Abstract

To determine the absolute flatness deviations of optical surfaces, a novel method using two optical plates to achieve the absolute flatness test is presented. Absolute deviations of three surfaces, the rear surface of plate I and the front and rear surfaces of plate II, are obtained by four measurements. Wavefront error due to the inhomogeneity of plate II is measured beforehand and is then subtracted from the test results. Vertical profiles of the three surfaces are compared with the measurement results obtained by Zygo’s three-flat application. Good agreement validates our method. The advantage of our method is that only one transmission flat is needed during the absolute test, which is especially useful for large-aperture interferometer calibration.

© 2009 Optical Society of America

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  1. Lord Rayleigh, “Interference bands and their application,” Nature 48, 212-214 (1893).
    [CrossRef]
  2. 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]
  3. I. Powell and E. Goulet, “Absolute figure measurements with a liquid-flat reference,” Appl. Opt. 37, 2579-2588 (1998).
    [CrossRef]
  4. M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42, 389-393(2005).
    [CrossRef]
  5. G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077-1084 (1967).
    [CrossRef] [PubMed]
  6. G. Schulz and J. Schwider, “Establishing an optical flatness,” Appl. Opt. 10, 929-934 (1971).
    [CrossRef] [PubMed]
  7. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379-383 (1984).
  8. J. Grzanna and G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107-112 (1990).
    [CrossRef]
  9. 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]
  10. G. Schulz, “Absolute flatness testing by an extended rotation method using two angles of rotation,” Appl. Opt. 32, 1055-1059 (1993).
    [CrossRef] [PubMed]
  11. C.Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698-4705 (1993).
    [CrossRef] [PubMed]
  12. C.Ai, J. C. Wyant, L.-Z. Shao, and R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent 5,502,566 (26 March 1996).
  13. 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]
  14. K.-E. Elssner, A. Vogel, J. Grzanna, and G. Schulz, “Establishing a flatness standard,” Appl. Opt. 33, 2437-2446 (1994).
    [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. C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015-1021 (1996).
    [CrossRef] [PubMed]
  17. V. Greco and G. Molesini, “Micro-temperature effects on absolute flatness test plates,” Pure Appl. Opt. 7, 1341-1346(1998).
    [CrossRef]
  18. 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]
  19. 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]
  20. K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40, 1637-1648(2001).
    [CrossRef]
  21. M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Jena) 112, 381-391 (2001).
  22. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856-5865 (2006).
    [CrossRef] [PubMed]
  23. F. Morin and S. Bouillet, “Absolute interferometric measurement of flatness: application of different methods to test a 600 mm diameter reference flat,” Proc. SPIE 6616, 1-11(2007).
  24. U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 1(2007).
    [CrossRef]
  25. M. Vannoni and G. Molesini, “Absolute planarity with three-flat test: an iterative approach with Zernike polynomials,” Opt. Express 16, 340-354 (2008).
    [CrossRef] [PubMed]
  26. M. Vannoni and G. Molesini, “Three-flat test with plates in horizonatal posture,” Appl. Opt. 47, 2133-2145 (2008).
    [CrossRef] [PubMed]
  27. J. Schwider, R. Burow, K.-E. Elssner, R. Spolaczyk, and J. Grzanna, “Homogeneity testing by phase sampling interferometry,” Appl. Opt. 24, 3059-3061 (1985).
    [CrossRef] [PubMed]
  28. V. B. Gubin and V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147-148 (1990).

2008 (2)

2007 (2)

F. Morin and S. Bouillet, “Absolute interferometric measurement of flatness: application of different methods to test a 600 mm diameter reference flat,” Proc. SPIE 6616, 1-11(2007).

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 1(2007).
[CrossRef]

2006 (1)

2005 (1)

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

2001 (2)

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

M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Jena) 112, 381-391 (2001).

2000 (1)

1999 (1)

1998 (3)

1996 (2)

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]

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

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, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147-148 (1990).

1985 (1)

1984 (1)

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

1971 (1)

1967 (1)

1893 (1)

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

Ai, C.

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.Ai, J. C. Wyant, L.-Z. Shao, and R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent 5,502,566 (26 March 1996).

Bouillet, S.

F. Morin and S. Bouillet, “Absolute interferometric measurement of flatness: application of different methods to test a 600 mm diameter reference flat,” Proc. SPIE 6616, 1-11(2007).

Burow, 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]

Elssner, K.-E.

Evans, C. J.

Fairman, P. S.

Farrant, D. I.

Forbes, G.

Freischlad, K. R.

Fritz, B. S.

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

Goulet, E.

Greco, V.

Griesmann, U.

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 1(2007).
[CrossRef]

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

Grzanna, J.

Gubin, V. B.

V. B. Gubin and V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147-148 (1990).

Kestner, R. N.

Küchel, M. F.

M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Jena) 112, 381-391 (2001).

Molesini, G.

Morin, F.

F. Morin and S. Bouillet, “Absolute interferometric measurement of flatness: application of different methods to test a 600 mm diameter reference flat,” Proc. SPIE 6616, 1-11(2007).

Oreb, B. (B). F.

Parks, R. E.

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]

C.Ai, J. C. Wyant, L.-Z. Shao, and R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent 5,502,566 (26 March 1996).

Powell, I.

Rayleigh, Lord

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

Schulz, G.

Schwider, J.

Shao, L.-Z.

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]

C.Ai, J. C. Wyant, L.-Z. Shao, and R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent 5,502,566 (26 March 1996).

Sharonov, V. N.

V. B. Gubin and V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147-148 (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]

Soons, J.

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 1(2007).
[CrossRef]

Spolaczyk, R.

Trivi, M.

Tronconi, R.

Vannoni, M.

Vecchio, C. Del

Vogel, A.

Walsh, C. J.

Wang, Q.

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 1(2007).
[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]

Wyant, J. C.

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.Ai, J. C. Wyant, L.-Z. Shao, and R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent 5,502,566 (26 March 1996).

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. (16)

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]

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

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

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

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]

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]

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 and J. Schwider, “Establishing an optical flatness,” 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]

M. Vannoni and G. Molesini, “Three-flat test with plates in horizonatal posture,” Appl. Opt. 47, 2133-2145 (2008).
[CrossRef] [PubMed]

J. Schwider, R. Burow, K.-E. Elssner, R. Spolaczyk, and J. Grzanna, “Homogeneity testing by phase sampling interferometry,” Appl. Opt. 24, 3059-3061 (1985).
[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 (1)

Lord Rayleigh, “Interference bands and their application,” Nature 48, 212-214 (1893).
[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. (3)

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 1(2007).
[CrossRef]

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

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]

Opt. Express (1)

Optik (Jena) (1)

M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Jena) 112, 381-391 (2001).

Proc. SPIE (1)

F. Morin and S. Bouillet, “Absolute interferometric measurement of flatness: application of different methods to test a 600 mm diameter reference flat,” Proc. SPIE 6616, 1-11(2007).

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, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147-148 (1990).

Other (1)

C.Ai, J. C. Wyant, L.-Z. Shao, and R. E. Parks, “Method and apparatus for absolute measurement of entire surfaces of flats,” U.S. patent 5,502,566 (26 March 1996).

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Figures (5)

Fig. 1
Fig. 1

Reference plate I and test plate II in a Fizeau interferometer and the Cartesian coordinate system of the interferometer.

Fig. 2
Fig. 2

Four configurations and corresponding measurements, the arrows indicate which two surfaces are interfering. In M 4 , plate II is rotated by Φ.

Fig. 3
Fig. 3

Wavefront error distribution due to inhomogeneity of plate II.

Fig. 4
Fig. 4

Three-dimensional surface figure: (a) surface A, (b) surface B, (c) surface C.

Fig. 5
Fig. 5

Comparison between the two methods of vertical profiles of three surfaces A, B, and C (a) our two-plate method; (b) Zygo three-flat method.

Tables (2)

Tables Icon

Table 1 PV and RMS Values of Vertical Profiles of Three Surfaces with Different Methods

Tables Icon

Table 2 Influence of Different Homogeneity Grades of Glass

Equations (16)

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M ( x , y ) = h reference ( x , y ) + h test ( x , y ) + alignment terms,
M 1 ( x , y ) = A ( x , y ) + B ( x , y ) , M 2 ( x , y ) = A ( x , y ) + ( 1 n ) B ( x , y ) n C ( x , y ) + δ ( x , y ) , M 3 ( x , y ) = A ( x , y ) + C ( x , y ) , M 4 ( x , y ) = A ( x , y ) + C Φ ( x , y ) .
F r ( x , y ) = F r ( x , y ) = F r ( x , y ) = F r ( x , y ) = F r Φ ( x , y ) .
M 1 r ( x , y ) = A r ( x , y ) + B r ( x , y ) , M 2 r ( x , y ) = A r ( x , y ) + ( 1 n ) B r ( x , y ) n C r ( x , y ) + δ r ( x , y ) , M 3 r ( x , y ) = A r ( x , y ) + C r ( x , y ) , M 4 r ( x , y ) = A r ( x , y ) + C r ( x , y ) .
C r = 2 ( 1 n ) M 1 r ( x , y ) 2 M 2 r ( x , y ) + n M 3 r ( x , y ) + n M 4 r ( x , y ) + 2 δ r ( x , y ) 4 n ,
M 3 a ( x , y ) = A a ( x , y ) + C a ( x , y ) , M 4 a ( x , y ) = A a ( x , y ) + C a Φ ( x , y ) .
E ( x , y ) = M 3 a ( x , y ) M 4 a ( x , y ) = C a ( x , y ) C a Φ ( x , y ) .
E ( r , θ ) = n , l R n l ( r ) [ E n l cos ( l θ ) + E n l sin ( l θ ) ] , C a ( r , θ ) = n , l R n l ( r ) [ C n l cos ( l θ ) + C n l sin ( l θ ) ] , C a Φ ( r , θ ) = C a ( r , θ Φ ) = n , l R n l ( r ) [ C n l ¯ cos ( l θ ) + C n l ¯ sin ( l θ ) ] ,
C n l ¯ = C n l cos ( l ϕ ) C n l sin ( l ϕ ) , C n l ¯ = C n l cos ( l ϕ ) + C n l sin ( l ϕ ) .
C n l = 1 2 · [ E n l E n l · sin ( l ϕ ) 1 cos ( l ϕ ) ] , C n l = 1 2 · [ E n l + E n l · sin ( l ϕ ) 1 cos ( l ϕ ) ] .
C ( x , y ) = C r ( x , y ) + C a ( x , y ) .
A ( - x , y ) = M 3 ( x , y ) C ( x , y ) , B ( x , y ) = M 1 ( x , y ) M 3 ( x , y ) + C ( x , y ) .
σ = ε 2 n 2 + 2 ( n 1 ) 2 .
δ C r = ε 2 · [ 4 ( n 1 ) 2 + 2 n 2 + 4 ] + σ 2 / 4 n = ε · 12 ( n 1 ) 2 + 10 n 2 + 4 / 4 n = 0.00078 λ .
δ C ϕ = 1 2 δ ϕ n , l R n l ( r ) [ ( l C n l + l sin l ϕ 1 cos l ϕ C n l ) cos ( l θ ) + ( l sin l ϕ 1 cos l ϕ C n l l C n l ) sin ( l θ ) ] ,
δ C a = δ C ϕ 2 + ε 2 = 0.00091 λ .

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