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Non-null testing for standard quadric surfaces with subaperture stitching technique
Optics Communications 340 (2015) 159–164
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Non-null testing for standard quadric surfaces with subaperture stitching technique Lisong Yan a,b,n, Xiaokun Wang a, Ligong Zheng a, Fan Di a, Feng Zhang a, Donglin Xue a, Haixiang Hu a,b, Xuefeng Zeng a, Xuejun Zhang a a Key Laboratory of Optical System Advanced Manufacturing Technology, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China b Graduate School 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 1 September 2014 Received in revised form 17 November 2014 Accepted 23 November 2014 Available online 26 November 2014
A new method is proposed for testing standard quadric surfaces with several subapertures based on interferometry. Subapertures arrangement, best-fit sphere calculation and distortion correction about such surfaces are discussed in this paper. In addition, we provide an experimental demonstration by testing a Ø310 mm convex hyperboloid mirror. The experimental result shows that the proposed method can accomplish the testing of quadric mirrors without auxiliary compensation effectively. The analysis and proposed methods bring much to the application of non-null aspheric testing. & 2014 Elsevier B.V. All rights reserved.
Keywords: Interferometry Surface measurements Optical fabrication Testing
1. Introduction Aspheric surfaces are widely used in a variety of optical systems to improve the imaging quality with fewer optical elements. The common methods used for testing aspheric surfaces, such as the null testing with compensator or CGH, will introduce extra errors from auxiliary optics and need a high cost. Besides, for large aperture aspheric surfaces especially convex surfaces, the aperture of neither interferometer nor the auxiliary optics is large enough to cover the full aperture of the tested surfaces. A subaperture stitching method has been developed to measure aspheric surfaces with low cost. The basic idea of subaperture stitching method is to divide the tested surface into several smaller subapertures, which can be tested with a standard interferometer. After completing the measurement of each subaperture, we get the full aperture map of the tested surface with relative stitching algorithms. The subaperture testing method was first introduced in 1980s [1]. Many researchers have developed different stitching algorithms. Obvious improvements can be observed from the Kwon– Thunen method [2] and the simultaneous fit method [3], to the discrete phase method [4,5], the optical null technique from QED [6,7], and then to the maximum likelihood algorithm from Arizona
University [8,9]. Aiming to improving the efficiency of the stitching, we proposed a kind of stitching technique to test standard quadric surfaces in non-null configuration. Advantages of the technique were claimed and verified through experiments. In this paper, we focus on the non-null testing technique for standard quadric surfaces including the best-fit sphere calculation to each subaperture, subapertures arrangement method of the full aperture and distortion correction method. The stitching was accomplished with our previously mentioned algorithm [10]. The stitching technique has also been applied to a Ø310 mm convex hyperboloid mirror. This paper is organized as follows. In Section 2, the basic theory of stitching technique is introduced. In Section 3, we apply the above technique to the actual experiment and the relative result is introduced. The conclusion is given in Section 4.
2. Theory 2.1. Calculation of best-fit spheres for subapertures The sag of a standard quadric surface can be expressed as [11]
ccon (x2 + y2 )
z= n Corresponding author at: Key Laboratory of Optical System Advanced Manufacturing Technology, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China E-mail address: [email protected] (L. Yan).
http://dx.doi.org/10.1016/j.optcom.2014.11.075 0030-4018/& 2014 Elsevier B.V. All rights reserved.
1+
1 − (1 + κ) ccon (x2 + y2 )
(1)
where ccon is the conic' axial curvature and κ the conic constant. For the off-axis section of rotationally symmetric surfaces, it can be
160
L. Yan et al. / Optics Communications 340 (2015) 159–164
(x c , yc , zc ) is the center of the subaperture. F is the center of the best-fit sphere of subaperture ACB. The angle between FC and Z axis is β . It is assumed that the tested area is rotationally symmetric around FC, which means γ1 (∠AFC ) is equal to γ2 (∠BFC ). A spherical coordinate is established where F is the center and ⎯⎯⎯→ FC is the positive direction. A point P in the subaperture ACB can then expressed as
⎧ x p = r sin θ cos φ + x c ⎪ ⎪ ⎨ yp = − r cos β sin θ sin φ + r sin β cos θ − h sin β + yc ⎪ ⎪ ⎩ z p = − r sin β sin θ sin φ − r cos β cos θ + h cos β + z c
(2)
where h is the length of FC , r is the length of FP , θ and φ are the zenith angle and azimuth angle respectively. As the point P (x p, yp , z p ) is in the subaperture ACB, it meets Fig. 1. Sketch of non-null testing. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
(κ + 1) z p2 − 2pz + (x p2 + yp2 ) = 0
(3)
where p is the radius of curvature at the vertex of the aspheric mirror (p = 1/ccon ). Eq. (3) can be written as
Ar 2 + Br + C = 0
(4)
where ⎧ A = κ (sin β sin θ sin φ + cos β cos θ) 2 + 1 ⎪ ⎪ B = [ − 2hκ sin β cos β + ( − 2(κ + 1) z c + 2p) sin β − 2yc cos β] sin θ sin φ ⎪ ⎨ +[ − 2h (κ cos2β + 1) + ( − 2(κ + 1) z c + 2p) cos β + 2yc sin β] cos θ ⎪ ⎪ 2 2 ⎪ ⎩ C = h (κ cos β + 1) + (2(κ + 1) z c − 2p) h cos β − 2yc h sin β (5)
Fig. 2. Test configuration of subaperture.
regarded as part of the rotationally symmetric surface. When calculating the best-fit sphere of a subaperture, for the sake of conciseness, we just consider the subaperture whose center is in the Y–Z plane shown in Fig. 1. The analysis is also adapted to the subaperture whose center is not in the Y–Z plane, as it can be regarded as a rotation result around the Z axis. As shown in Fig. 1, the red line is the profile map of an aspherical mirror. ACB is the subaperture to be tested where point C
r can be calculated by
r=
B2 − 4AC − B 2A
(6)
∧ ∂r ∧ ∂r ∧ ∂r ρ + θ + φ ρ∂θ ρ sin θ ∂φ ∂ρ
(7)
and
∇r =
Fig. 3. Testing ring.
L. Yan et al. / Optics Communications 340 (2015) 159–164
161
Fig. 4. Overlapping relationship between adjacent testing rings.
Only the gradient in the tangential direction is considered, then
Δr = (∇r)2 =
⎡ 2⎤ 1 ⎢ ⎛⎜ ∂r ⎞⎟2 ⎛ ∂r ⎞ ⎥ +⎜ ⎟ 2 ⎝ sin θ ∂φ ⎠ ⎥⎦ ρ ⎢⎣ ⎝ ∂θ ⎠
(8)
max (Δr) = min
(11)
Hence the best-fit sphere of the subaperture with h and β can be calculated with the Generalized Reduced Gradient Method by minimizing the maximum of Δr .
where
⎧ ∂r r2 r ∂A ∂B ⎪ =− − 2Ar + B ∂θ 2Ar + B ∂θ ⎪ ∂θ ⎪ ∂A ∂B r2 r ⎪ ∂r =− − ⎪ sin θ ∂φ 2Ar + B sin θ ∂φ 2Ar + B sin ∂φ ⎪ ⎪ ∂A = 2κ (sin β sin θ sin φ + cos β cos θ) ⎪ ∂θ ⎪ × (sin β cos θ sin φ − cos β sin θ) ⎪ ⎪ ∂A ⎪ = 2κ (sin β sin θ sin φ + cos β cos θ) sin β ⎪ ⎪ sin θ ∂φ ⎨ cos φ ⎪ ⎪ ∂B = [ − 2hκ sin β cos β + (2p − 2(κ + 1) z c ) sin β ⎪ ⎪ ∂θ ⎪ − 2yc cos β] cos θ sin φ ⎪ ⎪ − [ − 2h (κ cos2β + 1) + ( − 2(κ + 1) z c + 2p) ⎪ cos β + 2yc sin β] sin θ ⎪ ⎪ B ∂ ⎪ = [ − 2hκ sin β cos β + ( − 2(κ + 1) z c + 2p) ⎪ sin θ ∂φ ⎪ ⎪ × sin β − 2y cos β] cos φ ⎩ c
2.2. Subapertures arrangement The arrangement of subapertures is basically determined by the requirements of full aperture covering capability and overlapping ratio [12]. After accomplishing the best-fit sphere calculation, when the center of a subaperture is determined, the tested area is known. As shown in Fig. 2, Area 1 is the area of actual tested subaperture. The area in the tangent plane of the center of the subaperture should be an approximate circle. The projection of this area (Area 2 in Fig. 2) in the X–Y plane should be an ellipse. If the tested subaperture is rotated around the optical axis of the mirror, the tested area will be a ring shown in Fig. 3. Overlapping ratio between the nth and (n + 1)th ring is defined as
ratio =
(9)
In actual testing, F # of the standard transmission sphere is the restriction of testing. It should meets that
⎛ 1 ⎞ γ1 ≤ arctan ⎜ # ⎟ ⎝ 2F ⎠
π (rn′2 − rn2 + 1) π (rn′2+ 1 − rn2 + 1)
(12)
where rn and rn′ are the inner radius and outer radius of the nth ring respectively; rn + 1 and rn′ + 1 are the inner radius and outer radius of the (n + 1)th ring respectively shown in Fig. 4. As we accomplish the full aperture covering in the X–Y projection plane, the tested surface is also full aperture covered by subapertures. 2.3. Distortion correction
(10)
Under the limitation of Eq. (10), to make the maximum of Δr minimum, it meets
Before applying relative stitching algorithm [10] to the actual subaperture tested results, distortion should be corrected to each subaperture tested result.
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L. Yan et al. / Optics Communications 340 (2015) 159–164
For standard quadric surface whose surface can be described in Eq. (1), distortion can be corrected analytically, without targets or marked points [10,13].
For a point P (x, y, z) in the testing pixel coordinate system, it should be projected into the spherical coordinate where F is the ⎯⎯⎯→ center and FC is the positive direction mentioned in Section 2.1 as shown in Fig. 5 in Eq. (13),
⎧r = z + h ⎪ ⎪ ⎛ 1 ⎞ x 2 + y2 ⎨θ = arctan ⎜ # ⎟ ⎝ 2F ⎠ rpix ⎪ ⎪ x y φ arctan ( , ) = ⎩
(13)
where rpix is the radius of tested area in the testing pixel coordinate, h is radius of the best-fit sphere for the subaperture, θ and φ are the zenith angle and azimuth angle in the relative spherical coordinate respectively. Taking (r , θ , φ) to Eq. (14), the coordinate of point P (x p, yp , z p ) in the physical coordinate O − XYZ is calculated.
⎧ x p = r sin θ cos φ + x c ⎪ ⎪ ⎨ yp = − r cos β sin θ sin φ + r sin β cos θ − h sin β + yc ⎪ ⎪ ⎩ z p = − r sin β sin θ sin φ − r cos β cos θ + h cos β + z c
(14)
Fig. 5. Distortion correction.
where (x c , yc , zc ) is the center coordinate of the tested subaperture in the physical coordinate, β is the angle between FC and Z axis as mentioned in Section 2.1. After finishing the best-fit sphere calculation, subapertures arrangement, actual testing and distortion correction to each tested map, the full aperture map can be stitched with the algorithm, which was described in detail in our previously published paper [10].
3. Experimental verification
Fig. 6. Experimental setup.
In this experiment, the clear aperture of the convex hyperboloid mirror is about 310 mm. Conic constant κ is 3.662 and the vertex radius of curvature R is 4087 mm. A 6-dof platform is built for the subaperture testing of the mirror as shown in Fig. 6. A Ø150 mm aperture interferometer and a standard transmission sphere with F# 30 are chosen to accomplish the subaperture stitching. Three rings (15 subapertures) are designed to accomplish the full aperture covering of the mirror shown in Fig. 7. The measured results are shown in Fig. 8 and the stitching result is shown in Fig. 9. From Fig. 9, the stitching result is smooth and continuous. To evaluate the stitching accuracy, the full aperture testing to the convex mirror is also accomplished with the compensator as
Fig. 7. Subapertures arrangement.
L. Yan et al. / Optics Communications 340 (2015) 159–164
163
Fig. 8. Testing subapertures.
Fig. 10. Experimental setup.
Fig. 9. Stitching result.
shown in Fig. 10 and the relative full aperture testing result is shown in Fig. 11. By subtracting the data between stitching result and full aperture tested result point by point, the residual map can be got shown in Fig. 12. The PV and RMS errors of the residual map with the stitching technique mentioned here are 0.049λ and 0.007λ respectively. Considering the errors of different states between the subaperture testing and full aperture testing (The mirror is horizontal in the subaperture testing while it is upright in the full aperture testing), the errors of compensator and environmental difference in the two testing, the residual is reasonable.
4. Conclusion We have provided a non-null stitching testing method applied to standard quadric surfaces. To evaluate the stitching accuracy, experimental verification was taken by testing a convex hyperboloid mirror and the stitching result was compared with the full aperture tested result with compensator. It is proved by the above experimental results that the reported technique can obtain the reconstructed full-aperture surface map with satisfactory accuracy. As the experimental study is only for aspheres in a standard quadric form, further theoretical and experimental development for aspheres with terms of higher order and freeform surfaces will be taken to accomplish the stitching testing to such surfaces.
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L. Yan et al. / Optics Communications 340 (2015) 159–164
Acknowledgments This work was supported by the National Natural Science Foundation of China (61036015) and the National High Technology Research & Development Program of China (O8663NJ090).
References
Fig. 11. Full aperture testing result.
Fig. 12. Residual map.
[1] C.J. Kim, Polynomial fit of interferograms,, Appl. Opt. 21 (24) (1982) 4521–4525. [2] J.G. Thunen, O.Y. Kwon, Full aperture testing with subaperture test optics, Proc. SPIE 351 (1982) 19–27. [3] W.W. Chow, G.N. Lawrence, Method for subaperture testing interferogram reduction, Opt. Lett. 8 (1983) 468–470. [4] T.W. Stuhlinger, Subaperture optical testing: experimental verification, Proc. SPIE 656 (1986) 118–127. [5] M. Otsubo, K. Okada, J. Tsujiuchi, Measurement of large plane surface shapes by connecting small-aperture interferograms, Opt. Eng. 33 (1994) 608–613. [6] A. Kulawiec, P. Murphy, M.D. Marco, Measurement of high-departure aspheres using subaperture stitching with the Variable Optical Null (VONTM), Proc. SPIE 7655 (2010) (765512-1–765512-4). [7] C. Supranowitz, C.M. Fee, P. Murphy, Asphere metrology using variable optical null technology, Proc. SPIE 8416 (2012) (841604-1–841604-5). [8] Peng Su, Absolute measurements of large mirrors (Ph.D. Dissertation), The University of Arizona, 2008. [9] P. Su, J.H. Burge, R.E. Parks, Application of maximum likelihood reconstruction of subaperture data for measurement of large flat mirrors, Appl. Opt. 49 (2010) 21–31. [10] L.S. Yan, X.K. Wang, L.G. Zheng, X.F. Zeng, H.X. Hu, X.J. Zhang, Experimental study on subaperture testing with iterative triangulation algorithm, Opt. Express 21 (19) (2013) 22628–22644. [11] G.W. Forbes, Characterizing the shape of freeform optics, Opt. Express 20 (3) (2012) 2483–2492. [12] S.Y. Chen, S.Y. Li, Y.F. Dai, Z.W. Zheng, Lattice design for subaperture stitching test of a concave paraboloid surface, Appl. Opt. 45 (3) (2006) 2280–2286. [13] C.Y. Zhao, J.H. Burge, Orthonormal vector polynomials in a unit circle, application: fitting mapping distortions in a null test, Proc. SPIE 7426 (2009) (74260V-1–74260V-8).
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Non-null testing for standard quadric surfaces with subaperture stitching technique Lisong Yan a,b,n, Xiaokun Wang a, Ligong Zheng a, Fan Di a, Feng Zhang a, Donglin Xue a, Haixiang Hu a,b, Xuefeng Zeng a, Xuejun Zhang a a Key Laboratory of Optical System Advanced Manufacturing Technology, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China b Graduate School 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 1 September 2014 Received in revised form 17 November 2014 Accepted 23 November 2014 Available online 26 November 2014
A new method is proposed for testing standard quadric surfaces with several subapertures based on interferometry. Subapertures arrangement, best-fit sphere calculation and distortion correction about such surfaces are discussed in this paper. In addition, we provide an experimental demonstration by testing a Ø310 mm convex hyperboloid mirror. The experimental result shows that the proposed method can accomplish the testing of quadric mirrors without auxiliary compensation effectively. The analysis and proposed methods bring much to the application of non-null aspheric testing. & 2014 Elsevier B.V. All rights reserved.
Keywords: Interferometry Surface measurements Optical fabrication Testing
1. Introduction Aspheric surfaces are widely used in a variety of optical systems to improve the imaging quality with fewer optical elements. The common methods used for testing aspheric surfaces, such as the null testing with compensator or CGH, will introduce extra errors from auxiliary optics and need a high cost. Besides, for large aperture aspheric surfaces especially convex surfaces, the aperture of neither interferometer nor the auxiliary optics is large enough to cover the full aperture of the tested surfaces. A subaperture stitching method has been developed to measure aspheric surfaces with low cost. The basic idea of subaperture stitching method is to divide the tested surface into several smaller subapertures, which can be tested with a standard interferometer. After completing the measurement of each subaperture, we get the full aperture map of the tested surface with relative stitching algorithms. The subaperture testing method was first introduced in 1980s [1]. Many researchers have developed different stitching algorithms. Obvious improvements can be observed from the Kwon– Thunen method [2] and the simultaneous fit method [3], to the discrete phase method [4,5], the optical null technique from QED [6,7], and then to the maximum likelihood algorithm from Arizona
University [8,9]. Aiming to improving the efficiency of the stitching, we proposed a kind of stitching technique to test standard quadric surfaces in non-null configuration. Advantages of the technique were claimed and verified through experiments. In this paper, we focus on the non-null testing technique for standard quadric surfaces including the best-fit sphere calculation to each subaperture, subapertures arrangement method of the full aperture and distortion correction method. The stitching was accomplished with our previously mentioned algorithm [10]. The stitching technique has also been applied to a Ø310 mm convex hyperboloid mirror. This paper is organized as follows. In Section 2, the basic theory of stitching technique is introduced. In Section 3, we apply the above technique to the actual experiment and the relative result is introduced. The conclusion is given in Section 4.
2. Theory 2.1. Calculation of best-fit spheres for subapertures The sag of a standard quadric surface can be expressed as [11]
ccon (x2 + y2 )
z= n Corresponding author at: Key Laboratory of Optical System Advanced Manufacturing Technology, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China E-mail address: [email protected] (L. Yan).
http://dx.doi.org/10.1016/j.optcom.2014.11.075 0030-4018/& 2014 Elsevier B.V. All rights reserved.
1+
1 − (1 + κ) ccon (x2 + y2 )
(1)
where ccon is the conic' axial curvature and κ the conic constant. For the off-axis section of rotationally symmetric surfaces, it can be
160
L. Yan et al. / Optics Communications 340 (2015) 159–164
(x c , yc , zc ) is the center of the subaperture. F is the center of the best-fit sphere of subaperture ACB. The angle between FC and Z axis is β . It is assumed that the tested area is rotationally symmetric around FC, which means γ1 (∠AFC ) is equal to γ2 (∠BFC ). A spherical coordinate is established where F is the center and ⎯⎯⎯→ FC is the positive direction. A point P in the subaperture ACB can then expressed as
⎧ x p = r sin θ cos φ + x c ⎪ ⎪ ⎨ yp = − r cos β sin θ sin φ + r sin β cos θ − h sin β + yc ⎪ ⎪ ⎩ z p = − r sin β sin θ sin φ − r cos β cos θ + h cos β + z c
(2)
where h is the length of FC , r is the length of FP , θ and φ are the zenith angle and azimuth angle respectively. As the point P (x p, yp , z p ) is in the subaperture ACB, it meets Fig. 1. Sketch of non-null testing. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
(κ + 1) z p2 − 2pz + (x p2 + yp2 ) = 0
(3)
where p is the radius of curvature at the vertex of the aspheric mirror (p = 1/ccon ). Eq. (3) can be written as
Ar 2 + Br + C = 0
(4)
where ⎧ A = κ (sin β sin θ sin φ + cos β cos θ) 2 + 1 ⎪ ⎪ B = [ − 2hκ sin β cos β + ( − 2(κ + 1) z c + 2p) sin β − 2yc cos β] sin θ sin φ ⎪ ⎨ +[ − 2h (κ cos2β + 1) + ( − 2(κ + 1) z c + 2p) cos β + 2yc sin β] cos θ ⎪ ⎪ 2 2 ⎪ ⎩ C = h (κ cos β + 1) + (2(κ + 1) z c − 2p) h cos β − 2yc h sin β (5)
Fig. 2. Test configuration of subaperture.
regarded as part of the rotationally symmetric surface. When calculating the best-fit sphere of a subaperture, for the sake of conciseness, we just consider the subaperture whose center is in the Y–Z plane shown in Fig. 1. The analysis is also adapted to the subaperture whose center is not in the Y–Z plane, as it can be regarded as a rotation result around the Z axis. As shown in Fig. 1, the red line is the profile map of an aspherical mirror. ACB is the subaperture to be tested where point C
r can be calculated by
r=
B2 − 4AC − B 2A
(6)
∧ ∂r ∧ ∂r ∧ ∂r ρ + θ + φ ρ∂θ ρ sin θ ∂φ ∂ρ
(7)
and
∇r =
Fig. 3. Testing ring.
L. Yan et al. / Optics Communications 340 (2015) 159–164
161
Fig. 4. Overlapping relationship between adjacent testing rings.
Only the gradient in the tangential direction is considered, then
Δr = (∇r)2 =
⎡ 2⎤ 1 ⎢ ⎛⎜ ∂r ⎞⎟2 ⎛ ∂r ⎞ ⎥ +⎜ ⎟ 2 ⎝ sin θ ∂φ ⎠ ⎥⎦ ρ ⎢⎣ ⎝ ∂θ ⎠
(8)
max (Δr) = min
(11)
Hence the best-fit sphere of the subaperture with h and β can be calculated with the Generalized Reduced Gradient Method by minimizing the maximum of Δr .
where
⎧ ∂r r2 r ∂A ∂B ⎪ =− − 2Ar + B ∂θ 2Ar + B ∂θ ⎪ ∂θ ⎪ ∂A ∂B r2 r ⎪ ∂r =− − ⎪ sin θ ∂φ 2Ar + B sin θ ∂φ 2Ar + B sin ∂φ ⎪ ⎪ ∂A = 2κ (sin β sin θ sin φ + cos β cos θ) ⎪ ∂θ ⎪ × (sin β cos θ sin φ − cos β sin θ) ⎪ ⎪ ∂A ⎪ = 2κ (sin β sin θ sin φ + cos β cos θ) sin β ⎪ ⎪ sin θ ∂φ ⎨ cos φ ⎪ ⎪ ∂B = [ − 2hκ sin β cos β + (2p − 2(κ + 1) z c ) sin β ⎪ ⎪ ∂θ ⎪ − 2yc cos β] cos θ sin φ ⎪ ⎪ − [ − 2h (κ cos2β + 1) + ( − 2(κ + 1) z c + 2p) ⎪ cos β + 2yc sin β] sin θ ⎪ ⎪ B ∂ ⎪ = [ − 2hκ sin β cos β + ( − 2(κ + 1) z c + 2p) ⎪ sin θ ∂φ ⎪ ⎪ × sin β − 2y cos β] cos φ ⎩ c
2.2. Subapertures arrangement The arrangement of subapertures is basically determined by the requirements of full aperture covering capability and overlapping ratio [12]. After accomplishing the best-fit sphere calculation, when the center of a subaperture is determined, the tested area is known. As shown in Fig. 2, Area 1 is the area of actual tested subaperture. The area in the tangent plane of the center of the subaperture should be an approximate circle. The projection of this area (Area 2 in Fig. 2) in the X–Y plane should be an ellipse. If the tested subaperture is rotated around the optical axis of the mirror, the tested area will be a ring shown in Fig. 3. Overlapping ratio between the nth and (n + 1)th ring is defined as
ratio =
(9)
In actual testing, F # of the standard transmission sphere is the restriction of testing. It should meets that
⎛ 1 ⎞ γ1 ≤ arctan ⎜ # ⎟ ⎝ 2F ⎠
π (rn′2 − rn2 + 1) π (rn′2+ 1 − rn2 + 1)
(12)
where rn and rn′ are the inner radius and outer radius of the nth ring respectively; rn + 1 and rn′ + 1 are the inner radius and outer radius of the (n + 1)th ring respectively shown in Fig. 4. As we accomplish the full aperture covering in the X–Y projection plane, the tested surface is also full aperture covered by subapertures. 2.3. Distortion correction
(10)
Under the limitation of Eq. (10), to make the maximum of Δr minimum, it meets
Before applying relative stitching algorithm [10] to the actual subaperture tested results, distortion should be corrected to each subaperture tested result.
162
L. Yan et al. / Optics Communications 340 (2015) 159–164
For standard quadric surface whose surface can be described in Eq. (1), distortion can be corrected analytically, without targets or marked points [10,13].
For a point P (x, y, z) in the testing pixel coordinate system, it should be projected into the spherical coordinate where F is the ⎯⎯⎯→ center and FC is the positive direction mentioned in Section 2.1 as shown in Fig. 5 in Eq. (13),
⎧r = z + h ⎪ ⎪ ⎛ 1 ⎞ x 2 + y2 ⎨θ = arctan ⎜ # ⎟ ⎝ 2F ⎠ rpix ⎪ ⎪ x y φ arctan ( , ) = ⎩
(13)
where rpix is the radius of tested area in the testing pixel coordinate, h is radius of the best-fit sphere for the subaperture, θ and φ are the zenith angle and azimuth angle in the relative spherical coordinate respectively. Taking (r , θ , φ) to Eq. (14), the coordinate of point P (x p, yp , z p ) in the physical coordinate O − XYZ is calculated.
⎧ x p = r sin θ cos φ + x c ⎪ ⎪ ⎨ yp = − r cos β sin θ sin φ + r sin β cos θ − h sin β + yc ⎪ ⎪ ⎩ z p = − r sin β sin θ sin φ − r cos β cos θ + h cos β + z c
(14)
Fig. 5. Distortion correction.
where (x c , yc , zc ) is the center coordinate of the tested subaperture in the physical coordinate, β is the angle between FC and Z axis as mentioned in Section 2.1. After finishing the best-fit sphere calculation, subapertures arrangement, actual testing and distortion correction to each tested map, the full aperture map can be stitched with the algorithm, which was described in detail in our previously published paper [10].
3. Experimental verification
Fig. 6. Experimental setup.
In this experiment, the clear aperture of the convex hyperboloid mirror is about 310 mm. Conic constant κ is 3.662 and the vertex radius of curvature R is 4087 mm. A 6-dof platform is built for the subaperture testing of the mirror as shown in Fig. 6. A Ø150 mm aperture interferometer and a standard transmission sphere with F# 30 are chosen to accomplish the subaperture stitching. Three rings (15 subapertures) are designed to accomplish the full aperture covering of the mirror shown in Fig. 7. The measured results are shown in Fig. 8 and the stitching result is shown in Fig. 9. From Fig. 9, the stitching result is smooth and continuous. To evaluate the stitching accuracy, the full aperture testing to the convex mirror is also accomplished with the compensator as
Fig. 7. Subapertures arrangement.
L. Yan et al. / Optics Communications 340 (2015) 159–164
163
Fig. 8. Testing subapertures.
Fig. 10. Experimental setup.
Fig. 9. Stitching result.
shown in Fig. 10 and the relative full aperture testing result is shown in Fig. 11. By subtracting the data between stitching result and full aperture tested result point by point, the residual map can be got shown in Fig. 12. The PV and RMS errors of the residual map with the stitching technique mentioned here are 0.049λ and 0.007λ respectively. Considering the errors of different states between the subaperture testing and full aperture testing (The mirror is horizontal in the subaperture testing while it is upright in the full aperture testing), the errors of compensator and environmental difference in the two testing, the residual is reasonable.
4. Conclusion We have provided a non-null stitching testing method applied to standard quadric surfaces. To evaluate the stitching accuracy, experimental verification was taken by testing a convex hyperboloid mirror and the stitching result was compared with the full aperture tested result with compensator. It is proved by the above experimental results that the reported technique can obtain the reconstructed full-aperture surface map with satisfactory accuracy. As the experimental study is only for aspheres in a standard quadric form, further theoretical and experimental development for aspheres with terms of higher order and freeform surfaces will be taken to accomplish the stitching testing to such surfaces.
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L. Yan et al. / Optics Communications 340 (2015) 159–164
Acknowledgments This work was supported by the National Natural Science Foundation of China (61036015) and the National High Technology Research & Development Program of China (O8663NJ090).
References
Fig. 11. Full aperture testing result.
Fig. 12. Residual map.
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