Abstract

Recently we have proposed an iterative algorithm for subaperture stitching interferometry. It was referred to as the subaperture stitching and localization (SASL) algorithm. The limitation of the algorithm is that three-dimensional Cartesian coordinates are required, whereas the standard spherical interferometer can read out only the phase differences on the pixels. On the basis of the SASL algorithm, we propose an iterative algorithm for a spherical subaperture stitching test. It deals with data directly from the spherical interferometer. Unknown radii of best-fit spheres for a null test of subapertures are included in the optimization variables. The developed algorithm inherits the advantages of the SASL algorithm.

© 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, "An 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, 1587 (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).
  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. (Bellingham) 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).
  9. W. X. Liu, C. X. Liu, B. H. Hua, and J. D. Zheng, Solution Methods for Large-Scale Sparse Linear Equations (National Defense Industry, 1981) (in Chinese).

2005 (1)

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).

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. (Bellingham) 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).

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).

1982 (1)

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).

1981 (2)

C. Kim and J. Wyant, "Subaperture test of a large flat on a fast aspheric surface," J. Opt. Soc. Am. 71, 1587 (1981).

W. X. Liu, C. X. Liu, B. H. Hua, and J. D. Zheng, Solution Methods for Large-Scale Sparse Linear Equations (National Defense Industry, 1981) (in Chinese).

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. (Bellingham) 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).

Chen, S. Y.

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. (Bellingham) 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).

Dai, Y. F.

Dumas, P.

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).

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]

Fleig, J.

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).

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).

Hua, B. H.

W. X. Liu, C. X. Liu, B. H. Hua, and J. D. Zheng, Solution Methods for Large-Scale Sparse Linear Equations (National Defense Industry, 1981) (in Chinese).

Kim, C.

C. Kim and J. Wyant, "Subaperture test of a large flat on a fast aspheric surface," J. Opt. Soc. Am. 71, 1587 (1981).

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.

Liu, C. X.

W. X. Liu, C. X. Liu, B. H. Hua, and J. D. Zheng, Solution Methods for Large-Scale Sparse Linear Equations (National Defense Industry, 1981) (in Chinese).

Liu, W. X.

W. X. Liu, C. X. Liu, B. H. Hua, and J. D. Zheng, Solution Methods for Large-Scale Sparse Linear Equations (National Defense Industry, 1981) (in Chinese).

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).

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).

Wyant, J.

C. Kim and J. Wyant, "Subaperture test of a large flat on a fast aspheric surface," J. Opt. Soc. Am. 71, 1587 (1981).

Zheng, J. D.

W. X. Liu, C. X. Liu, B. H. Hua, and J. D. Zheng, Solution Methods for Large-Scale Sparse Linear Equations (National Defense Industry, 1981) (in Chinese).

J. Opt. Soc. Am. (1)

C. Kim and J. Wyant, "Subaperture test of a large flat on a fast aspheric surface," J. Opt. Soc. Am. 71, 1587 (1981).

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

Opt. Eng. (Bellingham) (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. (Bellingham) 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]

Other (5)

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).

W. X. Liu, C. X. Liu, B. H. Hua, and J. D. Zheng, Solution Methods for Large-Scale Sparse Linear Equations (National Defense Industry, 1981) (in Chinese).

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).

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).

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Figures (5)

Fig. 1
Fig. 1

Test geometry of the spherical Fizeau interferometer.

Fig. 2
Fig. 2

Lattice definition for a paraboloid.

Fig. 3
Fig. 3

Subinterferograms.

Fig. 4
Fig. 4

Track of iterations.

Fig. 5
Fig. 5

Stitched map using initial and optimal radii and configurations.

Tables (1)

Tables Icon

Table 1 Computing Time for Various Resolutions

Equations (34)

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[ x y z ] = [ r + ϕ r ts u r + ϕ r ts v r ts r + ϕ r ts r ts 2 u 2 v 2 ] ,
W = { w j , i = ( u j , i , v j , i , ϕ j , i ) R 3 j = 1 , , N i ; i = 1 , , s } ,
g * = { g i * SE ( 3 ) i = 1 , , s } ,
SE ( 3 ) = { exp ( t = 1 6 m t ξ ̂ t ) } ,
ξ ̂ = [ w ̂ v 0 0 ]
SE ( 3 ) G 0 = { g G 0 = exp ( t = 1 5 m t ξ ̂ t ) G 0 } ,
f i w j , i = g i 1 [ r i + ϕ j , i r ts u j , i , r i + ϕ j , i r ts u j , i , r ts r i + ϕ j , i r ts r ts 2 u j , i 2 v j , i 2 , 1 ] T .
d ( j , i ) = w j , i x j , i , n j , i ,
e j o , k = d j o , k d j o , i .
min F = μ 1 σ 2 + μ 2 σ o 2 ,
σ 2 = i = 1 s j = 1 N i f i w j , i x j , i , n j , i 2 i = 1 s N i .
σ o 2 = i = 1 s 1 k = i + 1 s j o = 1 N o i k ( f k w j o , k i k x j o , k i k , n j o , k i k ) ( f i w j o , i i k x j o , k i k , n j o , k i k ) 2 N o ,
g 1 l + 1 = g 1 l exp ( t = 1 h m t , 1 η ̂ ) g 1 l ( I + t = 1 h m t , 1 η ̂ ) , g i l + 1 = g i l exp ( t = 1 6 m t , i ξ ̂ ) g i l ( I + t = 1 6 m t , i ξ ̂ ) , i = 2 , , s r i l + 1 = r i l + r ̃ i , i = 1 , , s , }
A m = b ,
min F = μ 1 j = 1 N 1 ( r ̃ 1 r ts n j , 1 T g 1 1 [ u j , 1 , v j , 1 r ts 2 u j , 1 2 v j , 1 2 , 0 ] T + d j , 1 t = 1 h m t , 1 n j , 1 T η ̂ t x j , 1 ) 2 i = 1 s N i + μ 1 i = 2 s j = 1 N i ( r ̃ i r ts n j , i T g i 1 [ u j , i , v j , i r ts 2 u j , i 2 v j , i 2 , 0 ] T + d j , i t = 1 6 m t , i n j , i T ξ ̂ t x j , i ) 2 i = 1 s N i + μ 2 k = 2 s j o = 1 N o 1 k ( d j o , k 1 k d j o , 1 1 k t = 1 6 m t , k n j o , k T 1 k ξ ̂ t x j o , k 1 k + t = 1 h m t , 1 n j o , k T 1 k η ̂ t x j o , k 1 k + r ̃ k r ts n j o , k T 1 k g k 1 [ u j o , k 1 k , v j o , k 1 k r ts 2 u j o , k 2 1 k v j o , k 2 1 k , 0 ] T r ̃ 1 r ts n j o , k T 1 k g 1 1 [ u j o , 1 1 k , v j o , 1 1 k r ts 2 u j o , 1 2 1 k v j o , 1 2 1 k , 0 ] T ) 2 N o + μ 2 i = 2 s 1 k = i + 1 s j o = 1 N o i k ( d j o , k i k d j o , i i k t = 1 6 m t , k n j o , k T i k ξ ̂ t x j o , k i k + t = 1 6 m t , i n j o , k T i k ξ ̂ t x j o , k i k ( + r ̃ k r ts n j o , k T i k g k 1 [ u j o , k i k , v j o , k i k r ts 2 u j o , k 2 i k v j o , k 2 i k , 0 ] T r ̃ i r ts n j o , k T 1 k g i 1 [ u j o , i i k , v j o , i i k r ts 2 u j o , i 2 1 k v j o , i 2 i k , 0 ] T ) 2 N o .
m = [ m 1 , 1 , , m h , 1 , r ̃ 1 , m 1 , 2 , , m 6 , 2 , r ̃ 2 , m 1 , s , , m 6 , s , r ̃ s ] T .
A 1 = { λ 1 n j , 1 T η ̂ t x j , 1 1 t h λ 1 r ts n j , 1 T g 1 1 [ u j , 1 , v j , 1 r ts 2 u j , 1 2 v j , 1 2 , 0 ] T t = h + 1 0 otherwise } ,
b 1 = ( λ 2 d j , 1 ) ,
j = 1 , , N 1 ; t = 0 , , L ;
A i = { λ 1 n j , i T ξ ̂ t ( h + 1 ) 7 ( i 2 ) x j , i h + 1 + 7 ( i 2 ) + 1 t < h + 1 + 7 ( i 1 ) λ 1 r ts n j , i T g i 1 [ u j , i , v j , i r ts 2 u j , i 2 v j , i 2 , 0 ] T t = h + 1 + 7 ( i 1 ) 0 otherwise } ,
b i = ( λ 1 d j , i ) ,
j = 1 , , N i ; t = 1 , , L ; i = 2 , , s ;
A o 1 k = { λ 2 n j o , k T 1 k η ̂ t x j o , k 1 k t h λ 2 r ts n j o , k T 1 k g 1 1 [ u j o , 1 1 k , v j o , 1 1 k r ts 2 u j o , 1 2 1 k v j o , 1 2 1 k , 0 ] T t = h + 1 λ 2 n j o , k T 1 k ξ ̂ t ( h + 1 ) 7 ( k 2 ) x j o , k 1 k h + 1 + 7 ( k 2 ) + 1 t < h + 1 + 7 ( k 1 ) λ 2 r ts n j o , k T 1 k g k 1 [ u j o , k 1 k , v j o , k 1 k r ts 2 u j o , k 2 1 k v j o , k 2 1 k , 0 ] T t = h + 1 + 7 ( k 1 ) 0 otherwise } ,
b o 1 k = ( λ 2 ( d j o , k 1 k d j o , 1 1 k ) ) ,
j o = 1 , , N o 1 k ; t = 1 , , L ; k = 2 , , s ;
A o i k = { λ 2 n j o , k T i k ξ ̂ t ( h + 1 ) 7 ( i 2 ) x j o , k i k h + 1 + 7 ( i 2 ) + 1 t < h + 1 + 7 ( i 1 ) λ 2 r ts n j o , k T i k g i 1 [ u j o , i i k , v j o , i i k r ts 2 u j o , i 2 i k v j o , i 2 i k , 0 ] T t = h + 1 + 7 ( i 1 ) λ 2 n j o , k T i k ξ ̂ t ( h + 1 ) 7 ( k 2 ) x j o , k i k h + 1 + 7 ( k 2 ) + 1 t < h + 1 + 7 ( k 1 ) λ 2 r ts n j o , k T i k g k 1 [ u j o , k i k , v j o , k i k r ts 2 u j o , k 2 i k v j o , k 2 i k , 0 ] T t = h + 1 + 7 ( k 1 ) 0 otherwise } ,
b o i k = ( λ 2 ( d j o , k i k d j o , i i k ) ) ,
j o = 1 , , N o i k ; t = 1 , , L ; i = 2 , , s 1 ; k = i + 1 , , s .
A = [ A 1 T , A 2 T , , A s T , , ( A o i k ) T , , ( A o ( s 1 ) s ) T ] T ,
b = [ b 1 , b 2 , , b s , , b o i k , , b o ( s 1 ) s ] T .
min A m b 2 .
M i = [ R i 1 c i 1 A i b i ] .
Q i M i = [ R i c i 0 0 ] .
[ R N A c N A ] = [ R c 0 e ] ,

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