Abstract

Limited by the f-number of the transmission sphere, it is impossible to test the whole surface of a hyper-hemisphere using a standard interferometer directly. This paper presents an extension of the subaperture stitching test method to hyper hemispheres. The stitching algorithm is based on the coordinate mapping from local measurement frame to a global frame, and overlapping correspondence is calculated by virtue of coordinates of latitude and longitude. The reference surface error is represented by Zernike polynomials and self-calibrated during the stitching to achieve higher accuracy. Then the stitched surface error distribution is presented by map projection. To realize accessibility to the whole surface of a hyper-hemisphere, we also propose a design for the subaperture test platform, according to the subaperture lattice design. Finally, a hemisphere and a full sphere are tested and figured, respectively, to validate the method and the experimental setup.

© 2012 Optical Society of America

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2010 (1)

2009 (1)

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

2006 (1)

2004 (1)

U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” CIRP Ann.-Manuf. Tech. 53, 451–454 (2004).
[CrossRef]

2003 (1)

2000 (1)

S. Buchman, C. W. F. Everitt, B. Parkinson, J. P. Turneaure, and G. M. Keiser, “Cryogenic gyroscopes for the relativity mission,” Physica B 280, 497–498 (2000).
[CrossRef]

1998 (1)

Z. Li, J. Gou, and Y. Chu, “Geometric algorithm for workpiece localization,” IEEE Trans. Robot. Autom. 14, 864–878 (1998).
[CrossRef]

1997 (1)

T. Kanada, “Estimation of sphericity by means of statistical processing for roundness of spherical parts,” Precis. Eng. 20, 117–122 (1997).
[CrossRef]

1987 (1)

1980 (1)

J. A. Lipa and G. J. Siddal, “High precision measurement of gyro rotor sphericity,” Precis. Eng. 2, 123–128 (1980).
[CrossRef]

Buchman, S.

S. Buchman, C. W. F. Everitt, B. Parkinson, J. P. Turneaure, and G. M. Keiser, “Cryogenic gyroscopes for the relativity mission,” Physica B 280, 497–498 (2000).
[CrossRef]

Burge, J. H.

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

Chen, S. Y.

Chu, Y.

Z. Li, J. Gou, and Y. Chu, “Geometric algorithm for workpiece localization,” IEEE Trans. Robot. Autom. 14, 864–878 (1998).
[CrossRef]

Dai, Y. F.

Day, R. D.

Deck, L. L.

doiron, T.

J. Stoup and T. doiron, “High accuracy CMM measurements of large silicon spheres,” in ASPE Topical Meetings: Coordinate Measuring Machines (2003).

Everitt, C. W. F.

S. Buchman, C. W. F. Everitt, B. Parkinson, J. P. Turneaure, and G. M. Keiser, “Cryogenic gyroscopes for the relativity mission,” Physica B 280, 497–498 (2000).
[CrossRef]

Forbes, G.

D. Golini, G. Forbes, and P. Murphy, “Method for self-calibrated sub-aperture stitching for surface figure measurement,” U.S. patent 6,956,657 B2 (18October2005).

Golini, D.

D. Golini, G. Forbes, and P. Murphy, “Method for self-calibrated sub-aperture stitching for surface figure measurement,” U.S. patent 6,956,657 B2 (18October2005).

Gou, J.

Z. Li, J. Gou, and Y. Chu, “Geometric algorithm for workpiece localization,” IEEE Trans. Robot. Autom. 14, 864–878 (1998).
[CrossRef]

Griesmann, U.

U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” CIRP Ann.-Manuf. Tech. 53, 451–454 (2004).
[CrossRef]

Kanada, T.

T. Kanada, “Estimation of sphericity by means of statistical processing for roundness of spherical parts,” Precis. Eng. 20, 117–122 (1997).
[CrossRef]

H. Kawa, T. Kanada, and T. Watanabe, “Development of spherical form errors measuring system: design and some experimental results,” in ASPE 17th Annual Meeting (2002).

Kawa, H.

H. Kawa, T. Kanada, and T. Watanabe, “Development of spherical form errors measuring system: design and some experimental results,” in ASPE 17th Annual Meeting (2002).

Keiser, G. M.

S. Buchman, C. W. F. Everitt, B. Parkinson, J. P. Turneaure, and G. M. Keiser, “Cryogenic gyroscopes for the relativity mission,” Physica B 280, 497–498 (2000).
[CrossRef]

Lawrence, G. N.

Li, S. Y.

Li, Z.

Z. Li, J. Gou, and Y. Chu, “Geometric algorithm for workpiece localization,” IEEE Trans. Robot. Autom. 14, 864–878 (1998).
[CrossRef]

Liao, W. L.

Lipa, J. A.

J. A. Lipa and G. J. Siddal, “High precision measurement of gyro rotor sphericity,” Precis. Eng. 2, 123–128 (1980).
[CrossRef]

Murphy, P.

D. Golini, G. Forbes, and P. Murphy, “Method for self-calibrated sub-aperture stitching for surface figure measurement,” U.S. patent 6,956,657 B2 (18October2005).

Parkinson, B.

S. Buchman, C. W. F. Everitt, B. Parkinson, J. P. Turneaure, and G. M. Keiser, “Cryogenic gyroscopes for the relativity mission,” Physica B 280, 497–498 (2000).
[CrossRef]

Siddal, G. J.

J. A. Lipa and G. J. Siddal, “High precision measurement of gyro rotor sphericity,” Precis. Eng. 2, 123–128 (1980).
[CrossRef]

Soons, J.

U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” CIRP Ann.-Manuf. Tech. 53, 451–454 (2004).
[CrossRef]

Stoup, J.

J. Stoup and T. doiron, “High accuracy CMM measurements of large silicon spheres,” in ASPE Topical Meetings: Coordinate Measuring Machines (2003).

Turneaure, J. P.

S. Buchman, C. W. F. Everitt, B. Parkinson, J. P. Turneaure, and G. M. Keiser, “Cryogenic gyroscopes for the relativity mission,” Physica B 280, 497–498 (2000).
[CrossRef]

Wang, Q.

U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” CIRP Ann.-Manuf. Tech. 53, 451–454 (2004).
[CrossRef]

Watanabe, T.

H. Kawa, T. Kanada, and T. Watanabe, “Development of spherical form errors measuring system: design and some experimental results,” in ASPE 17th Annual Meeting (2002).

Zheng, Z. W.

Zhou, L.

Zhou, P.

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

Appl. Opt. (3)

CIRP Ann.-Manuf. Tech. (1)

U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” CIRP Ann.-Manuf. Tech. 53, 451–454 (2004).
[CrossRef]

IEEE Trans. Robot. Autom. (1)

Z. Li, J. Gou, and Y. Chu, “Geometric algorithm for workpiece localization,” IEEE Trans. Robot. Autom. 14, 864–878 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

Physica B (1)

S. Buchman, C. W. F. Everitt, B. Parkinson, J. P. Turneaure, and G. M. Keiser, “Cryogenic gyroscopes for the relativity mission,” Physica B 280, 497–498 (2000).
[CrossRef]

Precis. Eng. (2)

T. Kanada, “Estimation of sphericity by means of statistical processing for roundness of spherical parts,” Precis. Eng. 20, 117–122 (1997).
[CrossRef]

J. A. Lipa and G. J. Siddal, “High precision measurement of gyro rotor sphericity,” Precis. Eng. 2, 123–128 (1980).
[CrossRef]

Proc. SPIE (1)

P. Zhou and J. H. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

Other (4)

H. Kawa, T. Kanada, and T. Watanabe, “Development of spherical form errors measuring system: design and some experimental results,” in ASPE 17th Annual Meeting (2002).

J. Stoup and T. doiron, “High accuracy CMM measurements of large silicon spheres,” in ASPE Topical Meetings: Coordinate Measuring Machines (2003).

MathWorks, “Mapping Toolbox User’s Guide,” http://www.mathworks.com/help/toolbox/map/bq0w1eg.html .

D. Golini, G. Forbes, and P. Murphy, “Method for self-calibrated sub-aperture stitching for surface figure measurement,” U.S. patent 6,956,657 B2 (18October2005).

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Figures (16)

Fig. 1.
Fig. 1.

Test geometry with a spherical interferometer.

Fig. 2.
Fig. 2.

Projection onto the plane of latitude and longitude.

Fig. 3.
Fig. 3.

Overlap calculation on the plane of latitude and longitude.

Fig. 4.
Fig. 4.

Subaperture lattice design for the hemisphere.

Fig. 5.
Fig. 5.

Subaperture lattice design for the full sphere.

Fig. 6.
Fig. 6.

Test platform design for the hemisphere.

Fig. 7.
Fig. 7.

Test platform design for the full sphere.

Fig. 8.
Fig. 8.

Experimental setup for stitching test of the hemisphere.

Fig. 9.
Fig. 9.

Initial error distribution of the hemisphere (a) with reference error included, (b) with reference error self-calibrated, and (c) with globe show.

Fig. 10.
Fig. 10.

Experimental setup for figuring of the hemisphere.

Fig. 11.
Fig. 11.

Final error distribution of the hemisphere (a) with reference error self-calibrated and (b) with globe show.

Fig. 12.
Fig. 12.

Reference error. (a) Test report from Zygo, (b) calibration with higher-quality transmission sphere, (c) polynomial description of the calibration result, (d) retrieved by stitching of the hemisphere, and (e) retrieved by stitching of the full sphere.

Fig. 13.
Fig. 13.

Experimental setup for stitching test of the full sphere.

Fig. 14.
Fig. 14.

Experimental setup for figuring of the full sphere.

Fig. 15.
Fig. 15.

Initial error distribution of the full sphere (a) with reference error self-calibrated and (b) with orthogonal projection of the northern hemisphere.

Fig. 16.
Fig. 16.

Final error distribution of the full sphere (a) with reference error self-calibrated and (b) with orthogonal projection of the northern hemisphere.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

[xyz]=[(r+φ)βu(r+φ)βvrts+(r+φ)1β2(u2+v2)],
Z(u,v)=q=1QaqZq,
[xyz]=[(r+φZ(u,v))βu(r+φZ(u,v))βvrts+(r+φZ(u,v))1β2(u2+v2)].
fiwj,i=gi1[xj,i,yj,i,zj,i,1]T,
gi{exp(t=1hmtξ^t)G0}.
{gil+1=gilexp(t=1hmt,iξ^t)gil(I+t=1hmt,iξ^t)i=1,,sril+1=ril+r˜iβil+1=βil+β˜iaql+1=aql+a˜qq=1,,Q,
fiwj,ixj,i,nj,i=dj,it=1hmt,inj,iTξ^txj,i+r˜inj,iTgi1[βiuj,i,βivj,i,1βi2(uj,i2+vj,i2),0]T+β˜inj,iTgi1(ri+φj,iZ(uj,i,vj,i))[u,v,(uj,i2+vj,i2)βi1βi2(uj,i2+vj,i2),0]Tq=1Qa˜qZq(uj,i,vj,i)nj,iTgi1[βiuj,i,βivj,i,1βi2(uj,i2+vj,i2),0]T,
{lo=arctan(y/x)la=arcsin(z/x2+y2).

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