Abstract

Lattice design is subtle and complicated for the subaperture stitching test of aspheric surfaces. Methods are described in this paper for the collection and arrangement of subapertures, and calculation of the best-fit sphere for each subaperture. The best-fit sphere is determined by minimizing the mean-square aspheric deviations in the form of a surface integral. Finally, a numerical example is given to illustrate the procedure, and also to verify the validity of our proposed methods.

© 2006 Optical Society of America

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References

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  1. S. Y. Chen, S. Y. Li, and Y. F. Dai, "Iterative algorithm for subaperture stitching interferometry for general surfaces," J. Opt. Soc. Am. A 22, 1929-1936 (2005).
    [CrossRef]
  2. C. Kim and J. Wyant, "Subaperture test of a large flat on a fast aspheric surface," J. Opt. Soc. Am. 71, 15-87 (1981).
  3. J. G. Thunen and O. Y. Kwon, "Full aperture testing with subaperture test optics," in Wavefront Sensing, N. Bareket and C. L. Koliopoulos, eds., Proc. SPIE 351, 19-27 (1982).
  4. T. W. Stuhlinger, "Subaperture optical testing experimental verification," in Contemporary Optical Instrument Design, Fabrication, and Testing, L. H. J. F. Beckmann, J. D. Briers, and P. R. Yoder Jr., eds., Proc. SPIE 656, 118-127 (1986).
  5. M. Y. Chen, W. M. Cheng, and C. W. Wang, "Multiaperture overlap-scanning technique for large-aperture test." in Laser Interferometry IV: Computer-Aided. Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE 1553, 626-635 (1991).
    [CrossRef]
  6. W. M. Cheng and M. Y. Chen, "Transformation and connection of subapertures in the multiaperture overlap-scanning technique for large optics tests," Opt Eng. 32, 1947-1950 (1993).
    [CrossRef]
  7. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
    [CrossRef]
  8. J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, "An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparré, and B. Singh, eds., Proc. SPIE 5188, 296-307 (2003).
    [CrossRef]
  9. W. Gander and W. Gautschi, "Adaptive quadrature--revisited," BIT 40, 84-101 (2000).
    [CrossRef]

2005 (1)

S. Y. Chen, S. Y. Li, and Y. F. Dai, "Iterative algorithm for subaperture stitching interferometry for general surfaces," J. Opt. Soc. Am. A 22, 1929-1936 (2005).
[CrossRef]

2003 (2)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[CrossRef]

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, "An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparré, and B. Singh, eds., Proc. SPIE 5188, 296-307 (2003).
[CrossRef]

2000 (1)

W. Gander and W. Gautschi, "Adaptive quadrature--revisited," BIT 40, 84-101 (2000).
[CrossRef]

1993 (1)

W. M. Cheng and M. Y. Chen, "Transformation and connection of subapertures in the multiaperture overlap-scanning technique for large optics tests," Opt Eng. 32, 1947-1950 (1993).
[CrossRef]

1991 (1)

M. Y. Chen, W. M. Cheng, and C. W. Wang, "Multiaperture overlap-scanning technique for large-aperture test." in Laser Interferometry IV: Computer-Aided. Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE 1553, 626-635 (1991).
[CrossRef]

1986 (1)

T. W. Stuhlinger, "Subaperture optical testing experimental verification," in Contemporary Optical Instrument Design, Fabrication, and Testing, L. H. J. F. Beckmann, J. D. Briers, and P. R. Yoder Jr., eds., Proc. SPIE 656, 118-127 (1986).

1981 (1)

Chen, M. Y.

W. M. Cheng and M. Y. Chen, "Transformation and connection of subapertures in the multiaperture overlap-scanning technique for large optics tests," Opt Eng. 32, 1947-1950 (1993).
[CrossRef]

M. Y. Chen, W. M. Cheng, and C. W. Wang, "Multiaperture overlap-scanning technique for large-aperture test." in Laser Interferometry IV: Computer-Aided. Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE 1553, 626-635 (1991).
[CrossRef]

Chen, S. Y.

S. Y. Chen, S. Y. Li, and Y. F. Dai, "Iterative algorithm for subaperture stitching interferometry for general surfaces," J. Opt. Soc. Am. A 22, 1929-1936 (2005).
[CrossRef]

Cheng, W. M.

W. M. Cheng and M. Y. Chen, "Transformation and connection of subapertures in the multiaperture overlap-scanning technique for large optics tests," Opt Eng. 32, 1947-1950 (1993).
[CrossRef]

M. Y. Chen, W. M. Cheng, and C. W. Wang, "Multiaperture overlap-scanning technique for large-aperture test." in Laser Interferometry IV: Computer-Aided. Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE 1553, 626-635 (1991).
[CrossRef]

Dai, Y. F.

S. Y. Chen, S. Y. Li, and Y. F. Dai, "Iterative algorithm for subaperture stitching interferometry for general surfaces," J. Opt. Soc. Am. A 22, 1929-1936 (2005).
[CrossRef]

Dumas, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[CrossRef]

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, "An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparré, and B. Singh, eds., Proc. SPIE 5188, 296-307 (2003).
[CrossRef]

Fleig, J.

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, "An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparré, and B. Singh, eds., Proc. SPIE 5188, 296-307 (2003).
[CrossRef]

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[CrossRef]

Forbes, G.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[CrossRef]

Forbes, G. W.

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, "An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparré, and B. Singh, eds., Proc. SPIE 5188, 296-307 (2003).
[CrossRef]

Gander, W.

W. Gander and W. Gautschi, "Adaptive quadrature--revisited," BIT 40, 84-101 (2000).
[CrossRef]

Gautschi, W.

W. Gander and W. Gautschi, "Adaptive quadrature--revisited," BIT 40, 84-101 (2000).
[CrossRef]

Kim, C.

Kwon, O. Y.

J. G. Thunen and O. Y. Kwon, "Full aperture testing with subaperture test optics," in Wavefront Sensing, N. Bareket and C. L. Koliopoulos, eds., Proc. SPIE 351, 19-27 (1982).

Li, S. Y.

S. Y. Chen, S. Y. Li, and Y. F. Dai, "Iterative algorithm for subaperture stitching interferometry for general surfaces," J. Opt. Soc. Am. A 22, 1929-1936 (2005).
[CrossRef]

Murphy, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[CrossRef]

Murphy, P. E.

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, "An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparré, and B. Singh, eds., Proc. SPIE 5188, 296-307 (2003).
[CrossRef]

Stuhlinger, T. W.

T. W. Stuhlinger, "Subaperture optical testing experimental verification," in Contemporary Optical Instrument Design, Fabrication, and Testing, L. H. J. F. Beckmann, J. D. Briers, and P. R. Yoder Jr., eds., Proc. SPIE 656, 118-127 (1986).

Thunen, J. G.

J. G. Thunen and O. Y. Kwon, "Full aperture testing with subaperture test optics," in Wavefront Sensing, N. Bareket and C. L. Koliopoulos, eds., Proc. SPIE 351, 19-27 (1982).

Tricard, M.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[CrossRef]

Wang, C. W.

M. Y. Chen, W. M. Cheng, and C. W. Wang, "Multiaperture overlap-scanning technique for large-aperture test." in Laser Interferometry IV: Computer-Aided. Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE 1553, 626-635 (1991).
[CrossRef]

Wyant, J.

BIT (1)

W. Gander and W. Gautschi, "Adaptive quadrature--revisited," BIT 40, 84-101 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

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

S. Y. Chen, S. Y. Li, and Y. F. Dai, "Iterative algorithm for subaperture stitching interferometry for general surfaces," J. Opt. Soc. Am. A 22, 1929-1936 (2005).
[CrossRef]

Opt Eng. (1)

W. M. Cheng and M. Y. Chen, "Transformation and connection of subapertures in the multiaperture overlap-scanning technique for large optics tests," Opt Eng. 32, 1947-1950 (1993).
[CrossRef]

Opt. Photon. News (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[CrossRef]

Proc. SPIE (2)

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, "An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparré, and B. Singh, eds., Proc. SPIE 5188, 296-307 (2003).
[CrossRef]

M. Y. Chen, W. M. Cheng, and C. W. Wang, "Multiaperture overlap-scanning technique for large-aperture test." in Laser Interferometry IV: Computer-Aided. Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE 1553, 626-635 (1991).
[CrossRef]

Other (2)

J. G. Thunen and O. Y. Kwon, "Full aperture testing with subaperture test optics," in Wavefront Sensing, N. Bareket and C. L. Koliopoulos, eds., Proc. SPIE 351, 19-27 (1982).

T. W. Stuhlinger, "Subaperture optical testing experimental verification," in Contemporary Optical Instrument Design, Fabrication, and Testing, L. H. J. F. Beckmann, J. D. Briers, and P. R. Yoder Jr., eds., Proc. SPIE 656, 118-127 (1986).

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Figures (8)

Fig. 1
Fig. 1

Overlapping ratio between two longitudinally adjacent subapertures.

Fig. 2
Fig. 2

Overlapping ratio between two latitudinally adjacent subapertures.

Fig. 3
Fig. 3

Test configuration of the central subaperture.

Fig. 4
Fig. 4

Test configuration of the off-axis subaperture.

Fig. 5
Fig. 5

Test configuration of the outer off-axis subaperture.

Fig. 6
Fig. 6

Geometry of the spherical Fizeau interferometer.

Fig. 7
Fig. 7

Lattice definition for a paraboloid.

Fig. 8
Fig. 8

Subinterferograms.

Tables (1)

Tables Icon

Table 1 Best-Fit Spheres and Overlapping Ratios

Equations (29)

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r ˜ = r 0 sin θ ,
x 2 + y 2 = 2 p z ,
r 0 = p ( 1 + x 0     2 p 2 ) 3 / 2 ,
β i = arctan x 0 i p ,     β k = arctan x 0 k p .
T ik = T i     1 T k ,
T i ( k ) = [ cos β i ( k ) 0 sin β i ( k ) x 0 i ( k ) 0 1 0 0 sin β i ( k ) 0 cos β i ( k ) x 0 i ( k ) 2 2 p 0 0 0 1 ]
( cos β k x k sin β k z k + x 0 k ) 2 + y k     2 = 2 p ( sin β k x k + cos β k z k + x 0 k        2 2 p ) .
T ik = [ x 0 cos β R 0 x 0 sin β 0 0 0 1 ] [ 1 0 0 x 0 cos β 0 1 0 0 0 0 1 x 0 sin β 0 0 0 1 ] ,
ω ^ = [ 0 ω 3 ω 2 ω 3 0 ω 1 ω 2 ω 1 0 ] .
S = Q [ ρ 2 + ( ρ 2 2 p c ) 2 r bs ] 2 d q ,
d q = 1 + ( z x ) 2 + ( z y ) 2 d x d y .
d q = 1 + x 2 + y 2 p 2 d x d y = 1 + ρ 2 p 2 ρ d ρ d φ .
S = 2 π 0 r [ ρ 2 + ( ρ 2 2 p c ) 2 r bs ] 2 1 + ρ 2 p 2 ρ d ρ ,
r = ( c r 2 2 p ) tan θ .
A = Q d q = 2 π p 2 3 [ ( 1 + r 2 p 2 ) 3 / 2 1 ] .
min c , r bs f ( c , r bs ) = S / A .
T i = [ cos β 0 sin β x 0 0 1 0 0 sin β 0 cos β x 0     2 2 p 0 0 0 1 ] .
S = Q [ ( x + c sin β x 0 ) 2 + y 2 + ( x 2 + y 2 2 p x 0     2 2 p c cos β ) 2 r bs ] 2 d q = Q h ( x , y ) d q ,
S = 2 x l x u 0 y ( x ) h ( x , y ) 1 + x 2 + y 2 p 2 d x d y .
A = 2 x l x u 0 y ( x ) 1 + x 2 + y 2 p 2 d x d y .
min c , r bs , β f ( c , r bs , β ) = S / A .
{ [ ( x x 0 ) cos β + x 2 + y 2 x 0 2 2 p sin β ] 2 + y 2 = [ c + ( x x 0 ) sin β x 2 + y 2 x 0 2 2 p cos β ] tan θ . z = 0
[ x y z ] = [                     r bs + ϕ r ts u                       r bs + ϕ r ts v r ts r bs + ϕ r ts r ts 2 u 2 v 2 ] ,
g 1 = [ 1 0 0 0 0 1 0 0 0 0 1 r ts c 0 0 0 1 ] .
g i = [ cos β 0 sin β x 0 ( c r ts ) sin β 0 1 0 0 sin β 0 cos β x 0     2 2 p + ( c r ts ) cos β 0 0 0 1 ] .
g k = g i k       1 g i ,
g i k = [ [ x 0 ( c r ts ) sin β ] cos β R 0 [ x 0 ( c r ts ) sin β ] sin β 0 0 0 1 ] [ 1 0 0 [ x 0 ( c r ts ) sin β ] cos β 0 1 0 0 0 0 1 [ x 0 ( c r ts ) sin β ] sin β 0 0 0 1 ] ,
[ x M y M z M 1 ] = g i - 1 [ x y z 1 ] .
I ( u , v ) = a + b cos 4 π λ ϕ ,

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