Abstract

We demonstrate a method of performing the absolute three-flat test by using reflection symmetries of the surfaces and an algorithm for generating the rotation of arrays of pixel data. Most of the operations involve left/right and top/bottom flips of data arrays, operations that are very fast on most frame grabbers and are available on most commercial phase-measuring interferometers. We demonstrate the method with simulated data as well as with actual data from 150-mm-diameter surfaces that are flat to less than 25 nm peak to valley.

© 1998 Optical Society of America

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  1. F. H. Rolt, Gauges and Fine Measurement (Macmillan, London, 1928).
  2. G. Schulz, J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077–1084 (1967).
    [CrossRef] [PubMed]
  3. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
    [CrossRef]
  4. C. Ai, J. C. Wyant, “Absolute testing of flats decomposed into even and odd functions,” in Interferometry Surface Characterization and Testing, K. Creath, E. Grievenkamp, eds., Proc. SPIE1776, 73–83 (1992).
    [CrossRef]
  5. C. J. Evans, R. E. Parks, “Absolute calibration of spherical surfaces,” in Optical Fabrication and Testing, Vol. 13 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 185–187.
  6. Lord Raleigh, “Interference bands and their application,” Nature (London) 48, 212–214 (1893);“The interferometer,” Nature (London) 59, 533 (1899).
    [CrossRef]
  7. R. Bunnagel, H. Oehring, K. Steiner, “Fizeau interferometer for measuring the flatness of optical surfaces,” Appl. Opt. 7, 331–335 (1968).
    [CrossRef] [PubMed]
  8. H. Barrell, R. Marriner, “Liquid surface interferometry,” Nature (London) 162, 529–530 (1948).
    [CrossRef]
  9. G. D. Dew, “The measurement of optical flatness,” J. Sci. Instrum. 43, 409–415 (1966).
    [CrossRef] [PubMed]
  10. R. W. Porter, Amateur Telescope Making (Scientific American, New York, 1933), Vol. 3, p. 234.
  11. W. B. Emerson, “Determination of planeness and bending of optical flats,” J. Res. Natl. Bur. Stand. 49, 241 (1952).
    [CrossRef]
  12. J. B. Saunders, Precision Measurements, A. C. S. Van Heel, ed. (North-Holland, Amsterdam, 1967), pp. 8–11.
  13. B. E. Truax, “Absolute calibration method for laser Twyman–Green wavefront testing interferometers,” J. Opt. Soc. Am. 63, 1313–1316 (1973).
  14. K-E. Elssner, R. Burow, J. Grzanna, R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649–4661 (1989).
    [CrossRef] [PubMed]
  15. R. Cobleigh, Handy Farm Devices and How to Make Them (1909; reprinted by Lyons and Burford, New York, 1996).
  16. L. Tschirf, “Pruefung von Abricht-Linealen,” Arch. Technisch. Messen 8224-1 (1941).
  17. C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation and ‘absolute testing,’” CIRP Ann. 45/2, 617–634 (1996).
    [CrossRef]
  18. G. Schulz, J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31, 3767–3780 (1992).
    [CrossRef] [PubMed]
  19. See, for example, J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Workshop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.
  20. C. Ai, J. C. Wyant, “Absolute testing of flats using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993).
    [CrossRef] [PubMed]
  21. C. Ai, J. C. Wyant, “Modified three-flat method using even and odd functions,” in Optical Fabrication and Testing, Vol. 24 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), 71–74.
  22. C. J. Evans, R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996).
    [CrossRef] [PubMed]
  23. R. E. Parks, “Removal of test optics errors,” in Advances in Optical Metrology, N. Balasubramanian, J. C. Wyant, eds., Proc. SPIE153, 56–63 (1978).
    [CrossRef]
  24. R. E. Spero, S. E. Whitcomb, “The Laser Interferometer Gravitational-wave Observatory (LIGO),” Opt. Photon. News35–39 (July1995).
  25. W. T. Estler, C. J. Evans, L. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
    [CrossRef]

1997 (1)

W. T. Estler, C. J. Evans, L. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
[CrossRef]

1996 (2)

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

C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation and ‘absolute testing,’” CIRP Ann. 45/2, 617–634 (1996).
[CrossRef]

1995 (1)

R. E. Spero, S. E. Whitcomb, “The Laser Interferometer Gravitational-wave Observatory (LIGO),” Opt. Photon. News35–39 (July1995).

1993 (1)

1992 (1)

1989 (1)

1984 (1)

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

1973 (1)

B. E. Truax, “Absolute calibration method for laser Twyman–Green wavefront testing interferometers,” J. Opt. Soc. Am. 63, 1313–1316 (1973).

1968 (1)

1967 (1)

1966 (1)

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

1952 (1)

W. B. Emerson, “Determination of planeness and bending of optical flats,” J. Res. Natl. Bur. Stand. 49, 241 (1952).
[CrossRef]

1948 (1)

H. Barrell, R. Marriner, “Liquid surface interferometry,” Nature (London) 162, 529–530 (1948).
[CrossRef]

1941 (1)

L. Tschirf, “Pruefung von Abricht-Linealen,” Arch. Technisch. Messen 8224-1 (1941).

1893 (1)

Lord Raleigh, “Interference bands and their application,” Nature (London) 48, 212–214 (1893);“The interferometer,” Nature (London) 59, 533 (1899).
[CrossRef]

Ai, C.

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

C. Ai, J. C. Wyant, “Modified three-flat method using even and odd functions,” in Optical Fabrication and Testing, Vol. 24 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), 71–74.

C. Ai, J. C. Wyant, “Absolute testing of flats decomposed into even and odd functions,” in Interferometry Surface Characterization and Testing, K. Creath, E. Grievenkamp, eds., Proc. SPIE1776, 73–83 (1992).
[CrossRef]

Barrell, H.

H. Barrell, R. Marriner, “Liquid surface interferometry,” Nature (London) 162, 529–530 (1948).
[CrossRef]

Bruning, J. H.

See, for example, J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Workshop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

Bunnagel, R.

Burow, R.

Cobleigh, R.

R. Cobleigh, Handy Farm Devices and How to Make Them (1909; reprinted by Lyons and Burford, New York, 1996).

Dew, G. D.

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

Elssner, K-E.

Emerson, W. B.

W. B. Emerson, “Determination of planeness and bending of optical flats,” J. Res. Natl. Bur. Stand. 49, 241 (1952).
[CrossRef]

Estler, W. T.

W. T. Estler, C. J. Evans, L. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
[CrossRef]

C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation and ‘absolute testing,’” CIRP Ann. 45/2, 617–634 (1996).
[CrossRef]

Evans, C. J.

W. T. Estler, C. J. Evans, L. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
[CrossRef]

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

C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation and ‘absolute testing,’” CIRP Ann. 45/2, 617–634 (1996).
[CrossRef]

C. J. Evans, R. E. Parks, “Absolute calibration of spherical surfaces,” in Optical Fabrication and Testing, Vol. 13 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 185–187.

Fritz, B. S.

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

Greivenkamp, J. E.

See, for example, J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Workshop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

Grzanna, J.

Hocken, R. J.

C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation and ‘absolute testing,’” CIRP Ann. 45/2, 617–634 (1996).
[CrossRef]

Kestner, R. N.

Marriner, R.

H. Barrell, R. Marriner, “Liquid surface interferometry,” Nature (London) 162, 529–530 (1948).
[CrossRef]

Oehring, H.

Parks, R. E.

C. J. Evans, R. E. Parks, “Absolute calibration of spherical surfaces,” in Optical Fabrication and Testing, Vol. 13 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 185–187.

R. E. Parks, “Removal of test optics errors,” in Advances in Optical Metrology, N. Balasubramanian, J. C. Wyant, eds., Proc. SPIE153, 56–63 (1978).
[CrossRef]

Porter, R. W.

R. W. Porter, Amateur Telescope Making (Scientific American, New York, 1933), Vol. 3, p. 234.

Raleigh, Lord

Lord Raleigh, “Interference bands and their application,” Nature (London) 48, 212–214 (1893);“The interferometer,” Nature (London) 59, 533 (1899).
[CrossRef]

Rolt, F. H.

F. H. Rolt, Gauges and Fine Measurement (Macmillan, London, 1928).

Saunders, J. B.

J. B. Saunders, Precision Measurements, A. C. S. Van Heel, ed. (North-Holland, Amsterdam, 1967), pp. 8–11.

Schulz, G.

Schwider, J.

Shao, L.

W. T. Estler, C. J. Evans, L. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
[CrossRef]

Spero, R. E.

R. E. Spero, S. E. Whitcomb, “The Laser Interferometer Gravitational-wave Observatory (LIGO),” Opt. Photon. News35–39 (July1995).

Spolaczyk, R.

Steiner, K.

Truax, B. E.

B. E. Truax, “Absolute calibration method for laser Twyman–Green wavefront testing interferometers,” J. Opt. Soc. Am. 63, 1313–1316 (1973).

Tschirf, L.

L. Tschirf, “Pruefung von Abricht-Linealen,” Arch. Technisch. Messen 8224-1 (1941).

Whitcomb, S. E.

R. E. Spero, S. E. Whitcomb, “The Laser Interferometer Gravitational-wave Observatory (LIGO),” Opt. Photon. News35–39 (July1995).

Wyant, J. C.

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

C. Ai, J. C. Wyant, “Modified three-flat method using even and odd functions,” in Optical Fabrication and Testing, Vol. 24 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), 71–74.

C. Ai, J. C. Wyant, “Absolute testing of flats decomposed into even and odd functions,” in Interferometry Surface Characterization and Testing, K. Creath, E. Grievenkamp, eds., Proc. SPIE1776, 73–83 (1992).
[CrossRef]

Appl. Opt. (6)

Arch. Technisch. Messen 8224-1 (1)

L. Tschirf, “Pruefung von Abricht-Linealen,” Arch. Technisch. Messen 8224-1 (1941).

CIRP Ann. (1)

C. J. Evans, R. J. Hocken, W. T. Estler, “Self-calibration: reversal, redundancy, error separation and ‘absolute testing,’” CIRP Ann. 45/2, 617–634 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

B. E. Truax, “Absolute calibration method for laser Twyman–Green wavefront testing interferometers,” J. Opt. Soc. Am. 63, 1313–1316 (1973).

J. Res. Natl. Bur. Stand. (1)

W. B. Emerson, “Determination of planeness and bending of optical flats,” J. Res. Natl. Bur. Stand. 49, 241 (1952).
[CrossRef]

J. Sci. Instrum. (1)

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

Nature (London) (2)

Lord Raleigh, “Interference bands and their application,” Nature (London) 48, 212–214 (1893);“The interferometer,” Nature (London) 59, 533 (1899).
[CrossRef]

H. Barrell, R. Marriner, “Liquid surface interferometry,” Nature (London) 162, 529–530 (1948).
[CrossRef]

Opt. Eng. (1)

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

Opt. Photon. News (1)

R. E. Spero, S. E. Whitcomb, “The Laser Interferometer Gravitational-wave Observatory (LIGO),” Opt. Photon. News35–39 (July1995).

Precis. Eng. (1)

W. T. Estler, C. J. Evans, L. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
[CrossRef]

Other (9)

R. Cobleigh, Handy Farm Devices and How to Make Them (1909; reprinted by Lyons and Burford, New York, 1996).

F. H. Rolt, Gauges and Fine Measurement (Macmillan, London, 1928).

See, for example, J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Workshop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

C. Ai, J. C. Wyant, “Modified three-flat method using even and odd functions,” in Optical Fabrication and Testing, Vol. 24 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), 71–74.

R. E. Parks, “Removal of test optics errors,” in Advances in Optical Metrology, N. Balasubramanian, J. C. Wyant, eds., Proc. SPIE153, 56–63 (1978).
[CrossRef]

C. Ai, J. C. Wyant, “Absolute testing of flats decomposed into even and odd functions,” in Interferometry Surface Characterization and Testing, K. Creath, E. Grievenkamp, eds., Proc. SPIE1776, 73–83 (1992).
[CrossRef]

C. J. Evans, R. E. Parks, “Absolute calibration of spherical surfaces,” in Optical Fabrication and Testing, Vol. 13 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 185–187.

R. W. Porter, Amateur Telescope Making (Scientific American, New York, 1933), Vol. 3, p. 234.

J. B. Saunders, Precision Measurements, A. C. S. Van Heel, ed. (North-Holland, Amsterdam, 1967), pp. 8–11.

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Figures (7)

Fig. 1
Fig. 1

Two wave fronts obtained from the sum of a reference and a test surface, first in a starting azimuthal orientation and then after rotation of the reference surface by ϕ (72)°.

Fig. 2
Fig. 2

Additional wave fronts generated at multiples of ϕ by use of the two original wave fronts.

Fig. 3
Fig. 3

Three synthetic original surfaces used (a) to form the four necessary wave fronts and (b) to determine the topography of the original three surfaces. (c) The maps are the result of using the procedure described, and they show how well the procedure reproduces the noise-free input data.

Fig. 4
Fig. 4

Example of how one of the surfaces is obtained from three readily calculated components.

Fig. 5
Fig. 5

Upper row of surfaces calculated from four interferograms by the method described above. The lower row is the result of using 32 data sets and the full LIGO averaging and analysis. Units are in micrometers.

Fig. 6
Fig. 6

Original wave front F(x, y) and its four two-dimensional symmetry group components. All contour maps are plotted with the same scale height to show the division of the topography.

Fig. 7
Fig. 7

How one symmetry component can be obtained from a signed sum of the original wave front flipped in the four allowed positions by a change in the sign of the coordinates.

Equations (16)

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

z = F x ,   y = F ee + F eo + F oe + F oo .
S 1 + S 2 + S 3 + S 4 + S 5 / 5 = R RI + T RI + R RS + T RS order   of   nk ϕ .
T RS ϕ = 0 S 1 - S 1 + S 2 + S 3 + S 4 + S 5 / 5
S 1 = R x ,   y + T - x ,   y = R + T x , S 2 = R ϕ x ,   y + T - x ,   y = R ϕ + T x ,
S 2 - S 1 ϕ + S 2 = R ϕ + T x - R - T x ϕ + R ϕ + T x = R 2 ϕ - R ϕ + R ϕ + T x = R 2 ϕ + T x = S 3 ,
S 2 - S 1 2 ϕ + S 3 = R 3 ϕ + T x = S 4 , S 2 - S 1 3 ϕ + S 4 = R 4 ϕ + T x = S 5 ,   etc .
A x ,   y + C - x ,   y = W 1 ,     B x ,   y + C - x ,   y = W 2 , B x ,   y + A - x ,   y = W 3 ,     B x ,   y ϕ + C - x ,   y = W 4 ,
W 1 = W 1 ee + W 1 eo + W 1 oe + W 1 oo ,     W 2 = etc .
A ee = W 1 ee + W 3 ee / 2 - W 2 ee + W 4 ee / 4 , B ee = W 3 ee - A ee ,     C ee = W 1 ee - A ee ,
W 5 = B 2 ϕ + C x = W 4 - W 2 ϕ + W 4 .
B RS = W 2 - W 2 + W 4 + W 5 + W 6 + W 7 / 5 = B + C x - B + B ϕ + B 2 ϕ + B 3 ϕ + B 4 ϕ + 5 C x / 5 = B + C x - 5 B RI + 5 C RI x + 5 C RS x / 5 = B + C x - B RI + C RI x + C RS x ,
B ee = b ee RI + b ee RS ,
B RS = b ee RS + B eo + B oe + B oo .
b ee RS = B RS + B RS x + B RS y + B RS xy / 4 .
B = b ee RI + B RS = B ee - b ee RS + B RS ,
F ee = F x ,   y + F - x ,   y + F x ,   - y + F - x ,   - y / 4 , F eo = F x ,   y + F - x ,   y - F x ,   - y - F - x ,   - y / 4 , F oe = F x ,   y - F - x ,   y + F x ,   - y - F - x ,   - y / 4 , F oo = F x ,   y - F - x ,   y - F x ,   - y + F - x ,   - y / 4 .

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