Abstract

Limited by the aperture and f/number of the transmission sphere (TS), large convex spheres with very small R/number (ratio of the radius of curvature to the aperture) cannot be tested in a single measurement with a standard interferometer. We present the algorithm and troubleshooting for subaperture stitching test of a half meter-class convex sphere with R/0.61. Totally 90 off-axis subapertures are arranged on 5 rings around the central one. Since the subaperture is so small, its surface error is comparable with that of the TS reference error. Hence a self-calibrated stitching algorithm is proposed to separate the reference error from the measurements. Another serious problem is the nonlinear mapping between the subaperture’s local coordinates and the full aperture’s global coordinates. The nonlinearity increases remarkably with the off-axis angle. As a result, we cannot directly remove power from the full aperture error map as we usually do. Otherwise incorrect spherical aberration will be generated. We therefore propose the sphericity assessment algorithm to match the stitched surface error with a best-fit sphere. The residual is true surface error which can be used for corrective figuring or for tolerance assessment. The self-calibrated stitching and troubleshooting are demonstrated experimentally.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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2014 (2)

2013 (1)

2012 (2)

2010 (2)

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

2008 (1)

J. H. Burge, P. Su, and C. Zhao, “Optical metrology for very large convex aspheres,” Proc. SPIE 7018, 701818 (2008).
[Crossref]

2006 (1)

2005 (1)

1999 (1)

J. Gou, Y. Chu, and Z. Li, “A geometric theory of form, profile, and orientation tolerances,” Precis. Eng. 23(2), 79–93 (1999).
[Crossref]

1994 (1)

Bauer, M.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Bauman, B. J.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Bowden, G.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Burge, J. H.

J. H. Burge, P. Su, and C. Zhao, “Optical metrology for very large convex aspheres,” Proc. SPIE 7018, 701818 (2008).
[Crossref]

Chen, S.

Chu, Y.

J. Gou, Y. Chu, and Z. Li, “A geometric theory of form, profile, and orientation tolerances,” Precis. Eng. 23(2), 79–93 (1999).
[Crossref]

Creath, K.

Dai, Y.

DeVries, G.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Fleig, J.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Forbes, G.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Gilliand, Y. A.

Gilmore, D.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Gou, J.

J. Gou, Y. Chu, and Z. Li, “A geometric theory of form, profile, and orientation tolerances,” Precis. Eng. 23(2), 79–93 (1999).
[Crossref]

Hariharan, P.

Hu, H.

Kahn, S.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Kang, Y.

Ku, J.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Kulawiec, A.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Li, S.

Li, Z.

J. Gou, Y. Chu, and Z. Li, “A geometric theory of form, profile, and orientation tolerances,” Precis. Eng. 23(2), 79–93 (1999).
[Crossref]

Liao, W.

Liu, D.

Liu, W.

Miladinovich, D.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Murphy, P.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Nordby, M.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Nurita, N.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Olivier, S.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Rasmussen, A.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Riot, V.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Seppala, L.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Shen, Y.

Shi, T.

Su, P.

J. H. Burge, P. Su, and C. Zhao, “Optical metrology for very large convex aspheres,” Proc. SPIE 7018, 701818 (2008).
[Crossref]

Sui, Y.

Tian, C.

Tian, Z.

Tricard, M.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Wang, X.

Wu, H.

Xiao, H.

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

Yan, L.

Yang, H.

Yang, W.

Yang, Y.

Zeng, X.

Zhang, L.

Zhang, X.

Zhao, C.

Zheng, L.

Zheng, Z.

Appl. Opt. (3)

CIRP Ann. (1)

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Express (3)

Precis. Eng. (1)

J. Gou, Y. Chu, and Z. Li, “A geometric theory of form, profile, and orientation tolerances,” Precis. Eng. 23(2), 79–93 (1999).
[Crossref]

Proc. SPIE (2)

B. J. Bauman, G. Bowden, J. Ku, M. Nordby, S. Olivier, V. Riot, A. Rasmussen, L. Seppala, H. Xiao, N. Nurita, D. Gilmore, and S. Kahn, “Update and image quality error budget for the LSST camera optical design,” Proc. SPIE 7733, 77332W (2010).
[Crossref]

J. H. Burge, P. Su, and C. Zhao, “Optical metrology for very large convex aspheres,” Proc. SPIE 7018, 701818 (2008).
[Crossref]

Other (2)

D. Golini, G. Forbes, and P. Murphy, “Method for self-calibrated sub-aperture stitching for surface figure measurement,” United States Patent 6,956,657 B2 (October 18, 2005).

G. H. Gloub and C. F. Van Loan, Matrix Computations (3rd. Edition. The Johns Hopkins University, 1996).

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Figures (9)

Fig. 1
Fig. 1 Drawing of the steep convex sphere under test.
Fig. 2
Fig. 2 Subaperture layout.
Fig. 3
Fig. 3 Experimental setup.
Fig. 4
Fig. 4 Subaperture measurements. (a) central, (b) ring 1, (c) ring 2, (d) ring 3, (e) ring 4, (f) ring 5.
Fig. 5
Fig. 5 Subaperture test geometry.
Fig. 6
Fig. 6 Verification of self-calibrated stitching. (a) Uncalibrated stitching result, (b) Self-calibrated stitching result.
Fig. 7
Fig. 7 Calibrated TS reference error. (a) Self-calibrated result, (b) Zygo’s final report (PVr and Zernike fit).
Fig. 8
Fig. 8 Stitching results with power removed directly. (a) Without ball constraints. (b) With ball constraint α = 0.1, (c) Outmost subaperture without constraints, (d) Outmost subaperture with α = 0.1.
Fig. 9
Fig. 9 Stitching results with sphericity assessment. (a) Without ball constraints, (b) With ball constraint α = 0.1

Equations (11)

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( x 0 , y 0 ) = r r t s d t s d p ( u , v ) .
z 0 = r 2 x 0 2 y 0 2 r .
[ x y z 1 ] T = R z T 1 R y T 2 [ x 0 y 0 z 0 1 ] T .
R z = [ cos γ sin γ 0 0 sin λ cos γ 0 0 0 0 1 0 0 0 0 1 ] , R y = [ cos β 0 sin β 0 0 1 0 0 sin β 0 cos β 0 0 0 0 1 ] T 1 = [ 1 0 0 0 0 1 0 0 0 0 1 r 0 0 0 1 ] , T 2 = [ 1 0 0 0 0 1 0 0 0 0 1 r 0 0 0 1 ] .
Δ φ = a + b u + c v + p ( u 2 + v 2 ) + s φ u + t φ v + θ ( u φ v v φ u ) i = 1 M q i Z i .
[ x M , y M , z M ] = [ x , y , z ] + ( φ + p ( u 2 + v 2 ) i = 1 M q i Z i ) n .
min σ o 2 = i = 1 s 1 k = i + 1 s j o = 1 N i k o ( g k H i k j o , k h i k j o , k , n i k j o , k g i H i k j o , i h i k j o , k , n i k j o , k ) 2 N o .
A x = b s .t . x α .
{ x = x 0 cos β + ( r + z 0 ) sin β y = y 0 .
x 2 + y 2 = cos 2 β + 1 2 ( x 0 2 + y 0 2 ) + cos 2 β 1 2 ( x 0 2 y 0 2 ) + ( r + z 0 ) 2 sin 2 β + x 0 ( r + z 0 ) sin 2 β .
f ( g ) = i = 1 n ( d ( g , p i ) C ) 2 .

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