Abstract

Absolute test needs test part rotation to separate errors of the interferometer itself from errors due to the test surfaces. At this time, previous absolute test algorithms assume no azimuthal position error during part rotation. For large optics whose diameters are 0.6 m and over, however, exact rotations are physically difficult. Motivated by this, we propose a new algorithm that adopts least squares technique to determine the true azimuthal positions of part rotation and consequently eliminates testing errors caused by rotation inaccuracy.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara, "Phase shifting interferometry," in Optical Shop Testing, 2nd ed., (Wiley, New York, 1992), Chap.14.
  2. A. E. Jensen, "Absolute calibration method for laser Twyman-Green wave-front testing interferometers," J. Opt. Soc. Am. 63, 1313A (1973).
  3. K. L. Shu, "Ray-trace analysis and data reduction methods for the Ritchey-Common test," Appl. Opt. 22, 1879-1886 (1983).
    [CrossRef] [PubMed]
  4. B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).
  5. K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, "Absolute sphericity measurement," Appl. Opt. 28, 4649-4661 (1989).
    [CrossRef] [PubMed]
  6. G. Shulz and J. Grzanna, "Absolute flatness testing by the rotation method with optimal measuring error compensation," Appl. Opt. 31, 3767-3780 (1992).
    [CrossRef]
  7. C. Ai and J. C. Wyant, "Absolute testing of flats using even and odd functions," Appl. Opt. 32, 4698-4703 (1993).
    [CrossRef] [PubMed]
  8. C. Evans, R. Kestner, "Test optics error removal," Appl. Opt. 35, 1015-1021 (1996).
    [CrossRef] [PubMed]
  9. P. Hariharan, "Interferometric testing of optical surfaces: absolute measurement of flatness," Opt. Eng. 36, 2478-2481 (1997).
    [CrossRef]
  10. W. T. Estler, C. J. Evans, and L. Z. Shao, "Uncertainty estimation for multi-position form error metrology," Prec. Eng. 21, 72-82 (1997).
    [CrossRef]
  11. R. E. Parks, L. Shao, and C. J. Evans, "Pixel-based absolute topography test for three flats," Appl. Opt. 37, 5951-5956 (1998).
    [CrossRef]
  12. V. Greco, R. Tronconi, C. D. Vecchio, M. Trivi, and G. Molesini, "Absolute measurement of planarity with Fritz’s method: uncertainty evaluation," Appl. Opt. 38, 2018-2027 (1999).
    [CrossRef]
  13. P. E. Murphy, T. G. Brown, and D. T. Moore, "Interference imaging for aspheric surface testing," Appl. Opt. 39, 2122-2129 (2000).
    [CrossRef]
  14. K. R. Freischlad, "Absolute interferomtric testing based on reconstruction of rotational shear," Appl. Opt. 401637-1648 (2001).
    [CrossRef]
  15. S. Reichelt, C. Pruss, and H. J. Tiziani, "Absolute interferometric test of aspheres by use of twin computer-generated holograms," Appl. Opt. 42, 4468-4479 (2003).
    [CrossRef] [PubMed]
  16. Ulf Griesmann, "Three-flat test solutions based on simple mirror symmetry" Appl. Opt. 45, 5856-5865 (2006).
    [CrossRef] [PubMed]
  17. C. Evans, R. Hocken, and W. Estler, "Self-calibration: reversal, redundancy, error separation, and absolute testing," Annals of the CIRP 45, 617-634 (1996).
    [CrossRef]
  18. S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, "Merit Function regression method for efficient alignment control of two-mirror optical system," Opt. Express (to be published).
    [PubMed]
  19. H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, "Null Hartmann test for fabrication of large aspheric surfaces," Opt. Express 6, 1839-1847 (2005).
    [CrossRef]

2006 (1)

2005 (1)

H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, "Null Hartmann test for fabrication of large aspheric surfaces," Opt. Express 6, 1839-1847 (2005).
[CrossRef]

2003 (1)

2001 (1)

2000 (1)

1999 (1)

1998 (1)

1997 (2)

P. Hariharan, "Interferometric testing of optical surfaces: absolute measurement of flatness," Opt. Eng. 36, 2478-2481 (1997).
[CrossRef]

W. T. Estler, C. J. Evans, and L. Z. Shao, "Uncertainty estimation for multi-position form error metrology," Prec. Eng. 21, 72-82 (1997).
[CrossRef]

1996 (2)

C. Evans, R. Kestner, "Test optics error removal," Appl. Opt. 35, 1015-1021 (1996).
[CrossRef] [PubMed]

C. Evans, R. Hocken, and W. Estler, "Self-calibration: reversal, redundancy, error separation, and absolute testing," Annals of the CIRP 45, 617-634 (1996).
[CrossRef]

1993 (1)

1992 (1)

1989 (1)

1984 (1)

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).

1983 (1)

1973 (1)

A. E. Jensen, "Absolute calibration method for laser Twyman-Green wave-front testing interferometers," J. Opt. Soc. Am. 63, 1313A (1973).

Ai, C.

Brown, T. G.

Burow, R.

Elssner, K.-E.

Estler, W.

C. Evans, R. Hocken, and W. Estler, "Self-calibration: reversal, redundancy, error separation, and absolute testing," Annals of the CIRP 45, 617-634 (1996).
[CrossRef]

Estler, W. T.

W. T. Estler, C. J. Evans, and L. Z. Shao, "Uncertainty estimation for multi-position form error metrology," Prec. Eng. 21, 72-82 (1997).
[CrossRef]

Evans, C.

C. Evans, R. Kestner, "Test optics error removal," Appl. Opt. 35, 1015-1021 (1996).
[CrossRef] [PubMed]

C. Evans, R. Hocken, and W. Estler, "Self-calibration: reversal, redundancy, error separation, and absolute testing," Annals of the CIRP 45, 617-634 (1996).
[CrossRef]

Evans, C. J.

R. E. Parks, L. Shao, and C. J. Evans, "Pixel-based absolute topography test for three flats," Appl. Opt. 37, 5951-5956 (1998).
[CrossRef]

W. T. Estler, C. J. Evans, and L. Z. Shao, "Uncertainty estimation for multi-position form error metrology," Prec. Eng. 21, 72-82 (1997).
[CrossRef]

Freischlad, K. R.

Fritz, B. S.

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).

Greco, V.

Grzanna, J.

Hariharan, P.

P. Hariharan, "Interferometric testing of optical surfaces: absolute measurement of flatness," Opt. Eng. 36, 2478-2481 (1997).
[CrossRef]

Hocken, R.

C. Evans, R. Hocken, and W. Estler, "Self-calibration: reversal, redundancy, error separation, and absolute testing," Annals of the CIRP 45, 617-634 (1996).
[CrossRef]

Jensen, A. E.

A. E. Jensen, "Absolute calibration method for laser Twyman-Green wave-front testing interferometers," J. Opt. Soc. Am. 63, 1313A (1973).

Kestner, R.

Kim, S.

S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, "Merit Function regression method for efficient alignment control of two-mirror optical system," Opt. Express (to be published).
[PubMed]

Kim, S. W.

S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, "Merit Function regression method for efficient alignment control of two-mirror optical system," Opt. Express (to be published).
[PubMed]

Lee, I. W.

H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, "Null Hartmann test for fabrication of large aspheric surfaces," Opt. Express 6, 1839-1847 (2005).
[CrossRef]

Lee, Y. W.

H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, "Null Hartmann test for fabrication of large aspheric surfaces," Opt. Express 6, 1839-1847 (2005).
[CrossRef]

S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, "Merit Function regression method for efficient alignment control of two-mirror optical system," Opt. Express (to be published).
[PubMed]

Molesini, G.

Moore, D. T.

Murphy, P. E.

Parks, R. E.

Pruss, C.

Reichelt, S.

Shao, L.

Shao, L. Z.

W. T. Estler, C. J. Evans, and L. Z. Shao, "Uncertainty estimation for multi-position form error metrology," Prec. Eng. 21, 72-82 (1997).
[CrossRef]

Shu, K. L.

Shulz, G.

Song, J. B.

H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, "Null Hartmann test for fabrication of large aspheric surfaces," Opt. Express 6, 1839-1847 (2005).
[CrossRef]

Spolaczyk, R.

Tiziani, H. J.

Trivi, M.

Tronconi, R.

Vecchio, C. D.

Wyant, J. C.

Yang, H. S.

H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, "Null Hartmann test for fabrication of large aspheric surfaces," Opt. Express 6, 1839-1847 (2005).
[CrossRef]

S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, "Merit Function regression method for efficient alignment control of two-mirror optical system," Opt. Express (to be published).
[PubMed]

Annals of the CIRP (1)

C. Evans, R. Hocken, and W. Estler, "Self-calibration: reversal, redundancy, error separation, and absolute testing," Annals of the CIRP 45, 617-634 (1996).
[CrossRef]

Appl. Opt. (11)

R. E. Parks, L. Shao, and C. J. Evans, "Pixel-based absolute topography test for three flats," Appl. Opt. 37, 5951-5956 (1998).
[CrossRef]

V. Greco, R. Tronconi, C. D. Vecchio, M. Trivi, and G. Molesini, "Absolute measurement of planarity with Fritz’s method: uncertainty evaluation," Appl. Opt. 38, 2018-2027 (1999).
[CrossRef]

P. E. Murphy, T. G. Brown, and D. T. Moore, "Interference imaging for aspheric surface testing," Appl. Opt. 39, 2122-2129 (2000).
[CrossRef]

K. R. Freischlad, "Absolute interferomtric testing based on reconstruction of rotational shear," Appl. Opt. 401637-1648 (2001).
[CrossRef]

S. Reichelt, C. Pruss, and H. J. Tiziani, "Absolute interferometric test of aspheres by use of twin computer-generated holograms," Appl. Opt. 42, 4468-4479 (2003).
[CrossRef] [PubMed]

Ulf Griesmann, "Three-flat test solutions based on simple mirror symmetry" Appl. Opt. 45, 5856-5865 (2006).
[CrossRef] [PubMed]

K. L. Shu, "Ray-trace analysis and data reduction methods for the Ritchey-Common test," Appl. Opt. 22, 1879-1886 (1983).
[CrossRef] [PubMed]

K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, "Absolute sphericity measurement," Appl. Opt. 28, 4649-4661 (1989).
[CrossRef] [PubMed]

G. Shulz and J. Grzanna, "Absolute flatness testing by the rotation method with optimal measuring error compensation," Appl. Opt. 31, 3767-3780 (1992).
[CrossRef]

C. Ai and J. C. Wyant, "Absolute testing of flats using even and odd functions," Appl. Opt. 32, 4698-4703 (1993).
[CrossRef] [PubMed]

C. Evans, R. Kestner, "Test optics error removal," Appl. Opt. 35, 1015-1021 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

A. E. Jensen, "Absolute calibration method for laser Twyman-Green wave-front testing interferometers," J. Opt. Soc. Am. 63, 1313A (1973).

Opt. Eng. (2)

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).

P. Hariharan, "Interferometric testing of optical surfaces: absolute measurement of flatness," Opt. Eng. 36, 2478-2481 (1997).
[CrossRef]

Opt. Express (2)

S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, "Merit Function regression method for efficient alignment control of two-mirror optical system," Opt. Express (to be published).
[PubMed]

H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, "Null Hartmann test for fabrication of large aspheric surfaces," Opt. Express 6, 1839-1847 (2005).
[CrossRef]

Prec. Eng. (1)

W. T. Estler, C. J. Evans, and L. Z. Shao, "Uncertainty estimation for multi-position form error metrology," Prec. Eng. 21, 72-82 (1997).
[CrossRef]

Other (1)

D. Malacara, "Phase shifting interferometry," in Optical Shop Testing, 2nd ed., (Wiley, New York, 1992), Chap.14.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Comparison of simulation results when true values of αj are not equally spaced mistakenly as 0°, 61°, 121°, 179°, 241°, 299°. (a) Original wavefront generated for simulation with all the Zernike coefficients being 0.1λ for k=1-5. (b) Fringe map of the original wavefront (P-V: 1.118µm, rms: 0.110λ). (c) The wavefront error extracted by the 6-step averaging (P-V: 0.017µm, rms: 0.001λ). (d) The wavefront error computed by the least-squares algorithm (P-V: 0.001µm, rms: 3×10-7λ).

Fig. 2.
Fig. 2.

Suppression capabilities of high frequency components when αj are intentionally taken as 0°, 61°, 121°, 179°, 241°, 299°. (a) Original wavefront generated for simulation with all the Zernike coefficients being 0.1λ for k=1-8 (P-V: 1.763µm, rms: 0.11λ). (b) The wavefront error extracted by the 6-step averaging (P-V: 0.409µm, rms: 0.001λ). (c) The wavefront error computed by the least squares algorithm (P-V: 0.012µm, rms: 2×10-5λ).

Fig. 3.
Fig. 3.

Simulated result after random noise inserting.

Tables (1)

Tables Icon

Table 1. Typical aberration coefficients before and after absolute tests. The unit is λ (632.8 nm).

Equations (25)

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

W = T + P
D j = W j W 0 = ( T + P j ) ( T + P 0 ) = P j P 0 .
P 0 = P ( r , θ ) = l , k R l k ( r ) [ c lk cos ( k θ ) + d lk sin ( k θ ) ] .
P j = P ( r , θ + α j ) = l , k R l k ( r ) [ c lk cos ( k θ + α j ) + d lk sin ( k θ + α j ) ]
= l , k R l k ( r ) [ c lk cos ( k θ ) + d lk sin ( k θ ) ]
where c lk = c lk cos ( k α j ) + d lk sin ( k α j ) and
d lk = d lk cos ( k α j ) c lk sin ( k α j ) .
c lk = 1 2 [ Δ c lk + Δ d lk sin ( k α j ) ( 1 cos ( k α j ) ) ] , and d lk = 1 2 [ Δ d lk + Δ d lk sin ( k α j ) ( 1 cos ( k α j ) ) ] .
D j = P 0 [ cos ( k α j ) 1 ] + P ˜ 0 sin ( k α j ) ,
where P ˜ 0 = l , k R l k ( r ) [ d lk cos ( k θ ) c lk sin ( k θ ) ] .
j = 0 N 1 D j = P 0 [ j = 0 N 1 cos ( k α j ) N ] + P ˜ 0 j = 0 N 1 sin ( k α j ) .
j = 0 N 1 sin ( k α j ) = 0 for all k ,and
j = 0 N 1 cos ( k α j ) = 0 for k not being integer multiplies of N , otherwise
j = 0 N 1 cos ( k α j ) = N .
P 0 = 1 N j = 0 N 1 D j .
p k ( r , θ ) = l L ( k ) R l k ( r ) [ c lk cos ( k θ ) + d lk sin ( k θ ) ] = l L ( k ) ξ l k Z l k ( r , θ )
where Z l k ( r , θ ) R l k ( r ) { cos sin } ( k θ )
D j k ( r , θ ) = P 0 k ( r , θ ) [ cos ( k α j ) 1 ] + P ˜ 0 k ( r , θ ) sin ( k α j )
= l L ( k ) { ξ 0 l k Z l k ( r , θ ) [ cos ( k α j ) 1 ] + ξ ˜ 0 l k Z l k ( r , θ ) sin ( k α j ) }
= l L ( k ) X lj k Z l k ( r , θ ) .
E l k = j = 0 N 1 { X lj k X ̂ lj k } 2 = j = 0 N 1 { ξ 0 l k [ cos ( k α j ) 1 ] + ξ ˜ 0 l k sin ( k α j ) X ̂ lj k } 2 ,
E j k = l L ( k ) { X lj k X ̂ lj k } 2 = l L ( k ) { ξ 0 l k [ cos ( k α j ) 1 ] + ξ ˜ 0 l k sin ( k α j ) X ̂ lj k } 2 .
α E l k ξ 0 l k = 0 , E l k ξ ˜ 0 l k = 0 , E j k cos ( k α j ) = 0 , and E j k sin ( k α j ) = 0 .
[ j = 0 N 1 [ cos ( k α j ) 1 ] 2 j = 0 N 1 sin ( k α j ) [ cos ( k α j ) 1 ] j = 0 N 1 sin ( k α j ) [ cos ( k α j ) 1 ] j = 0 N 1 sin 2 ( k α j ) ] [ ξ 0 l k ξ ̃ 0 l k ] = [ j = 0 N 1 X ̂ lj k [ cos ( k α j ) 1 ] j = 0 N 1 X ̂ lj k sin ( k α j ) ]
[ l L ( k ) [ ξ 0 l k ] 2 l L ( k ) ξ 0 l k ξ ˜ 0 l k l L ( k ) ξ 0 l k ξ ˜ 0 l k l L ( k ) [ ξ ˜ 0 l k ] 2 ] [ cos ( k α j ) sin ( k α j ) ] = [ l L ( k ) { X ̂ lj k ξ 0 l k + [ ξ 0 l k ] 2 } l L ( k ) { X ̂ lj k ξ 0 l k + ξ 0 l k ξ ˜ 0 l k } ]

Metrics