Abstract

A further-developed three-plate rotation method [Appl. Opt. 31, 3767 (1992)] for the absolute testing of flats is used as a measuring method for establishing flatness standards. A special phase-stepping Fizeau interferometer was built that permits flats of diameters up to 200 mm to be tested. First results show a mean error between 0.002 and 0.003 λ in the determination of the absolute flatness deviations (λ = 632.8 nm). These deviations are obtained at 500–600 points on each plate. A number of experimental conditions connected with the thermal and mechanical stability of the plates must be fulfilled.

© 1994 Optical Society of America

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References

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  1. G. Schulz, J. Schwider, C. Hiller, B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10, 929–934 (1971).
    [CrossRef] [PubMed]
  2. G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
    [CrossRef]
  3. G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 93–167.
    [CrossRef]
  4. P. Hariharan, “Interferometry with lasers,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1987), Vol. 24, pp. 103–164.
    [CrossRef]
  5. K. Creath, “Phase-measurement interferometric techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. 26, pp. 348–393.
    [CrossRef]
  6. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 271–359.
    [CrossRef]
  7. 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).
  8. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
  9. K. Küchel, “Absolute measurement of flat mirrors in the Ritchey–Common test,” in Optical Fabrication and Testing, R. E. Parks, ed., OSA 1986 Technical Digest Series (Optical Society of America, Washington, D.C., 1986), pp. 114–119.
  10. A. E. Jensen, “Absolute calibration method for laser Twyman–Green wave-front testing interferometers,” J. Opt. Soc. Am. 63, 1313 (A) (1973).
  11. Ref. 3, p. 136.
  12. J. Schwider, “Fizeau- and Michelson-type interferograms and their relation to the absolute testing of optical surfaces,” Optik (Stuttgart) 89, 113–117 (1992).
  13. J. Schwider, K. E. Elssner, R. Spolaczyk, J. Grzanna, A. Bode, “Verfahren zur Messung von Flächenabweichungen eines Planflächennormals von einer Ebene mittels eines Twyman–Green interferometers,” German patentG01B 254, 774 (1December1986).
  14. J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
    [CrossRef]
  15. K. E. Elssner, J. Grzanna, A. Vogel, “Verfahren zur absoluten Planflächenprüfung,” German patentG01B 9/02 295, 412 (19June1990).
  16. G. Schulz, J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31, 3767–3780 (1992).
    [CrossRef] [PubMed]
  17. Recently a paper on an extended rotation method using five positional combinations appeared: G. Schulz, “Absolute flatness testing by an extended rotation method using two angles of rotation,” Appl. Opt. 32, 1055–1059 (1993). This extension is not considered here.
    [CrossRef] [PubMed]
  18. Mean measuring errors are calculated from the measured values according to Ref. 16. This calculation is carried out by taking advantage of the fact that the measured values have to fulfill certain conditions, because the system of equations for the determination of the unknown flatness deviations is overdetermined (see Ref. 2). From the (more or less good) fulfillment of those conditions by the measured values, the mean measuring error can be determined quantitatively.
  19. A. F. Slomba, J. W. Figoski, “A coaxial interferometer with low mapping distortion,” in Advances in Optical Metrology I, N. Balasubramanian, J. C. Wyant, eds., Proc. Soc. Photo-Opt. Instrum. Eng.153, 156–161 (1978).
  20. K. E. Elssner, S. Wallburg, “Abbildungim Twyman–Green interferometer,” Beitr. Optik Quantenelektronik 7, 142–143 (1982).
  21. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  22. S. A. Rodionov, P. Agurok, “Effect of errors in the optical system of an interferometer on the accuracy of measuring the shape of a surface,” Opt. Mekh. Prom. 55, 3–5 (1988).
  23. J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
    [CrossRef] [PubMed]
  24. G. W. Dew, “The measurement of optical flatness,” J. Sci. Instrum. 43, 409–415 (1966).
    [CrossRef] [PubMed]
  25. A. F. Slomba, “Application of lasers to measure the optical thermal and mechanical properties of materials,” in Practical Applications of Low Power Lasers, J. S. Chivian, D. Eden, eds., Proc. Soc. Photo-Opt. Instrum. Eng.92, 32–40 (1976).
  26. H. Beuchel, B. Schuhmacher, R. Hoffmann, V. Eberhardt, “Einfluss der Lagefixierung auf die Geometrie der optisch wirksamen Flächen von grossen Spiegelbauelementen,” Feingerätetechnik 34, 54–55 (1985).
  27. P. Hariharan, “Digital phase-stepping interferometry: effects of multiple reflected beams,” Appl. Opt. 26, 2506–2507 (1987).
    [CrossRef] [PubMed]
  28. This correspondence is based on the following. According to Section 2, steps 3 and 5, an interpolation is carried out from the square grid into the polar grid and then back into the square grid. For optimal interpolations, the point spacings of the two grids should be matched. Therefore we have chosen the radial point spacing in the polar grid equal to the point spacing in the square grid, and we have adapted the latter spacing and the rotation angle to each other in such a way that the mean-square value of the azimuthal point spacing in the polar grid is of the order of the point spacing in the square grid. This results (for a rotation angle of 84° with 2N = 60 points on each circle of the polar grid) in a square grid of approximately 30 × 30 points.
  29. The factor 1.4 is the factor of the total error propagation from the measurements to the results, determined theoretically and by computer simulations in Ref. 16 in a number of steps. That factor can be taken from Fig. 11 of Ref. 16 (step IV of that Fig. 11 is also confirmed by Fig. 7 of Ref. 16).

1993 (1)

1992 (2)

J. Schwider, “Fizeau- and Michelson-type interferograms and their relation to the absolute testing of optical surfaces,” Optik (Stuttgart) 89, 113–117 (1992).

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

1990 (1)

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

1989 (1)

1988 (1)

S. A. Rodionov, P. Agurok, “Effect of errors in the optical system of an interferometer on the accuracy of measuring the shape of a surface,” Opt. Mekh. Prom. 55, 3–5 (1988).

1987 (1)

1985 (1)

H. Beuchel, B. Schuhmacher, R. Hoffmann, V. Eberhardt, “Einfluss der Lagefixierung auf die Geometrie der optisch wirksamen Flächen von grossen Spiegelbauelementen,” Feingerätetechnik 34, 54–55 (1985).

1984 (1)

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

1983 (1)

1982 (1)

K. E. Elssner, S. Wallburg, “Abbildungim Twyman–Green interferometer,” Beitr. Optik Quantenelektronik 7, 142–143 (1982).

1973 (1)

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

1971 (1)

1967 (1)

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

1966 (1)

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

Agurok, P.

S. A. Rodionov, P. Agurok, “Effect of errors in the optical system of an interferometer on the accuracy of measuring the shape of a surface,” Opt. Mekh. Prom. 55, 3–5 (1988).

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).

Beuchel, H.

H. Beuchel, B. Schuhmacher, R. Hoffmann, V. Eberhardt, “Einfluss der Lagefixierung auf die Geometrie der optisch wirksamen Flächen von grossen Spiegelbauelementen,” Feingerätetechnik 34, 54–55 (1985).

Bode, A.

J. Schwider, K. E. Elssner, R. Spolaczyk, J. Grzanna, A. Bode, “Verfahren zur Messung von Flächenabweichungen eines Planflächennormals von einer Ebene mittels eines Twyman–Green interferometers,” German patentG01B 254, 774 (1December1986).

Burow, R.

Creath, K.

K. Creath, “Phase-measurement interferometric techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. 26, pp. 348–393.
[CrossRef]

Dew, G. W.

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

Eberhardt, V.

H. Beuchel, B. Schuhmacher, R. Hoffmann, V. Eberhardt, “Einfluss der Lagefixierung auf die Geometrie der optisch wirksamen Flächen von grossen Spiegelbauelementen,” Feingerätetechnik 34, 54–55 (1985).

Elssner, K. E.

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

K. E. Elssner, S. Wallburg, “Abbildungim Twyman–Green interferometer,” Beitr. Optik Quantenelektronik 7, 142–143 (1982).

K. E. Elssner, J. Grzanna, A. Vogel, “Verfahren zur absoluten Planflächenprüfung,” German patentG01B 9/02 295, 412 (19June1990).

J. Schwider, K. E. Elssner, R. Spolaczyk, J. Grzanna, A. Bode, “Verfahren zur Messung von Flächenabweichungen eines Planflächennormals von einer Ebene mittels eines Twyman–Green interferometers,” German patentG01B 254, 774 (1December1986).

Figoski, J. W.

A. F. Slomba, J. W. Figoski, “A coaxial interferometer with low mapping distortion,” in Advances in Optical Metrology I, N. Balasubramanian, J. C. Wyant, eds., Proc. Soc. Photo-Opt. Instrum. Eng.153, 156–161 (1978).

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. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

K. E. Elssner, J. Grzanna, A. Vogel, “Verfahren zur absoluten Planflächenprüfung,” German patentG01B 9/02 295, 412 (19June1990).

J. Schwider, K. E. Elssner, R. Spolaczyk, J. Grzanna, A. Bode, “Verfahren zur Messung von Flächenabweichungen eines Planflächennormals von einer Ebene mittels eines Twyman–Green interferometers,” German patentG01B 254, 774 (1December1986).

Hariharan, P.

P. Hariharan, “Digital phase-stepping interferometry: effects of multiple reflected beams,” Appl. Opt. 26, 2506–2507 (1987).
[CrossRef] [PubMed]

P. Hariharan, “Interferometry with lasers,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1987), Vol. 24, pp. 103–164.
[CrossRef]

Hiller, C.

Hoffmann, R.

H. Beuchel, B. Schuhmacher, R. Hoffmann, V. Eberhardt, “Einfluss der Lagefixierung auf die Geometrie der optisch wirksamen Flächen von grossen Spiegelbauelementen,” Feingerätetechnik 34, 54–55 (1985).

Jensen, A. E.

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

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.

Küchel, K.

K. Küchel, “Absolute measurement of flat mirrors in the Ritchey–Common test,” in Optical Fabrication and Testing, R. E. Parks, ed., OSA 1986 Technical Digest Series (Optical Society of America, Washington, D.C., 1986), pp. 114–119.

Merkel, K.

Rodionov, S. A.

S. A. Rodionov, P. Agurok, “Effect of errors in the optical system of an interferometer on the accuracy of measuring the shape of a surface,” Opt. Mekh. Prom. 55, 3–5 (1988).

Schuhmacher, B.

H. Beuchel, B. Schuhmacher, R. Hoffmann, V. Eberhardt, “Einfluss der Lagefixierung auf die Geometrie der optisch wirksamen Flächen von grossen Spiegelbauelementen,” Feingerätetechnik 34, 54–55 (1985).

Schulz, G.

Recently a paper on an extended rotation method using five positional combinations appeared: G. Schulz, “Absolute flatness testing by an extended rotation method using two angles of rotation,” Appl. Opt. 32, 1055–1059 (1993). This extension is not considered here.
[CrossRef] [PubMed]

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]

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. 13, pp. 93–167.
[CrossRef]

Schwider, J.

J. Schwider, “Fizeau- and Michelson-type interferograms and their relation to the absolute testing of optical surfaces,” Optik (Stuttgart) 89, 113–117 (1992).

J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
[CrossRef] [PubMed]

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

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

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 271–359.
[CrossRef]

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 93–167.
[CrossRef]

J. Schwider, K. E. Elssner, R. Spolaczyk, J. Grzanna, A. Bode, “Verfahren zur Messung von Flächenabweichungen eines Planflächennormals von einer Ebene mittels eines Twyman–Green interferometers,” German patentG01B 254, 774 (1December1986).

Slomba, A. F.

A. F. Slomba, J. W. Figoski, “A coaxial interferometer with low mapping distortion,” in Advances in Optical Metrology I, N. Balasubramanian, J. C. Wyant, eds., Proc. Soc. Photo-Opt. Instrum. Eng.153, 156–161 (1978).

A. F. Slomba, “Application of lasers to measure the optical thermal and mechanical properties of materials,” in Practical Applications of Low Power Lasers, J. S. Chivian, D. Eden, eds., Proc. Soc. Photo-Opt. Instrum. Eng.92, 32–40 (1976).

Spolaczyk, R.

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

J. Schwider, K. E. Elssner, R. Spolaczyk, J. Grzanna, A. Bode, “Verfahren zur Messung von Flächenabweichungen eines Planflächennormals von einer Ebene mittels eines Twyman–Green interferometers,” German patentG01B 254, 774 (1December1986).

Vogel, A.

K. E. Elssner, J. Grzanna, A. Vogel, “Verfahren zur absoluten Planflächenprüfung,” German patentG01B 9/02 295, 412 (19June1990).

Wallburg, S.

K. E. Elssner, S. Wallburg, “Abbildungim Twyman–Green interferometer,” Beitr. Optik Quantenelektronik 7, 142–143 (1982).

Appl. Opt. (6)

Beitr. Optik Quantenelektronik (1)

K. E. Elssner, S. Wallburg, “Abbildungim Twyman–Green interferometer,” Beitr. Optik Quantenelektronik 7, 142–143 (1982).

Feingerätetechnik (1)

H. Beuchel, B. Schuhmacher, R. Hoffmann, V. Eberhardt, “Einfluss der Lagefixierung auf die Geometrie der optisch wirksamen Flächen von grossen Spiegelbauelementen,” Feingerätetechnik 34, 54–55 (1985).

J. Opt. Soc. Am. (1)

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

J. Sci. Instrum. (1)

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

Opt. Acta (1)

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).

Opt. Mekh. Prom. (1)

S. A. Rodionov, P. Agurok, “Effect of errors in the optical system of an interferometer on the accuracy of measuring the shape of a surface,” Opt. Mekh. Prom. 55, 3–5 (1988).

Optik (Stuttgart) (1)

J. Schwider, “Fizeau- and Michelson-type interferograms and their relation to the absolute testing of optical surfaces,” Optik (Stuttgart) 89, 113–117 (1992).

Other (14)

J. Schwider, K. E. Elssner, R. Spolaczyk, J. Grzanna, A. Bode, “Verfahren zur Messung von Flächenabweichungen eines Planflächennormals von einer Ebene mittels eines Twyman–Green interferometers,” German patentG01B 254, 774 (1December1986).

K. E. Elssner, J. Grzanna, A. Vogel, “Verfahren zur absoluten Planflächenprüfung,” German patentG01B 9/02 295, 412 (19June1990).

Mean measuring errors are calculated from the measured values according to Ref. 16. This calculation is carried out by taking advantage of the fact that the measured values have to fulfill certain conditions, because the system of equations for the determination of the unknown flatness deviations is overdetermined (see Ref. 2). From the (more or less good) fulfillment of those conditions by the measured values, the mean measuring error can be determined quantitatively.

A. F. Slomba, J. W. Figoski, “A coaxial interferometer with low mapping distortion,” in Advances in Optical Metrology I, N. Balasubramanian, J. C. Wyant, eds., Proc. Soc. Photo-Opt. Instrum. Eng.153, 156–161 (1978).

K. Küchel, “Absolute measurement of flat mirrors in the Ritchey–Common test,” in Optical Fabrication and Testing, R. E. Parks, ed., OSA 1986 Technical Digest Series (Optical Society of America, Washington, D.C., 1986), pp. 114–119.

Ref. 3, p. 136.

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 93–167.
[CrossRef]

P. Hariharan, “Interferometry with lasers,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1987), Vol. 24, pp. 103–164.
[CrossRef]

K. Creath, “Phase-measurement interferometric techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. 26, pp. 348–393.
[CrossRef]

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 271–359.
[CrossRef]

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).

A. F. Slomba, “Application of lasers to measure the optical thermal and mechanical properties of materials,” in Practical Applications of Low Power Lasers, J. S. Chivian, D. Eden, eds., Proc. Soc. Photo-Opt. Instrum. Eng.92, 32–40 (1976).

This correspondence is based on the following. According to Section 2, steps 3 and 5, an interpolation is carried out from the square grid into the polar grid and then back into the square grid. For optimal interpolations, the point spacings of the two grids should be matched. Therefore we have chosen the radial point spacing in the polar grid equal to the point spacing in the square grid, and we have adapted the latter spacing and the rotation angle to each other in such a way that the mean-square value of the azimuthal point spacing in the polar grid is of the order of the point spacing in the square grid. This results (for a rotation angle of 84° with 2N = 60 points on each circle of the polar grid) in a square grid of approximately 30 × 30 points.

The factor 1.4 is the factor of the total error propagation from the measurements to the results, determined theoretically and by computer simulations in Ref. 16 in a number of steps. That factor can be taken from Fig. 11 of Ref. 16 (step IV of that Fig. 11 is also confirmed by Fig. 7 of Ref. 16).

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Figures (12)

Fig. 1
Fig. 1

One of the three flats (A, B, or C) in top view (above) and side view (below). The polar coordinate system (ρ, ϑ) is appropriate here. The unknown deviations of the optical surface from the reference plane (dashed line) are to be determined.

Fig. 2
Fig. 2

Combination AB in side view. The surface distance dν is known by the interference measurement, i.e., it is known apart from an additive constant, which is not of interest. Dν is the distance between the reference planes. The unknown flatness deviations xν and yν are to be determined. For each value of ν (Fig. 1) the azimuth ϑ= νΦ/2 of flat A coincides with the azimuth ϑ = −νΦ/2 of flat B if the coordinate systems are oriented correspondingly; dν, Dν, xν, and yν are functions of the position.

Fig. 3
Fig. 3

Top view of the four positional combinations of the rotation method (first described in Fig. 3 of Ref. 2). In the fourth combination (ABΦ), flat B has been rotated from its position in the first combination (AB) by the angle Φ. The coordinate systems of all three surfaces are oriented so that in the first three combinations, which are the basic combinations, the azimuths ϑ = 0 of both flats coincide (coincidence at ν = 0). Then in the fourth combination, which is the rotational combination, the azimuths ϑ = Φ/2 coincide (coincidence at ν = 1).

Fig. 4
Fig. 4

Optical scheme of the Fizeau interferometer: Mo, microobjective; Mc, microcomputer.

Fig. 5
Fig. 5

Top view of the interferometer, thermal insulation housing removed.

Fig. 6
Fig. 6

Front view of the interferometer onto plate I, surrounded by mechanical elements for plate support and adjustment.

Fig. 7
Fig. 7

Support system of the plate. The plate is held in its neutral layer on short edges.

Fig. 8
Fig. 8

Three-dimensional plots and contour lines of the absolute flatness deviations of the three 200-mm-diameter, 35-mm-thick plates: (a) Zerodur, (b) fused silica, (c) BK-7 glass. PV, peak-to-valley deviation (shown by vertical lines on the left-hand side of the plots); CD, difference of level between contour lines; D, diameter (mm); R, radius of curvature of the surface. At some places the contour lines show peculiar patterns, e.g., squares, triangles, or short line portions, indicating surface parts with small gradients. This is a consequence of the contour-line representation software used.

Fig. 9
Fig. 9

Independent determinations of the Zerodur plate of Fig. 8(a): (a) is identical to Fig. 8(a); the measurements for (b) were carried out three months after those for (a), and those for (c), 2 h after those for (b). PV, CD, D, and R are defined as in Fig. 8.

Fig. 10
Fig. 10

Difference between (a) Figs. 9(a) and 9(c), (b) Figs. 9(b) and 9(c). PV, CD, D, and R are defined as in Fig. 8.

Fig. 11
Fig. 11

Determination of the Zerodur plate in a three-plate set with the fused-silica plate of Fig. 8(b) but with another BK-7 plate, different from that of Fig. 8(c). PV, CD, D, and R are defined in Fig. 8.

Fig. 12
Fig. 12

Difference between Figs. 11 and 9(a). PV, CD, D, and R are defined as in Fig. 8.

Tables (1)

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Table 1 Maximum Allowable Tolerances of Section 3 Conditions for M = 7, N = 30, and 256 × 256 Pixels

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