Abstract

Absolute planarity measurements by interferometry are classically made using three flats, compared two by two in the course of four or more tests. Data reduction is performed with various analytical methods. Here we present instead a data processing algorithm that converges to solution numerically by iteration. Examples are presented both on synthetic interferograms and on experimental data. High accuracy and versatility of the approach are demonstrated.

© 2007 Optical Society of America

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References

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  1. Lord Rayleigh, "Interference bands and their application," Nature (London) 48, 212-214 (1893).
    [CrossRef]
  2. H. Barrell and R. Marriner, "Liquid surface interferometry," Nature (London) 162, 529-530 (1948).
    [CrossRef]
  3. G. D. Dew, "The measurement of optical flatness," J. Sci. Instrum. 43, 409-415 (1966).
    [CrossRef] [PubMed]
  4. R. Brünnagel, H. -A. Oehring, and K. Steiner, "Fizeau interferometer for measuring the flatness of optical surfaces," Appl. Opt. 7, 331-335 (1967).
    [CrossRef]
  5. J. P. Marioge, B. Bonino, and M. Mullot, "Standard of flatness: its application to Fabry-Perot interferometers," Appl. Opt. 14, 2283-2285 (1975).
    [CrossRef] [PubMed]
  6. K.-E. Elssner, A. Vogel, J. Grzanna, and G. Schulz, "Establishing a flatness standard," Appl. Opt. 33, 2437-2446 (1994).
    [CrossRef] [PubMed]
  7. J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, "Large-aperture high-accuracy phase-shifting digital flat interferometer," Opt. Eng. 35, 1936-1942 (1996).
    [CrossRef]
  8. I. Powell and E. Goulet, "Absolute figure measurements with a liquid-flat reference," Appl. Opt. 37, 2579-2588 (1998).
    [CrossRef]
  9. M. Vannoni and G. Molesini, "Validation of absolute planarity reference plates with a liquid mirror," Metrologia 42, 389-393 (2005).
    [CrossRef]
  10. G. Schulz and J. Schwider, "Precise measurement of planeness," Appl. Opt. 6,1077-1084 (1967).
    [CrossRef] [PubMed]
  11. G. Schulz, J. Schwider, C. Hiller, and B. Kicker, "Establishing an optical flatness standard," Appl. Opt. 10, 929-934 (1971).
    [CrossRef] [PubMed]
  12. J. Grzanna and G. Schulz, "Absolute testing of flatness standards at square-grid points," Opt. Commun. 77, 107-112 (1990).
    [CrossRef]
  13. G. Schulz and J. Grzanna, "Absolute flatness testing by the rotation method with optimal measuring error compensation," Appl. Opt. 31, 3767-3780 (1992).
    [CrossRef] [PubMed]
  14. G. Schulz, "Absolute flatness testing by an extended rotation method using two angles of rotation," Appl. Opt. 32, 1055-1059 (1993).
    [CrossRef] [PubMed]
  15. J. Grzanna, "Absolute testing of optical flats at points on a square grid: error propagation," Appl. Opt. 33, 6654-6661 (1994).
    [CrossRef] [PubMed]
  16. B. (B). F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, "Calibration of a 300-mm-Aperture Phase-Shifting Fizeau Interferometer," Appl. Opt. 39, 5161-5171 (2000).
    [CrossRef]
  17. S. Sonozaki, K. Iwata, and Y. Iwahashi, "Measurement of profiles along a circle on two flat surfaces by use of a Fizeau interferometer with no standard," Appl. Opt. 42, 6853-6858 (2003).
    [CrossRef] [PubMed]
  18. B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 33, 379-383 (1984).
  19. C. Ai and J. C. Wyant, "Absolute testing of flats by using even and odd functions," Appl. Opt. 32, 4698-4705 (1993).
    [CrossRef] [PubMed]
  20. C. J. Evans and R. N. Kestner, "Test optics error removal," Appl. Opt. 35,1015-1021 (1996).
    [CrossRef] [PubMed]
  21. P. Hariharan, "Interferometric testing of optical surfaces: absolute measurement of flatness," Opt. Eng. 36, 2478-2481 (1997).
    [CrossRef]
  22. C. J. Evans, "Comment on the paper ‘Interferometric testing of optical surfaces: absolute measurement of flatness," Opt. Eng. 37, 1880-1882 (1998).
    [CrossRef]
  23. R. E. Parks, L.-Z. Shao, and C. J. Evans, "Pixel-based absolute topography test for three flats," Appl. Opt. 37, 5951-5956 (1998).
    [CrossRef]
  24. V. Greco, R. Tronconi, C. Del Vecchio, M. Trivi, and G. Molesini, "Absolute measurement of planarity with Fritz’s method: uncertainty evaluation," Appl. Opt. 38, 2018-2027 (1999).
    [CrossRef]
  25. K. R. Freischlad, "Absolute interferometric testing based on reconstruction of rotational shear," Appl. Opt. 40, 1637-1648 (2001).
    [CrossRef]
  26. M. F. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
    [CrossRef]
  27. U. Griesmann, "Three-flat test solutions based on simple mirror symmetry," Appl. Opt. 45, 5856-5865 (2006).
    [CrossRef] [PubMed]
  28. W. Gao, P.S. Huang, T. Yamada, S. Kiyono, "A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers," Precision Engineering 26, 396-404 (2002).
    [CrossRef]
  29. M. Schulz and C. Elster, "Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution," Opt. Eng. 45, 060503 (2006).
    [CrossRef]
  30. V. Greco and G. Molesini, "Micro-temperature effects on absolute flatness test plates," Pure Appl. Opt. 7, 1341-1346 (1998).
    [CrossRef]
  31. V. B. Gubin and V. N. Sharonov, "Absolute calibration of spherical surfaces," Sov. J. Opt. Technol. 57, 554-555 (1990).
  32. .International Bureau of Weights and Measures, International Electrotechnical Commission, International Federation of Clinical Chemistry, International Organization for Standardization, International Union of Pure and Applied Chemistry, International Union of Pure and Applied Physics, and International Organization of Legal Metrology, Guide to the Expression of Uncertainty in Measurements (International Organization for Standardization, Geneva, 1993).

2006 (2)

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

M. Schulz and C. Elster, "Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution," Opt. Eng. 45, 060503 (2006).
[CrossRef]

2005 (1)

M. Vannoni and G. Molesini, "Validation of absolute planarity reference plates with a liquid mirror," Metrologia 42, 389-393 (2005).
[CrossRef]

2003 (1)

2002 (1)

W. Gao, P.S. Huang, T. Yamada, S. Kiyono, "A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers," Precision Engineering 26, 396-404 (2002).
[CrossRef]

2001 (2)

2000 (1)

1999 (1)

1998 (4)

C. J. Evans, "Comment on the paper ‘Interferometric testing of optical surfaces: absolute measurement of flatness," Opt. Eng. 37, 1880-1882 (1998).
[CrossRef]

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

V. Greco and G. Molesini, "Micro-temperature effects on absolute flatness test plates," Pure Appl. Opt. 7, 1341-1346 (1998).
[CrossRef]

I. Powell and E. Goulet, "Absolute figure measurements with a liquid-flat reference," Appl. Opt. 37, 2579-2588 (1998).
[CrossRef]

1997 (1)

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

1996 (2)

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

J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, "Large-aperture high-accuracy phase-shifting digital flat interferometer," Opt. Eng. 35, 1936-1942 (1996).
[CrossRef]

1994 (2)

1993 (2)

1992 (1)

1990 (2)

J. Grzanna and G. Schulz, "Absolute testing of flatness standards at square-grid points," Opt. Commun. 77, 107-112 (1990).
[CrossRef]

V. B. Gubin and V. N. Sharonov, "Absolute calibration of spherical surfaces," Sov. J. Opt. Technol. 57, 554-555 (1990).

1984 (1)

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 33, 379-383 (1984).

1975 (1)

1971 (1)

1967 (2)

1966 (1)

G. D. Dew, "The measurement of optical flatness," J. Sci. Instrum. 43, 409-415 (1966).
[CrossRef] [PubMed]

1948 (1)

H. Barrell and R. Marriner, "Liquid surface interferometry," Nature (London) 162, 529-530 (1948).
[CrossRef]

1893 (1)

Lord Rayleigh, "Interference bands and their application," Nature (London) 48, 212-214 (1893).
[CrossRef]

Ai, C.

Barrell, H.

H. Barrell and R. Marriner, "Liquid surface interferometry," Nature (London) 162, 529-530 (1948).
[CrossRef]

Bonino, B.

Brünnagel, R.

Chen, J.

J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, "Large-aperture high-accuracy phase-shifting digital flat interferometer," Opt. Eng. 35, 1936-1942 (1996).
[CrossRef]

Chen, L.

J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, "Large-aperture high-accuracy phase-shifting digital flat interferometer," Opt. Eng. 35, 1936-1942 (1996).
[CrossRef]

Del Vecchio, C.

Dew, G. D.

G. D. Dew, "The measurement of optical flatness," J. Sci. Instrum. 43, 409-415 (1966).
[CrossRef] [PubMed]

Elssner, K.-E.

Elster, C.

M. Schulz and C. Elster, "Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution," Opt. Eng. 45, 060503 (2006).
[CrossRef]

Evans, C. J.

Freischlad, K. R.

Fritz, B. S.

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 33, 379-383 (1984).

Gao, W.

W. Gao, P.S. Huang, T. Yamada, S. Kiyono, "A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers," Precision Engineering 26, 396-404 (2002).
[CrossRef]

Goulet, E.

Greco, V.

Griesmann, U.

Grzanna, J.

Gubin, V. B.

V. B. Gubin and V. N. Sharonov, "Absolute calibration of spherical surfaces," Sov. J. Opt. Technol. 57, 554-555 (1990).

Hariharan, P.

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

Hiller, C.

Huang, P.S.

W. Gao, P.S. Huang, T. Yamada, S. Kiyono, "A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers," Precision Engineering 26, 396-404 (2002).
[CrossRef]

Iwahashi, Y.

Iwata, K.

Kestner, R. N.

Kicker, B.

Kiyono, S.

W. Gao, P.S. Huang, T. Yamada, S. Kiyono, "A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers," Precision Engineering 26, 396-404 (2002).
[CrossRef]

Küchel, M. F.

M. F. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
[CrossRef]

Marioge, J. P.

Marriner, R.

H. Barrell and R. Marriner, "Liquid surface interferometry," Nature (London) 162, 529-530 (1948).
[CrossRef]

Molesini, G.

M. Vannoni and G. Molesini, "Validation of absolute planarity reference plates with a liquid mirror," Metrologia 42, 389-393 (2005).
[CrossRef]

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

V. Greco and G. Molesini, "Micro-temperature effects on absolute flatness test plates," Pure Appl. Opt. 7, 1341-1346 (1998).
[CrossRef]

Mullot, M.

Oehring, H. -A.

Parks, R. E.

Powell, I.

Schulz, G.

Schulz, M.

M. Schulz and C. Elster, "Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution," Opt. Eng. 45, 060503 (2006).
[CrossRef]

Schwider, J.

Shao, L.-Z.

Sharonov, V. N.

V. B. Gubin and V. N. Sharonov, "Absolute calibration of spherical surfaces," Sov. J. Opt. Technol. 57, 554-555 (1990).

Song, D.

J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, "Large-aperture high-accuracy phase-shifting digital flat interferometer," Opt. Eng. 35, 1936-1942 (1996).
[CrossRef]

Sonozaki, S.

Steiner, K.

Trivi, M.

Tronconi, R.

Vannoni, M.

M. Vannoni and G. Molesini, "Validation of absolute planarity reference plates with a liquid mirror," Metrologia 42, 389-393 (2005).
[CrossRef]

Vogel, A.

Wang, Q.

J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, "Large-aperture high-accuracy phase-shifting digital flat interferometer," Opt. Eng. 35, 1936-1942 (1996).
[CrossRef]

Wyant, J. C.

Yamada, T.

W. Gao, P.S. Huang, T. Yamada, S. Kiyono, "A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers," Precision Engineering 26, 396-404 (2002).
[CrossRef]

Zhu, R.

J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, "Large-aperture high-accuracy phase-shifting digital flat interferometer," Opt. Eng. 35, 1936-1942 (1996).
[CrossRef]

Appl. Opt. (17)

R. Brünnagel, H. -A. Oehring, and K. Steiner, "Fizeau interferometer for measuring the flatness of optical surfaces," Appl. Opt. 7, 331-335 (1967).
[CrossRef]

J. P. Marioge, B. Bonino, and M. Mullot, "Standard of flatness: its application to Fabry-Perot interferometers," Appl. Opt. 14, 2283-2285 (1975).
[CrossRef] [PubMed]

K.-E. Elssner, A. Vogel, J. Grzanna, and G. Schulz, "Establishing a flatness standard," Appl. Opt. 33, 2437-2446 (1994).
[CrossRef] [PubMed]

I. Powell and E. Goulet, "Absolute figure measurements with a liquid-flat reference," Appl. Opt. 37, 2579-2588 (1998).
[CrossRef]

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

G. Schulz, "Absolute flatness testing by an extended rotation method using two angles of rotation," Appl. Opt. 32, 1055-1059 (1993).
[CrossRef] [PubMed]

J. Grzanna, "Absolute testing of optical flats at points on a square grid: error propagation," Appl. Opt. 33, 6654-6661 (1994).
[CrossRef] [PubMed]

B. (B). F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, "Calibration of a 300-mm-Aperture Phase-Shifting Fizeau Interferometer," Appl. Opt. 39, 5161-5171 (2000).
[CrossRef]

S. Sonozaki, K. Iwata, and Y. Iwahashi, "Measurement of profiles along a circle on two flat surfaces by use of a Fizeau interferometer with no standard," Appl. Opt. 42, 6853-6858 (2003).
[CrossRef] [PubMed]

C. Ai and J. C. Wyant, "Absolute testing of flats by using even and odd functions," Appl. Opt. 32, 4698-4705 (1993).
[CrossRef] [PubMed]

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

R. E. Parks, L.-Z. 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. Del Vecchio, M. Trivi, and G. Molesini, "Absolute measurement of planarity with Fritz’s method: uncertainty evaluation," Appl. Opt. 38, 2018-2027 (1999).
[CrossRef]

K. R. Freischlad, "Absolute interferometric testing based on reconstruction of rotational shear," Appl. Opt. 40, 1637-1648 (2001).
[CrossRef]

G. Schulz and J. Schwider, "Precise measurement of planeness," Appl. Opt. 6,1077-1084 (1967).
[CrossRef] [PubMed]

G. Schulz, J. Schwider, C. Hiller, and B. Kicker, "Establishing an optical flatness standard," Appl. Opt. 10, 929-934 (1971).
[CrossRef] [PubMed]

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

J. Sci. Instrum. (1)

G. D. Dew, "The measurement of optical flatness," J. Sci. Instrum. 43, 409-415 (1966).
[CrossRef] [PubMed]

Metrologia (1)

M. Vannoni and G. Molesini, "Validation of absolute planarity reference plates with a liquid mirror," Metrologia 42, 389-393 (2005).
[CrossRef]

Nature (London) (2)

Lord Rayleigh, "Interference bands and their application," Nature (London) 48, 212-214 (1893).
[CrossRef]

H. Barrell and R. Marriner, "Liquid surface interferometry," Nature (London) 162, 529-530 (1948).
[CrossRef]

Opt. Commun. (1)

J. Grzanna and G. Schulz, "Absolute testing of flatness standards at square-grid points," Opt. Commun. 77, 107-112 (1990).
[CrossRef]

Opt. Eng. (5)

M. Schulz and C. Elster, "Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution," Opt. Eng. 45, 060503 (2006).
[CrossRef]

J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, "Large-aperture high-accuracy phase-shifting digital flat interferometer," Opt. Eng. 35, 1936-1942 (1996).
[CrossRef]

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

C. J. Evans, "Comment on the paper ‘Interferometric testing of optical surfaces: absolute measurement of flatness," Opt. Eng. 37, 1880-1882 (1998).
[CrossRef]

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 33, 379-383 (1984).

Optik (1)

M. F. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
[CrossRef]

Precision Engineering (1)

W. Gao, P.S. Huang, T. Yamada, S. Kiyono, "A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers," Precision Engineering 26, 396-404 (2002).
[CrossRef]

Pure Appl. Opt. (1)

V. Greco and G. Molesini, "Micro-temperature effects on absolute flatness test plates," Pure Appl. Opt. 7, 1341-1346 (1998).
[CrossRef]

Sov. J. Opt. Technol. (1)

V. B. Gubin and V. N. Sharonov, "Absolute calibration of spherical surfaces," Sov. J. Opt. Technol. 57, 554-555 (1990).

Other (1)

.International Bureau of Weights and Measures, International Electrotechnical Commission, International Federation of Clinical Chemistry, International Organization for Standardization, International Union of Pure and Applied Chemistry, International Union of Pure and Applied Physics, and International Organization of Legal Metrology, Guide to the Expression of Uncertainty in Measurements (International Organization for Standardization, Geneva, 1993).

Supplementary Material (1)

» Media 1: AVI (689 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

(0.69 MB) Movie of the evolution of the iterative algorithm. To the left the original surfaces, at the center the corresponding trial surfaces after 16 iterations, and to the right after 256 iterations. [Media 1]

Fig. 2.
Fig. 2.

Set of three surfaces recovered after the application of the modified Fritz’s approach to archival data (2003).

Fig. 3.
Fig. 3.

Surfaces reconstructed with the iterative algorithm, using the same archival data of Fig. 2.

Fig. 4.
Fig. 4.

High frequency (left) and low frequency residuals (right) of Fritz’s computation on the pair KM.

Fig. 5.
Fig. 5.

High frequency (left) and low frequency residuals (right) of the iterative algorithm on the pair KM as in Fig. 4.

Fig. 6.
Fig. 6.

Standard deviations for the average maps K, L, M for a collection of 40 measured data set.

Equations (18)

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

K F x y = K x y
K F M = K F + M
L F M = L F + M
L F M R = L F + M R
L F K = L F + K
Δ ( K F M ) = ( K F M ) exp K F M
Δ ( L F M ) = ( L F M ) exp L F M
Δ ( L F M R ) = ( L F M R ) exp L F M R
Δ ( L F K ) = ( L F K ) exp L F K
K new = K + 1 2 Δ ( K F M ) F 10 + 1 2 Δ ( L F K ) 10
L new = L + 1 3 Δ ( L F K ) F 10 + 1 3 Δ ( L F M ) F 10 + 1 3 Δ ( L F M R ) F 10
M new = M + 1 3 Δ ( K F M ) 10 + 1 3 Δ ( L F M ) 10 + 1 3 Δ ( L F M R ) -R 10 .
ρ = ( i i 0 ) 2 + ( j j 0 ) 2
θ = tan 1 j j 0 i i 0
ρ = ρ
θ = θ + φ
i = Int ( ρ cos θ ' )
j = Int ( ρ sin θ )

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