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Study on the Stressed Mirror Polishing with a Continuous Polishing Machine for Large Aperture Off-axis Aspheric Mirrors
CHINESE ASTRONOMY AND ASTROPHYSICS
ELSEVIER
Chinese Astronomy and Astrophysics 36 (2012) 435–444
Study on the Stressed Mirror Polishing with a Continuous Polishing Machine for Large Aperture Off-axis Aspheric Mirrors† ZHANG Hai-ying1,2 CUI Xiang-qun1,2 JIANG Zi-bo1,2 LI Xin-nan1,2 1,2 ZHENG Yi LIU Xing-tao1,2 NI Hou-kun1,2 1
National Astronomical Observatories/Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing 210042 2 Key Laboratory of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing 210042
Abstract A special stressed annular polishing technique is proposed to mill the off-axis aspheric sub-mirrors of a large segmented mirror with an annular polishing machine. Based on the basic principle of stressed annular polishing technique, a set of special stressing mechanisms are designed to convert milling the aspheric surfaces of sub-mirrors with different off-axis distances into milling the spherical surfaces with identical radii of curvature, so that they can be polished simultaneously on a continuous polishing machine. It took about continuous 40 hours to polish a scaled-down mirror of the planning Chinese Future Giant Telescope (CFGT) using this technique. This mirror has the 330 mm diameter, 3.6 m off-axis distance, and the 21.6 m radius of curvature, and its maximum asphericity is 16 micron. The experiment shows that this method has a high efficiency, suits batch manufacturing, especially the batch manufacturing of aspheric sub-mirrors of the segmented primary mirror of an extremely large aperture telescope. Key words: telescopes—instrumentation: detectors—methods: analytical— techniques: interferometric—techniques: miscellaneous
†
Supported by National Natural Science Foundation Received 2011–09-07; revised version 2011–11–28 A translation of Acta Astron. Sin. Vol. 53, No. 2, pp. 161–170, 2012 [email protected]
0275-1062/12/$-see front matter © 2012 Elsevier All rights reserved. c 2012B.V. 0275-1062/01/$-see front matter Elsevier Science B. V. All rights reserved. doi:10.1016/j.chinastron.2012.10.009 PII:
436
LI Xin-nan et al. / Chinese Astronomy and Astrophysics 36 (2012) 435–444
1. INTRODUCTION In the end of the 20th century, the segmented mirror of 10 m aperture came into use, implying that a major obstacle that restricts the light-gathering ability of a telescope has been overridden, and that to increase the aperture of an astronomical optical/infrared telescope to 30 m and more, even 100 m, is possible. After entering the 21th century, in response to the need of the rapid development of astronomical science, a series of projects of 30 m-class extremely-large aperture telescopes have been proposed in the different countries and areas of the world (refer to References [1-3] and the website http://en.wikipedia.org/wiki/European Extremely Large Telescope), and some fundamental work has been successively initiated in the United States and European countries. The light-gathering ability of the currently proposed extremely-large telescope is 10 times that of existing telescopes, and the number of off-axis aspheric sub-mirrors of its segmented primary mirror is also proportionally increased, reaching several hundreds even one thousand, the associated problem is the conflict between the manufacturing time of large-quantity high-precision optical surfaces and the required period for developing this telescope, therefore, how to mill and polish these mirrors with a high efficiency and precision becomes the key-important problem for developing an extremely-large aperture optical/infrared telescope. The stressed polishing technique for milling the off-axis aspheric surfaces with a largeaperture tool was developed in the course of manufacturing 10 m-class segmented mirrors. According to the calculation of elastic mechanics, an external force is imposed on the mirror blank to produce a deformation in opposition to the required aspheric surface shape, and under the condition of keeping this deformation, to mill and polish the mirror surface to be a spherical surface. After the external force is taken off, the required off-axis aspheric surface is obtained[4−6] . As the surface to be milled is a spherical surface, not an aspheric surface, so a large-aperture tool can be used, and therefore the efficiency is greatly improved. Compared with the conventional digit-controlled polishing with a small machine, the highfrequency surface error can be reduced. Before this, the development of the 36 sub-mirrors of the 10 m KECK primary mirror took about 3 years, and when the GTC telescope was developed, including the time for technological preparation, it took totally 6 years. It is estimated that if making no further improvement, the manufacturing ability of the existing technology will not exceed 20 blanks per year. Obviously, for an extremely-large aperture optical telescope, such a speed of mirror milling far lags behind the schedule for developing the whole-body telescope. The conventional polishing method with an annular polishing machine belongs to a kind of polishing technique of high-precision flat surfaces, the diameter of the polishing module of the machine is far greater than the size of the workpiece, the effective working region is an annulus as wide as a third of the diameter of the polishing module. An annular polishing machine generally has 2∼3 or even more operating positions. It can polish multiple circular or non-circular mirrors simultaneously, and it needs not to stop when a mirror is put in or taken out from it, in favor of batch fabrication. Because of these advantages, the annular polishing technique has been increasingly applied to the fabrication and production of practical optical elements, even the milling process of large-aperture planar and spherical surfaces, which has certain requirements on both quantity and consistency. For example, the sub-mirrors of the reflective Schmidt corrector of LAMOST, the segmented sub-mirrors
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in the American HET telescope and the SALT telescope of South Africa. In practical large-aperture mirror polishing, the annular polishing machine exhibits its advantages of high efficiency and consistency, and shows its precious prospects for applications. This paper has proposed a kind of stressed annular polishing method to mill the large-aperture off-axis aspheric surfaces by adopting an annular polishing machine in combination with the stressed polishing technique, and taking account of the parameters of the primary mirror of the planned 30 m CFGT (Chinese Future Giant Telescope), performed a polishing experiment on the model mirror with a very good experimental result.
2. CHARACTERISTICS AND MATHEMATICAL EXPRESSION OF AN OFF-AXIS ASPHERIC SURFACE The segmented primary mirror of a large optical telescope equivalents to the entrance pupil of the telescope in size, besides the segment unit on the symmetrical axis, other sub-mirrors are all of off-axis, deviating from the symmetrical axis. Fig.1 is a schematic diagram of the position distribution of sub-mirrors for an extremely large telescope, the left and right halves show the different approaches to segmentation. Assuming that the radius of the offaxis mirror is a, for an off-axis aspheric sub-mirror, its surface shape can be expressed by the local coordinates with the origin at its center[7,8] : m αseg (1) Z(ρ, θ) = mn ρ cos(nθ) , mn
in which m, n are integers, m ≥ n ≥ 0, m − n is an even number, ρ = x2 + y 2 /a, θ = arctan(y/x). The coefficients αmn are functions of mirror’s geometrical parameters, dependent on the off-axis distance, aperture, conic coefficient, and the vertex radius of curvature, and so on. 2 − Kε2 a2 α20 = , (2) k 4(1 − Kε2 )3/2 Kε2 a2 , (3) α22 = k 4(1 − Kε2 )3/2 a3 Kε[1 − (K + 1)ε2 ]1/2 (4 − Kε2 ) α31 = 2 , (4) k 8(1 − Kε2 )3 a3 K 2 ε3 [1 − (K + 1)ε2 ]1/2 α33 = 2 , (5) k 8(1 − Kε2 )3 a4 8(1 + K) − 24Kε2 + 3K 2 ε4 (1 − 3K) − K 3 ε6 (2 − K) , (6) α40 = 3 k 64(1 − Kε2 )9/2 a4 Kε2 [2(1 + 3K) − (9 + 7K)Kε2 + (2 + K)K 2 ε4 ] . (7) α42 = 3 k 64(1 − Kε2 )9/2 Here, ε = R/k, R is the off-axis distance, k is the radius of curvature at the vertex of surface of revolution, K is the coefficient of quadric surface. For the sub-mirrors of a segmented mirror, the radius of curvature at the mirror’s center depends on the off-axis position of the sub-mirror, varying from inside to outside.
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Fig. 1 Illustration for the segmentation of an extremely large telescope
When the stressed polishing technique is adopted, all sub-mirrors can be firstly milled to be the spherical surfaces with the same radius of curvature, the difference between the defined spherical surface and the off-axis aspheric surface can be realized by prestressing the sub-mirror, but the surface deformation produced by prestressing should be equal to the difference between the defined spherical surface and the required off-axis aspheric surface. If αseg denotes the coefficient of surface shape, αstr expresses the coefficient after prestressing, then the relation between the two can be expressed as follows: 2
seg seg seg a str αstr αstr 20 = 2L − α20 α22 = −α22 31 = −α31 4 seg seg seg a str αstr αstr 33 = −α33 40 = 8L3 − α40 α42 = −α42
in which L is the radius of curvature of the defined spherical surface. Taking the primary mirror in the proposed 30 m CFGT as an example, it has the outer radius of 30 m, the vertex radius of curvature of 72 m, the coefficient of quadric surface K =1.000863, and the maximum size of sub-mirrors of 1.1 m. The calculated relationship between the sub-mirror’s coefficient of quadric surface and its off-axis distance is shown in Fig.2. From Fig.2 we can find that the deviation of the off-axis aspheric surface from a spherical surface is determined mainly by the coefficients α20 , α22 , and α31 , which correspond respectively to the defocus, the astigmatism and coma aberration of the surface shape. The coefficients α33 , α40 , and α42 are very small, and negligible in the process of stressed polishing. 3. STRESSING MECHANISMS We consider the aspheric sub-mirror of a segmented mirror as a thin-slab structure. In theory, for realizing the conversion between an off-axis aspheric surface and a spherical surface, the shearing force and bending moment imposed on the mirror edge should be distributed continuously. In practical situations, they are substituted by the discrete forces
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Fig. 2 The relationship of the aspheric coefficients with the off-axis distance for the CFGT sub-mirrors
and moments, i.e., to approximate the continuous force and moment along the periphery by setting arch bars at limited discrete points. Assume that the number of loading points is N , the azimuth of each point is θn (n = 1, · · · , N ), θn = 2π(n − 1)/N , then the force and moment on the arch bar at the azimuth θn are: V (θn ) = V1 cos(θn ) + V2 cos(2θn ) ,
(8)
M (θn ) = M0 + M1 cos(θn ) + M2 cos(2θn ) ,
(9)
in which M and V are determined by the surface shape, E is the elastic modulus, ν is the Poisson ratio, h is the thickness. Eh3 , (10) D= 12(1 − ν 2 ) M0 = (D/a2 )[2(1 + ν)αstr 20 ] ,
(11)
M1 = (D/a2 )[2(3 + ν)αstr 31 ] ,
(12)
2
M2 = (D/a )[2(1 − V0 = 0 ,
ν)αstr 22 ] ,
(13) (14)
M1 V1 = − , (15) a 2M2 V2 = . (16) a Summarizing the previous several formulae, the surface shape and the imposed force or moment on the mirror edge are therefore one-by-one correspondent. If only the corresponding mechanisms are available in structure, the requirement on the mirror polishing is realizable.
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Fig.3 gives the overall structure of stressing mechanisms, which differs completely from that adopted by the KECK telescope. The stressing mechanisms of KECK are set along the flank of the mirror, and in Fig.3 all the stressing mechanisms are arranged on the back of the mirror, in favor of polishing on an annular polishing machine. Pistons are taken as the actuators, 12 equally-spaced stressing points are set along the back rim of the mirror.
Fig. 3
Sketch of the combined stressing mechanism
4. TEST OF ASPHERIC SURFACE MILLING 4.1 Test Object The test was conducted using a continuous polishing machine of 1 m aperture. According to the working ability of this machine, it can process the mirror with a diameter of 330 mm in maximum, the test adopted the scaled-down model of CFGT as the test object, which has the following parameters: the vertex radius of curvature 21.6 m, the off-axis aperture 330 mm, the off-axis distance 3.6 m, and considering that the coefficient of quadric surface of the scaled-down model is K =-1.000863, very close to a paraboloid, for simplifying the test equipment and without any loss of generality, the coefficient of quadric surface of the test object was taken to be K =-1. The radius of curvature of the polishing modulus was selected to be 22.08 m, and the maximum asphericity of this off-axis aspheric surface was calculated to be 16.8 μm. 4.2 Test Equipment of Prestress In order to obtain the required surface shape relying on the stressing mechanisms, a special equipment is necessary for the measurement and control. For the 1:3 scaled-down model mirror of CFGT, the maximum deviation of the aspheric surface from a spherical surface is greater than 20 μm, and the dynamical range of variations is rather large, for mea-
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suring the stressed deformation of its surface, an infrared interferometer of 10 μm wavelength or a contact measurement sensor can be adopted. The interferometric measurement has a rather high accuracy but requires a quite long optical path, as a compromise, the contact LVDT array with a compact structure was adopted. We studied the method to fit the surface shape by discrete measured data, including the distribution of sampling points, the fitting algorithm, etc.[9] The computer-aided data sampling and processing were selected to avoid effectively the effects of the measuring environment and other man-made factors on the measuring accuracy, thus the whole data sampling system can give the real-time surface error of the testing mirror, with a rms sampling accuracy of 0.3 μm. Fig.4 is the picture for the real stressing mechanisms and test equipment.
Fig. 4
Stressing mechanism and test equipment
Fig.5 gives the variation of mirror surface measured by the LVDT array during the test of stressing calibration, which shows respectively the variations of the coma aberration, defocus and astigmatism as a function of moment load, the result indicates that the stressing mechanisms exhibit a good linearity, satisfying the requirement. 4.3 Polishing Experiment And Its Result According to the parameters of the test mirror, the loading coefficients of corresponding
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Fig. 5
Curves of the typical aberration values versus the load
aberrations can be calculated as −6 αstr m, 20 = 8.226 × 10
α22str = 8.4 × 10−6 m,
−7 αstr m. 31 = 7.442 × 10
Take the elastic modulus of the mirror blank to be E =91.7 Gpa, the Poisson ratio ν =0.247, and the glass thickness 15 mm, we obtained the theoretical defocus loading force F0 =12.8 N, the coma-aberration loading force F1 =-2.6 N, and the astigmatism loading force F2 =14.8 N. They correspond to the theoretical defocus loading pressure P0 = 5.15 × 104 Pa, the coma-aberration loading pressure P1 = −1.03 × 104 Pa, and the astigmatism loading pressure P2 = −5.96 × 104 Pa. Fig.6 is the picture of the test mirror under the stressed polishing. After 40 hour continuous polishing, the surface error decreases rapidly and becomes stable. Fig.7 illustrates
Fig. 6 The test mirror in stressed polishing
the surface error of the mirror after polishing, the rms error is 0.327λ. Fig.8 shows the convergence process of the mirror surface error during the stressed polishing, and Fig.9
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displays interferogram of the mirror after the final figuring of surface and cutting into a hexagon, the diagonal length of the mirror is 312 mm, and the rms surface accuracy is better than 0.02λ (the operating wavelength of the interferometer is 632.8 nm).
Fig. 7 Result of the stressed mirror polishing with the continuous polishing machine. The left panel is the contour map of mirror surface, and the right panel is the corresponding interferogram
Fig. 8 Convergence curve of the polishing process
5. CONCLUSION AND PROSPECT Aiming at the need of batch manufacturing of off-axis aspheric sub-mirrors in an extremelylarge aperture optical/infrared telescope, and combining for the first time the stressed polishing technique with the annular polishing technique, with the model mirror of the 30 m CFGT as the test object, we have performed an experimental study on the stressed annular polishing technique of large-aperture off-axis aspheric surfaces. The test result indicates that
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Fig. 9
Interferogram of the mirror cut into hexagon
the stressed annular polishing technique is an effective method for milling large-aperture aspheric surfaces, and it has very good application prospects. In view of the current trend of relevant research in the world, the stressed annular polishing technique possesses simultaneously the technological advantages of the stressed method and annular polishing machine method, especially in the aspect of batch manufacturing, it has the advantages of high efficiency and stability. To continue the study and development of the stressed annular polishing technique will give an impetus to the independent development of the Chinese extremely-large aperture astronomical telescope, as well as the international cooperation in the field of mirror fabrication. References 1
Nelson J. E., Sunders G. H., SPIE, 2006, 6267, 1
2
Su D. Q., Wang Y. N., Cui X. Q., ChA&A, 2004, 28, 366
3
Su D. Q., Wang Y. N., Cui X. Q., SPIE, 2004, 5489, 440
4
Terry S. M., Nelson J. E., SPIE, 1985, 1236, 681
5
Alvarez P., Tarruella J. C. L., Espinosa J. M. R., SPIE, 2006, 6267, 10
6
Wang J. G., Pan J. H., Acta Optica Sinica, 1985, 5, 943
7
Lubliner J., Nelson J. E., Applied Optics, 1980, 19, 2340
8
Nelson J. E., Gabor G., Hunt L. K., et al., Applied Optics, 1980, 19, 2348
9
Zhou Y., Li X. N., Astronomical Research and Techniques, 2008, 5, 311
ELSEVIER
Chinese Astronomy and Astrophysics 36 (2012) 435–444
Study on the Stressed Mirror Polishing with a Continuous Polishing Machine for Large Aperture Off-axis Aspheric Mirrors† ZHANG Hai-ying1,2 CUI Xiang-qun1,2 JIANG Zi-bo1,2 LI Xin-nan1,2 1,2 ZHENG Yi LIU Xing-tao1,2 NI Hou-kun1,2 1
National Astronomical Observatories/Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing 210042 2 Key Laboratory of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing 210042
Abstract A special stressed annular polishing technique is proposed to mill the off-axis aspheric sub-mirrors of a large segmented mirror with an annular polishing machine. Based on the basic principle of stressed annular polishing technique, a set of special stressing mechanisms are designed to convert milling the aspheric surfaces of sub-mirrors with different off-axis distances into milling the spherical surfaces with identical radii of curvature, so that they can be polished simultaneously on a continuous polishing machine. It took about continuous 40 hours to polish a scaled-down mirror of the planning Chinese Future Giant Telescope (CFGT) using this technique. This mirror has the 330 mm diameter, 3.6 m off-axis distance, and the 21.6 m radius of curvature, and its maximum asphericity is 16 micron. The experiment shows that this method has a high efficiency, suits batch manufacturing, especially the batch manufacturing of aspheric sub-mirrors of the segmented primary mirror of an extremely large aperture telescope. Key words: telescopes—instrumentation: detectors—methods: analytical— techniques: interferometric—techniques: miscellaneous
†
Supported by National Natural Science Foundation Received 2011–09-07; revised version 2011–11–28 A translation of Acta Astron. Sin. Vol. 53, No. 2, pp. 161–170, 2012 [email protected]
0275-1062/12/$-see front matter © 2012 Elsevier All rights reserved. c 2012B.V. 0275-1062/01/$-see front matter Elsevier Science B. V. All rights reserved. doi:10.1016/j.chinastron.2012.10.009 PII:
436
LI Xin-nan et al. / Chinese Astronomy and Astrophysics 36 (2012) 435–444
1. INTRODUCTION In the end of the 20th century, the segmented mirror of 10 m aperture came into use, implying that a major obstacle that restricts the light-gathering ability of a telescope has been overridden, and that to increase the aperture of an astronomical optical/infrared telescope to 30 m and more, even 100 m, is possible. After entering the 21th century, in response to the need of the rapid development of astronomical science, a series of projects of 30 m-class extremely-large aperture telescopes have been proposed in the different countries and areas of the world (refer to References [1-3] and the website http://en.wikipedia.org/wiki/European Extremely Large Telescope), and some fundamental work has been successively initiated in the United States and European countries. The light-gathering ability of the currently proposed extremely-large telescope is 10 times that of existing telescopes, and the number of off-axis aspheric sub-mirrors of its segmented primary mirror is also proportionally increased, reaching several hundreds even one thousand, the associated problem is the conflict between the manufacturing time of large-quantity high-precision optical surfaces and the required period for developing this telescope, therefore, how to mill and polish these mirrors with a high efficiency and precision becomes the key-important problem for developing an extremely-large aperture optical/infrared telescope. The stressed polishing technique for milling the off-axis aspheric surfaces with a largeaperture tool was developed in the course of manufacturing 10 m-class segmented mirrors. According to the calculation of elastic mechanics, an external force is imposed on the mirror blank to produce a deformation in opposition to the required aspheric surface shape, and under the condition of keeping this deformation, to mill and polish the mirror surface to be a spherical surface. After the external force is taken off, the required off-axis aspheric surface is obtained[4−6] . As the surface to be milled is a spherical surface, not an aspheric surface, so a large-aperture tool can be used, and therefore the efficiency is greatly improved. Compared with the conventional digit-controlled polishing with a small machine, the highfrequency surface error can be reduced. Before this, the development of the 36 sub-mirrors of the 10 m KECK primary mirror took about 3 years, and when the GTC telescope was developed, including the time for technological preparation, it took totally 6 years. It is estimated that if making no further improvement, the manufacturing ability of the existing technology will not exceed 20 blanks per year. Obviously, for an extremely-large aperture optical telescope, such a speed of mirror milling far lags behind the schedule for developing the whole-body telescope. The conventional polishing method with an annular polishing machine belongs to a kind of polishing technique of high-precision flat surfaces, the diameter of the polishing module of the machine is far greater than the size of the workpiece, the effective working region is an annulus as wide as a third of the diameter of the polishing module. An annular polishing machine generally has 2∼3 or even more operating positions. It can polish multiple circular or non-circular mirrors simultaneously, and it needs not to stop when a mirror is put in or taken out from it, in favor of batch fabrication. Because of these advantages, the annular polishing technique has been increasingly applied to the fabrication and production of practical optical elements, even the milling process of large-aperture planar and spherical surfaces, which has certain requirements on both quantity and consistency. For example, the sub-mirrors of the reflective Schmidt corrector of LAMOST, the segmented sub-mirrors
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in the American HET telescope and the SALT telescope of South Africa. In practical large-aperture mirror polishing, the annular polishing machine exhibits its advantages of high efficiency and consistency, and shows its precious prospects for applications. This paper has proposed a kind of stressed annular polishing method to mill the large-aperture off-axis aspheric surfaces by adopting an annular polishing machine in combination with the stressed polishing technique, and taking account of the parameters of the primary mirror of the planned 30 m CFGT (Chinese Future Giant Telescope), performed a polishing experiment on the model mirror with a very good experimental result.
2. CHARACTERISTICS AND MATHEMATICAL EXPRESSION OF AN OFF-AXIS ASPHERIC SURFACE The segmented primary mirror of a large optical telescope equivalents to the entrance pupil of the telescope in size, besides the segment unit on the symmetrical axis, other sub-mirrors are all of off-axis, deviating from the symmetrical axis. Fig.1 is a schematic diagram of the position distribution of sub-mirrors for an extremely large telescope, the left and right halves show the different approaches to segmentation. Assuming that the radius of the offaxis mirror is a, for an off-axis aspheric sub-mirror, its surface shape can be expressed by the local coordinates with the origin at its center[7,8] : m αseg (1) Z(ρ, θ) = mn ρ cos(nθ) , mn
in which m, n are integers, m ≥ n ≥ 0, m − n is an even number, ρ = x2 + y 2 /a, θ = arctan(y/x). The coefficients αmn are functions of mirror’s geometrical parameters, dependent on the off-axis distance, aperture, conic coefficient, and the vertex radius of curvature, and so on. 2 − Kε2 a2 α20 = , (2) k 4(1 − Kε2 )3/2 Kε2 a2 , (3) α22 = k 4(1 − Kε2 )3/2 a3 Kε[1 − (K + 1)ε2 ]1/2 (4 − Kε2 ) α31 = 2 , (4) k 8(1 − Kε2 )3 a3 K 2 ε3 [1 − (K + 1)ε2 ]1/2 α33 = 2 , (5) k 8(1 − Kε2 )3 a4 8(1 + K) − 24Kε2 + 3K 2 ε4 (1 − 3K) − K 3 ε6 (2 − K) , (6) α40 = 3 k 64(1 − Kε2 )9/2 a4 Kε2 [2(1 + 3K) − (9 + 7K)Kε2 + (2 + K)K 2 ε4 ] . (7) α42 = 3 k 64(1 − Kε2 )9/2 Here, ε = R/k, R is the off-axis distance, k is the radius of curvature at the vertex of surface of revolution, K is the coefficient of quadric surface. For the sub-mirrors of a segmented mirror, the radius of curvature at the mirror’s center depends on the off-axis position of the sub-mirror, varying from inside to outside.
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Fig. 1 Illustration for the segmentation of an extremely large telescope
When the stressed polishing technique is adopted, all sub-mirrors can be firstly milled to be the spherical surfaces with the same radius of curvature, the difference between the defined spherical surface and the off-axis aspheric surface can be realized by prestressing the sub-mirror, but the surface deformation produced by prestressing should be equal to the difference between the defined spherical surface and the required off-axis aspheric surface. If αseg denotes the coefficient of surface shape, αstr expresses the coefficient after prestressing, then the relation between the two can be expressed as follows: 2
seg seg seg a str αstr αstr 20 = 2L − α20 α22 = −α22 31 = −α31 4 seg seg seg a str αstr αstr 33 = −α33 40 = 8L3 − α40 α42 = −α42
in which L is the radius of curvature of the defined spherical surface. Taking the primary mirror in the proposed 30 m CFGT as an example, it has the outer radius of 30 m, the vertex radius of curvature of 72 m, the coefficient of quadric surface K =1.000863, and the maximum size of sub-mirrors of 1.1 m. The calculated relationship between the sub-mirror’s coefficient of quadric surface and its off-axis distance is shown in Fig.2. From Fig.2 we can find that the deviation of the off-axis aspheric surface from a spherical surface is determined mainly by the coefficients α20 , α22 , and α31 , which correspond respectively to the defocus, the astigmatism and coma aberration of the surface shape. The coefficients α33 , α40 , and α42 are very small, and negligible in the process of stressed polishing. 3. STRESSING MECHANISMS We consider the aspheric sub-mirror of a segmented mirror as a thin-slab structure. In theory, for realizing the conversion between an off-axis aspheric surface and a spherical surface, the shearing force and bending moment imposed on the mirror edge should be distributed continuously. In practical situations, they are substituted by the discrete forces
439
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Fig. 2 The relationship of the aspheric coefficients with the off-axis distance for the CFGT sub-mirrors
and moments, i.e., to approximate the continuous force and moment along the periphery by setting arch bars at limited discrete points. Assume that the number of loading points is N , the azimuth of each point is θn (n = 1, · · · , N ), θn = 2π(n − 1)/N , then the force and moment on the arch bar at the azimuth θn are: V (θn ) = V1 cos(θn ) + V2 cos(2θn ) ,
(8)
M (θn ) = M0 + M1 cos(θn ) + M2 cos(2θn ) ,
(9)
in which M and V are determined by the surface shape, E is the elastic modulus, ν is the Poisson ratio, h is the thickness. Eh3 , (10) D= 12(1 − ν 2 ) M0 = (D/a2 )[2(1 + ν)αstr 20 ] ,
(11)
M1 = (D/a2 )[2(3 + ν)αstr 31 ] ,
(12)
2
M2 = (D/a )[2(1 − V0 = 0 ,
ν)αstr 22 ] ,
(13) (14)
M1 V1 = − , (15) a 2M2 V2 = . (16) a Summarizing the previous several formulae, the surface shape and the imposed force or moment on the mirror edge are therefore one-by-one correspondent. If only the corresponding mechanisms are available in structure, the requirement on the mirror polishing is realizable.
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Fig.3 gives the overall structure of stressing mechanisms, which differs completely from that adopted by the KECK telescope. The stressing mechanisms of KECK are set along the flank of the mirror, and in Fig.3 all the stressing mechanisms are arranged on the back of the mirror, in favor of polishing on an annular polishing machine. Pistons are taken as the actuators, 12 equally-spaced stressing points are set along the back rim of the mirror.
Fig. 3
Sketch of the combined stressing mechanism
4. TEST OF ASPHERIC SURFACE MILLING 4.1 Test Object The test was conducted using a continuous polishing machine of 1 m aperture. According to the working ability of this machine, it can process the mirror with a diameter of 330 mm in maximum, the test adopted the scaled-down model of CFGT as the test object, which has the following parameters: the vertex radius of curvature 21.6 m, the off-axis aperture 330 mm, the off-axis distance 3.6 m, and considering that the coefficient of quadric surface of the scaled-down model is K =-1.000863, very close to a paraboloid, for simplifying the test equipment and without any loss of generality, the coefficient of quadric surface of the test object was taken to be K =-1. The radius of curvature of the polishing modulus was selected to be 22.08 m, and the maximum asphericity of this off-axis aspheric surface was calculated to be 16.8 μm. 4.2 Test Equipment of Prestress In order to obtain the required surface shape relying on the stressing mechanisms, a special equipment is necessary for the measurement and control. For the 1:3 scaled-down model mirror of CFGT, the maximum deviation of the aspheric surface from a spherical surface is greater than 20 μm, and the dynamical range of variations is rather large, for mea-
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suring the stressed deformation of its surface, an infrared interferometer of 10 μm wavelength or a contact measurement sensor can be adopted. The interferometric measurement has a rather high accuracy but requires a quite long optical path, as a compromise, the contact LVDT array with a compact structure was adopted. We studied the method to fit the surface shape by discrete measured data, including the distribution of sampling points, the fitting algorithm, etc.[9] The computer-aided data sampling and processing were selected to avoid effectively the effects of the measuring environment and other man-made factors on the measuring accuracy, thus the whole data sampling system can give the real-time surface error of the testing mirror, with a rms sampling accuracy of 0.3 μm. Fig.4 is the picture for the real stressing mechanisms and test equipment.
Fig. 4
Stressing mechanism and test equipment
Fig.5 gives the variation of mirror surface measured by the LVDT array during the test of stressing calibration, which shows respectively the variations of the coma aberration, defocus and astigmatism as a function of moment load, the result indicates that the stressing mechanisms exhibit a good linearity, satisfying the requirement. 4.3 Polishing Experiment And Its Result According to the parameters of the test mirror, the loading coefficients of corresponding
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Fig. 5
Curves of the typical aberration values versus the load
aberrations can be calculated as −6 αstr m, 20 = 8.226 × 10
α22str = 8.4 × 10−6 m,
−7 αstr m. 31 = 7.442 × 10
Take the elastic modulus of the mirror blank to be E =91.7 Gpa, the Poisson ratio ν =0.247, and the glass thickness 15 mm, we obtained the theoretical defocus loading force F0 =12.8 N, the coma-aberration loading force F1 =-2.6 N, and the astigmatism loading force F2 =14.8 N. They correspond to the theoretical defocus loading pressure P0 = 5.15 × 104 Pa, the coma-aberration loading pressure P1 = −1.03 × 104 Pa, and the astigmatism loading pressure P2 = −5.96 × 104 Pa. Fig.6 is the picture of the test mirror under the stressed polishing. After 40 hour continuous polishing, the surface error decreases rapidly and becomes stable. Fig.7 illustrates
Fig. 6 The test mirror in stressed polishing
the surface error of the mirror after polishing, the rms error is 0.327λ. Fig.8 shows the convergence process of the mirror surface error during the stressed polishing, and Fig.9
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displays interferogram of the mirror after the final figuring of surface and cutting into a hexagon, the diagonal length of the mirror is 312 mm, and the rms surface accuracy is better than 0.02λ (the operating wavelength of the interferometer is 632.8 nm).
Fig. 7 Result of the stressed mirror polishing with the continuous polishing machine. The left panel is the contour map of mirror surface, and the right panel is the corresponding interferogram
Fig. 8 Convergence curve of the polishing process
5. CONCLUSION AND PROSPECT Aiming at the need of batch manufacturing of off-axis aspheric sub-mirrors in an extremelylarge aperture optical/infrared telescope, and combining for the first time the stressed polishing technique with the annular polishing technique, with the model mirror of the 30 m CFGT as the test object, we have performed an experimental study on the stressed annular polishing technique of large-aperture off-axis aspheric surfaces. The test result indicates that
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Fig. 9
Interferogram of the mirror cut into hexagon
the stressed annular polishing technique is an effective method for milling large-aperture aspheric surfaces, and it has very good application prospects. In view of the current trend of relevant research in the world, the stressed annular polishing technique possesses simultaneously the technological advantages of the stressed method and annular polishing machine method, especially in the aspect of batch manufacturing, it has the advantages of high efficiency and stability. To continue the study and development of the stressed annular polishing technique will give an impetus to the independent development of the Chinese extremely-large aperture astronomical telescope, as well as the international cooperation in the field of mirror fabrication. References 1
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