the large convex hyperbolic mirrors. For example, VISTA Telescope’s M2 mirror was measured by the Hindle Sphere method which needed a 2.4 m diameter sphere [4]. However, it was difficult to make such a huge sphere. The SAGEM Optronics and AirLand Systems—REOSC High Performance Optics had used the large optical gauge test method to test the VLT’s and Gemini Telescope’s M2 mirrors [5]. This method needed a large size high optical homogeneity materials and a concave aspheric surface which were very difficult to produce. The holographic test plate method was used to test the 1.7 m secondary mirror for the MMT [6,7]. However, this test used a full-aperture test plate with a computer-generated hologram (CGH) which was very difficult to be fabricated onto the spherical reference surface. The large aperture vibration insensitive sub-aperture Fizeau interferometer
n Corresponding author at:Chinese Academy of Science, Institute of Optics and Electronics, P.O.B.350, Chengdu 610209, China. Tel.: þ86 13408590346. E-mail address: [email protected] (F. Yan).
0143-8166/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlaseng.2013.01.020
combined with stitching software was also used in a 1.7-m secondary mirror test [8]. However, a 1-m sub-aperture aspheric Fizeau test plate was difficult to fabricate in this method. In 2003, QED Technologies had developed a general-purpose stitching interferometer workstation (SSI) [9] that can automatically carry out high-quality subaperture stitching of flat, spherical, and milddeparture aspheric surfaces. In 2006, the QED’s SSI-ATM added asphere metrology to the SSI’s capabilities [10]. In 2010, with the help of the Variable Optical Null (VON) technology, QED Technologies had developed the Aspheric Stitching Interferometer (ASI). It even could test a wider range of aspheres with up to 1000 waves of departure [11]. But the largest aperture that the workstation can test is only 200 mm in diameter. Chen [12,13] had used the iterative stitching algorithm to test the large optical surface by a small aperture interferometer. But it was only applied to the concave mirror. The stitching algorithm was very complex, and it cost a long time to get the full aperture. In this paper, we introduce a simple measurement method for the large convex hyperbolic mirror surfaces in high precision. It is based on the traditional Hindle-Sphere method and Subaperture stitching method. In this method the assistant element is only a small sphere and a 4 in. interferometer. This paper is organized as follows. In Section 2, the method configuration is described. The assistant sphere parameters that were used in the method and the subaperture stitching theory are particularly described in this section. In Section 3, the experimental setup and the results are described. Our measurement method has been used to test a convex hyperboloid mirror with 150-mm diameter and f/1.17; then we compare the results with those of the traditional optical
F. Yan et al. / Optics and Lasers in Engineering 51 (2013) 856–860
compensation method. In Section 4, a brief summary and conclusions are given. We also discuss some limitations of this measurement in this section.
2. Prototype for the large convex hyperbolic mirrors testing 2.1. Measurement configuration and the assistant sphere parameters Our measurement configuration is shown in Fig. 1. According to Fig. 1, in our method the light path to and from the convex hyperbolic mirror is the same as the Hindle-sphere method in the operational instrument. But our sphere’s diameter is much smaller than that used in the traditional Hindle-sphere method. A part of the convex hyperbolic mirror surface would be tested at one position, and then we get one subaperture surface shape. Rotating the convex hyperbolic mirror around the optical axis with a special angle to next position, we get another subaperture surface shape. When the subapertures cover the full aperture, we can get the full aperture surface shape by stitching the subapertures together. The sphere parameters are defined as a function, which not only comprises the convex mirrors, but also the overlap areas between two subapertures. The radius of curvature RH and the diameters of its aperture DH are calculated as follows. The diameter of the test mirror outer border is D and hole border is e. c and k are the radius of curvature and the conic constant of the convex mirror, respectively. Based on the theory in Ref. [14], we can obtain RH and DH respectively by 8 y2 ðf 1 f Þ2 ½y21 þ ðz1 f 2 Þ2 1=2 > > < RH ¼ y2 ðz1 f 2 Þy1 ðz2 f 1 Þ 1=2 ð1Þ ðf z2 Þðf 2 z1 Þ þ y1 y2 1=2 > > : DH ¼ 2 RH 1 ½y2 þ ðf 2z Þ2 1=2 ½y2 þ ðf z Þ2 1=2 2
2
2
1
2
1
where cy21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi, y1 ¼ e=2, y2 ¼ D=2, z1 ¼ 1þ 1ðk þ1Þc2 y21 cy2 2 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi, f1 ¼ , c 1ðkÞ1=2 1þ 1ðk þ1Þc2 y2 2 1 1 f2 ¼ c 1 þ ðkÞ1=2 z2 ¼
believable when the overlap area between two subapertures is more than 20% of one subaperture. In our method, the subapertures are like a circle. We suppose it as one circle and the test area is shown in Fig. 2. P is the overlap factor, which is obtained by dividing the overlap area by the area of a subaperture. It can be described by ! 0 00 AB O O UAB 2 2 P ¼ arctan 0 00 U Z0:2 ð2Þ p p O0 O00 2 þ AB2 OO We obtain 9AB9=9O0 O00 9 Z 1:0576. According to Fig. 2 and based on the geometry we can obtain ( AB ¼ ðDeÞ=2 0 00 ð3Þ O O ¼ ðD þ eÞ=2Utanð+BOO’Þ
y ¼ +O0 OO00 ¼ 2+BOO0 is the angle at which we rotate the test mirror to get another subaperture surface shape. We can obtain y by De y r2arctan 0:9456 ð4Þ Dþe The number of subapertuers is 2p/y. We take its integer plus 1 as the subapertuer number N. The rotation angle y is all equal to 2p/N. And the 9OC9 and 9OD9 should be used to confirm the sphere parameters; they could be described as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ðD þ eÞ2 Dþe De > < jOC j ¼ 4cosð y=2Þ 16cos2 ðy=2Þ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ ðD þ eÞ2 > Dþe De : jODj ¼ 4cosð y=2Þ þ 16cos2 ðy=2Þ 4 In Eq. (1), let y1 ¼ jOC j,y2 ¼ jODj, then we obtain the sphere parameters that our method needs. 2.2. Subaperture stitching theory The subaperture stitching theory has been developed for 30 years and has many stitching algorithms [12–13,15–22]. In order to get high precision result, our method is based on the Discrete Phase Method. Because the tested mirror is aspheric, we take the piston, tilt, power and coma as the alignment errors. All the subaperture surface shapes are compensated by the information of the overlap area to include all the subapertures into one
If we want to use a smaller sphere to test the hyperbolic mirror, we could divide the hyperbolic mirror in different zones. Then the spherical parameters are obtained by the zone parameters. The sphere parameters obtained from Eq. (1) only take into account the convex mirror parameters. The subaperture tested by this sphere could not cover the full convex mirror surface. Based on the subaperture stitching theory, the stitching results are
Fig. 1. Mainframe of our system configuration for large convex hyperbolic mirrors test.
857
Fig. 2. The position of the subapertures.
858
F. Yan et al. / Optics and Lasers in Engineering 51 (2013) 856–860
3. Experiment results
coordinate: W i ¼ W i þ Ai Z 1 ðr, yÞ þBi Z 2 ðr, yÞ þ C i Z 3 ðr, yÞ þ Di Z 4 ðr, yÞ þ Ei Z 7 ðr, yÞ þ F i Z 8 ðr, yÞ
ð6Þ
where W i is the compensation result of Wi, Ai, y, Fi are the stitching parameters for compensating the alignment error, Z1, y, Z8 are the Zernike polynomial. The least-squares method is used to calculate the stitching parameters. Ideally, the compensation results between the adjoining subapertures W i and W i þ 1 should be equal in the overlap area. We could get the matrix equation as " # ½Mi ð7Þ ½½Hi ,½Hi U þ ½W i W i þ 1 ¼ 0 ½M i þ 1 where Hi is the coefficient matrix in the overlap area, Mi and Mi þ 1 are the stitching parameters for compensating, Wi and Wi þ 1 are the alignment errors, respectively. If there are N subapertures when the mirror is tested by our measurement method, the matrix equations could be given by 2 6 6 6 6 6 6 4
½H 1 0 ^ 0 ½HN
½H1 ½H 2 ^ 0
0 ½H2 ^ 0
&
0 0 ^ ½HN1
0
0
0
3 3 2 3 2 ½M 1 0 W 1 W 2 7 6 ½M 7 6 W W 7 0 2 2 3 7 7 6 7 6 7 7 6 7 6 7¼0 7U6 ^ 7 þ 6 ^ ^ 7 7 6 7 6 7 7 6W 6 ½M ½HN1 7 W 5 4 N1 5 4 N1 N5 ½H N ½M N W N W 1
ð8Þ The least-squares method is used to obtain the stitching parameters M1, M2 y, MN. The compensation results (W 1 , W 2 ,y, W N ) can be worked out by Eq. (6). Then we stitch them to get the full tested mirrors surface shape.
As shown in Fig. 3, the experimental setup for testing the convex hyperbolic mirror is very simple, which consists of a Fizeau interferometer (ZYGO GPI XP 4’), a 6-axis adjustor and a small sphere. The f/1.17 and 150 mm convex hyperboloid mirror under test is mounted with a 6-axis adjustor. The subaperture surface shape can be accurately extracted by Fizeau interferometer. In this experiment, eight subapertures were required to cover the full-aperture surface. The experimental results of eight subapertures are shown in Fig. 4. In theory, if the interferometer, sphere and aspheric mirror are in the ideal position, the subaperture surface shape which we had obtained from interferometer is the convex aspheric mirror subaperture surface shape. Actually, we could not make them in the ideal position, so there are alignment errors of the piston, tilts, defocus and coma in each subaperture. Eq. (8) was used to compensate the subaperture alignment errors for the full-aperture shape. They would be removed in the stitched results. The stitched result without alignment errors and nominal aspheric prescription is shown in Fig. 5. The results of the traditional optical compensation method removed with the alignment errors of piston, tilts, defocus and coma are shown in Fig. 6. Compared with Fig. 5, we found that there was a big primary spherical error in the stitching result. We reanalyzed our measurement configuration and found that the primary spherical aberrations were mainly caused by the defocus between the interferometer and convex hyperbolic mirror. In recent years, the adaptive mirrors are widely used as secondary mirrors [23]. They can compensate the low order aberrations, so the tested low order aberrations could be flexible. If we use the laser tracker or computer-generated hologram (CGH) to make the
Fig. 3. Experimental setup for testing a convex hyperboloid mirror with our method.
Fig. 4. Experiment results: the eight subapertures in our measurement.
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Fig. 6. Traditional optical compensation method measurement results removed aligment errors like piston, tilts, defocus and coma. (a) Phase data (PV¼ 0.530l, RMS¼ 0.054l). (b) First 36-term Zernike fitting (PV ¼0.252l, RMS ¼0.035l). Fig. 5. Stitched results removed aligment errors like piston, tilts, defocus and coma. (a) Stitched phase data (PV¼ 0.880l, RMS ¼0.138l). (b) First 36-term Zernike fitting (PV ¼ 0.590l, RMS ¼0.140l).
interferometer, sphere and aspheric mirror in the ideal position, more low order aberrations can be measured. We compared the results again with our results from which piston, tilts, defocus, coma, and primary spherical aberrations were removed. Our results are shown in Fig. 7. The mean RMS of 30 measurements is 0.052l. The repeatability of the RMS is 0.002l (3s). The size of Fig. 6(a) is only 435 435 pixels. Our results are shown in Fig. 7(a), which has the size of 635 635 pixels. It is clear that our method has a higher lateral resolution than the traditional method. Comparing Fig. 6 with Fig. 7, the surface shapes are highly similar, and the differences of PV and RMS values are approximately 0.051l and 0.001l, respectively. In order to clearly compare the results of the two methods, we make the size of the figures depicting both the results of the same size and obtain a direct subtraction of Fig. 6 (a) and Fig. 7(a). This was shown in Fig. 8. The differences of PV and RMS values are 0.125l and 0.015l, respectively. As shown in Fig. 8, the differences looked like ripple and ‘‘swirling’’; the reason may be affected by the environment, as vibration and air disturbance [24]. In summary, the stitching results of our measurement method have good agreement with those of the traditional optical
compensation method measurement without the alignment errors (piston, tilts, defocus, coma and primary spherical).
4. Conclusion and discussion In this paper, a new measurement of large convex hyperbolic mirrors using Hindle and stitching methods was introduced. The measurement configuration and the stitching theory have been clearly described. This method used a small sphere to obtain the subaperture’s surface of the convex hyperbolic mirror, and then used the stitching method to get the full surface shape. The experimental demonstrations were performed on a convex hyperbolic mirror with our method. The results show that the repeatability of the RMS is 0.002l (3s). The experimental results prove that the proposed method can obtain the stitched full-aperture surface shape with satisfactory accuracy. But there are some problems. Firstly, it will consume a little longer time to test the large convex aspheric surface shape as it would be susceptible to the environment. Secondly, the defocus between the interferometer and the convex aspheric mirror is not easy to confirm that would cause primary spherical error. So we used the laser tracker or CGH to obtain the interferometer, sphere and aspheric mirror in
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Fig. 8. Direct subtraction of Fig. 6 (a) and Fig. 7(a) maps (PV¼ 0.125l, RMS¼0.015l).
Fig. 7. Stitched results removed aligment errors like piston, tilts, defocus, coma and primary spherical. (a) Stitched phase data (PV¼ 0.479l, RMS ¼0.053l). (b) First 36-term Zernike fitting (PV ¼0.216l, RMS ¼0.029l).
the ideal position; then the convex hyperbolic mirror surfaces could be measured for more low-order aberrations. In summary, it is shown that our results are in good agreement compared with those of the traditional optical compensation method. Our method can yield high-precision full-aperture surface shapes; furthermore, the cost of maintaining is relatively low and the operation is convenient. Further theoretical and experimental development of the proposed method will allow the characterization of some interesting higher precision full-aperture surface shapes in the full convex hyperbolic aperture stitching.
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