Abstract

The absolute testing of optical flats through the use of the rotation–shift method is discussed. Absolute deviations of three flats are determined by the evaluation of at least four interference patterns of pairs of these flats. In the rotation–shift method the absolute deviations are calculated at the points of a square grid. The propagation of error in measurements of the interference patterns to the magnitude of the error of the overall result is calculated with respect to special combinations of pairs of the flats. Proposals are made to minimize the error propagation. The mean-square value of the error-propagation factors is in the range of 0.75. Square grids of up to 151 points × 151 points are used.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 8, Chap. 4, pp. 95–167.
  2. G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
    [CrossRef]
  3. G. Schulz, J. Schwider, C. Hiller, B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10, 929–934 (1971).
    [CrossRef] [PubMed]
  4. G. Schulz, J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring-error compensation,” Appl. Opt. 31, 3767–3780 (1992).
    [CrossRef] [PubMed]
  5. J. Grzanna, R. Burow, G. Schulz, A. Vogel, “Absolute Ebenheitsprüfung über einem Quadratgitter,” Beitr. Optik Quantenelektron. (Berlin) 13, 93 (1988).
  6. J. Grzanna, A. Vogel, “Absolute testing of optical flats on a quadratic grid,” in Fringe 1989: Automatic Processing of Fringe Patterns, W. Osten, ed. (Akademie, Berlin, 1989), p. 72.
  7. J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
    [CrossRef]
  8. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
  9. G. D. Dew, “The measurement of optical flatness,” J. Sci. Instrum. 43, 409–415 (1966).
    [CrossRef] [PubMed]
  10. K. G. Birch, M. G. Cox, “Calculation of the flatness of a surface: a least-squares approach,” NPL Rep. MOM (National Physical Laboratory, Teddington, UK, 5December1973), pp. 1–47.
  11. D. A. Ketelsen, D. S. Anderson, “Optical testing with large liquid flats,” in Advances in Fabrication and Metrology for Optics and Large Optics, J. B. Arnold, R. E. Parks, eds., Proc. Soc. Photo-Opt. Instrum. Eng.966, 365–371 (1988).
  12. J. Schwider, “Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen. II,” Opt. Acta 14, 389–400 (1967).
    [CrossRef]

1992 (1)

1990 (1)

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

1988 (1)

J. Grzanna, R. Burow, G. Schulz, A. Vogel, “Absolute Ebenheitsprüfung über einem Quadratgitter,” Beitr. Optik Quantenelektron. (Berlin) 13, 93 (1988).

1984 (1)

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

1971 (1)

1967 (2)

J. Schwider, “Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen. II,” Opt. Acta 14, 389–400 (1967).
[CrossRef]

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

1966 (1)

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

Anderson, D. S.

D. A. Ketelsen, D. S. Anderson, “Optical testing with large liquid flats,” in Advances in Fabrication and Metrology for Optics and Large Optics, J. B. Arnold, R. E. Parks, eds., Proc. Soc. Photo-Opt. Instrum. Eng.966, 365–371 (1988).

Birch, K. G.

K. G. Birch, M. G. Cox, “Calculation of the flatness of a surface: a least-squares approach,” NPL Rep. MOM (National Physical Laboratory, Teddington, UK, 5December1973), pp. 1–47.

Burow, R.

J. Grzanna, R. Burow, G. Schulz, A. Vogel, “Absolute Ebenheitsprüfung über einem Quadratgitter,” Beitr. Optik Quantenelektron. (Berlin) 13, 93 (1988).

Cox, M. G.

K. G. Birch, M. G. Cox, “Calculation of the flatness of a surface: a least-squares approach,” NPL Rep. MOM (National Physical Laboratory, Teddington, UK, 5December1973), pp. 1–47.

Dew, G. D.

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

Fritz, B. S.

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

Grzanna, J.

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

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

J. Grzanna, R. Burow, G. Schulz, A. Vogel, “Absolute Ebenheitsprüfung über einem Quadratgitter,” Beitr. Optik Quantenelektron. (Berlin) 13, 93 (1988).

J. Grzanna, A. Vogel, “Absolute testing of optical flats on a quadratic grid,” in Fringe 1989: Automatic Processing of Fringe Patterns, W. Osten, ed. (Akademie, Berlin, 1989), p. 72.

Hiller, C.

Ketelsen, D. A.

D. A. Ketelsen, D. S. Anderson, “Optical testing with large liquid flats,” in Advances in Fabrication and Metrology for Optics and Large Optics, J. B. Arnold, R. E. Parks, eds., Proc. Soc. Photo-Opt. Instrum. Eng.966, 365–371 (1988).

Kicker, B.

Schulz, G.

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

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

J. Grzanna, R. Burow, G. Schulz, A. Vogel, “Absolute Ebenheitsprüfung über einem Quadratgitter,” Beitr. Optik Quantenelektron. (Berlin) 13, 93 (1988).

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

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 8, Chap. 4, pp. 95–167.

Schwider, J.

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

J. Schwider, “Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen. II,” Opt. Acta 14, 389–400 (1967).
[CrossRef]

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 8, Chap. 4, pp. 95–167.

Vogel, A.

J. Grzanna, R. Burow, G. Schulz, A. Vogel, “Absolute Ebenheitsprüfung über einem Quadratgitter,” Beitr. Optik Quantenelektron. (Berlin) 13, 93 (1988).

J. Grzanna, A. Vogel, “Absolute testing of optical flats on a quadratic grid,” in Fringe 1989: Automatic Processing of Fringe Patterns, W. Osten, ed. (Akademie, Berlin, 1989), p. 72.

Appl. Opt. (2)

Beitr. Optik Quantenelektron. (Berlin) (1)

J. Grzanna, R. Burow, G. Schulz, A. Vogel, “Absolute Ebenheitsprüfung über einem Quadratgitter,” Beitr. Optik Quantenelektron. (Berlin) 13, 93 (1988).

J. Sci. Instrum. (1)

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

Opt. Acta (2)

J. Schwider, “Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen. II,” Opt. Acta 14, 389–400 (1967).
[CrossRef]

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

Opt. Commun. (1)

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

Opt. Eng. (1)

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

Other (4)

J. Grzanna, A. Vogel, “Absolute testing of optical flats on a quadratic grid,” in Fringe 1989: Automatic Processing of Fringe Patterns, W. Osten, ed. (Akademie, Berlin, 1989), p. 72.

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 8, Chap. 4, pp. 95–167.

K. G. Birch, M. G. Cox, “Calculation of the flatness of a surface: a least-squares approach,” NPL Rep. MOM (National Physical Laboratory, Teddington, UK, 5December1973), pp. 1–47.

D. A. Ketelsen, D. S. Anderson, “Optical testing with large liquid flats,” in Advances in Fabrication and Metrology for Optics and Large Optics, J. B. Arnold, R. E. Parks, eds., Proc. Soc. Photo-Opt. Instrum. Eng.966, 365–371 (1988).

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 (5)

Fig. 1
Fig. 1

Three flats, A, B, and C, and their coordinate systems.

Fig. 2
Fig. 2

Five combinations of flats A, B, and C: Three basic combinations, a rotation–shift combination, and a long shift combination.

Fig. 3
Fig. 3

Side view of the first combination of flats A and B. RA and RB are the reference planes for A and B, respectively.

Fig. 4
Fig. 4

Error-propagation factors of flats A, B, and C generated by the combination (61, 30.9, y, −1, x, −1).

Fig. 5
Fig. 5

Optimal error-propagation factors of flats A, B, and C generated by the combination (61, 30.3, x, 1, x, 4), where n = 61 is kept constant.

Tables (5)

Tables Icon

Table 1 Methods for Absolute Testing of Optical Flats

Tables Icon

Table 2 Selected Combinations of Flats for Minimal Error Propagation

Tables Icon

Table 3 Influence of the Aperture Radius r on Error Propagationa

Tables Icon

Table 4 Influence of v2 of the Long Shift on Error Propagationa

Tables Icon

Table 5 Influence of n on the Error Propagationa

Equations (38)

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

A ( x , y ) + B ( x , y ) W 1 ( x , y ) = D 1 ( x , y ) , C ( x , y ) + A ( x , y ) W 2 ( x , y ) = D 2 ( x , y ) C ( x , y ) + B ( x , y ) W 3 ( x , y ) = D 3 ( x , y ) .
x = 0 , + 1 , 1 , + 2 , 2 , , y = 0 , + 1 , 1 , + 2 , 2 , ,
A ( x , y ) + B ( y + 1 , x ) W 4 ( x , y ) = D 4 ( x , y ) , C ( x , y ) + A ( y + υ , y ) W 5 ( x , y ) = D 5 ( x , y ) .
i = 1 m x i A ( x i , y i ) = 0 , i = 1 m y i A ( x i , y i ) = 0 , i = 1 m A ( x i , y i ) = 0 ; i = 1 m x i B ( x i , y i ) = 0 , i = 1 m y i B ( x i , y i ) = 0 , i = 1 m B ( x i , y i ) = 0 ; i = 1 m x i C ( x i , y i ) = 0 , i = 1 m y i C ( x i , y i ) = 0 , i = 1 m C ( x i , y i ) = 0 .
[ I R 0 W 0 0 0 0 R 0 I 0 W 0 0 0 0 R I 0 0 W 0 0 I 4 S 0 0 0 0 W 4 0 U 0 I 5 0 0 0 0 W 5 W T 0 0 0 0 0 0 0 0 W T 0 0 0 0 0 0 0 0 W T 0 0 0 0 0 ] × [ a b c w 1 w 2 w 3 w 4 w 5 ] = l . s . [ d 1 d 2 d 3 d 4 d 5 0 0 0 ] ,
W = [ x 1 y 1 1 x 2 y 2 1 x m y m 1 ] .
G x = l . s . d .
( G T G ) x = G T d .
[ I R 0 R 0 I 0 R I I 4 S 0 U 0 I 5 ] [ a b c ] = l . s . [ f 1 f 2 f 3 f 4 f 5 ] H x = l . s . f ,
( H T H ) x = H T f .
D ¯ 1 ( x , y ) + D ¯ 2 ( x , y ) D ¯ 3 ( x , y ) D ¯ 1 ( x , y ) D ¯ 2 ( x , y ) + D 3 ( x , y ) = 0 ,
x = 0 , + 1 , + 2 , , y = 0 , + 1 , 1 , + 2 , 2 , .
D i ¯ ( x , y ) = D i ( x , y ) + W i ( x , y ) , i = 1 , 2 , 3 .
F ( a i , b i , c i ) = x = 0 , + 1 , 1 , + 2 , 2 , , y = 0 , + 1 , 1 , + 2 , 2 , , x 2 + y 2 r 2 , [ D i ( x , y ) a i x b i y c i ] 2 = min ,
p = diag [ ( H T H ) 1 ] = ( p 1 , p 2 , , p 3 m ) ,
rms A = ( i = 1 m p i 2 ) / m ,
max A = max 1 i m p i , min A = min 1 i m p i .
rms B = ( i = m + 1 2 m p i 2 ) / m , max B = max m + 1 i 2 m p i , min B = min m + 1 i 2 m p i , rms C = ( i = 2 m + 1 3 m p i 2 ) / m , max C = max 2 m + 1 i 3 m p i , min C = min 2 m + 1 i 3 m p i .
A B rev , C A rev , C B rev , A B rev , rot , sh ( u 1 , υ 1 ) , C A rev , sh ( u 2 , υ 2 ) .
h r < h + 1 ,
h = ( n 1 ) / 2
( n , r , u 1 , υ 1 , u 2 , υ 2 ) .
( n , r , x , 1 , x , 4 ) .
min A = 0 . 62 , min B = 0 . 68 , min C = 0 . 68 .
u 1 = u 2 = x , υ 1 = 1 , υ 2 = 4 ,
h 2 + h 0 2 r < h 2 + ( h 0 + 1 ) 2 ,
W w i = l . s . d i , 1 i 3 ,
w i = ( W T W ) 1 W T d i , 1 i 3 .
f i = d i W w i , 1 i 3 .
F ( z ) = [ H x ( z ) g ( z ) , H x ( z ) g ( z ) ] ,
g ( z ) = [ f 1 f 2 f 3 d 4 d 5 ] + [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 W 4 0 0 0 0 0 W 5 ] [ 0 0 0 w 4 w 5 ]
F ( z * ) = min ,
Q ( z ) = i = 1 6 k i z i 2 + z 1 i = 2 6 k 5 + i z i + z 2 i = 3 6 k 9 + i z i + z 3 i = 4 6 k 12 + i z i + z 4 i = 5 6 k 14 + i + k 21 z 5 z 6 + i = 1 6 k 21 + i z i + k 28 = v ( z ) k ,
Q ( z ) = v ( z ) k = F ( z )
[ v ( z 1 0 ) v ( z 2 0 ) v ( z 28 0 ) ] [ k 1 k 2 k 28 ] = [ F ( z 1 0 ) F ( z 2 0 ) F ( z 28 0 ) ] ,
( 2 Q / z 2 ) z = Q / z .
z * = ( z 1 * , z 2 * , , z 6 * ) , w 4 = [ z 1 * z 2 * z 3 * ] , w 5 = [ z 4 * z 5 * z 6 * ] ,
f i = d i W w i , i = 4 , i = 5 .

Metrics