Abstract

Interferometers accurately measure the difference between two wavefronts, one from a reference surface and the other from an unknown surface. If the reference surface is near perfect or is accurately known from some other test, then the shape of the unknown surface can be determined. We investigate the case where neither the reference surface nor the surface under test is well known. By making multiple shear measurements where both surfaces are translated and/or rotated, we obtain sufficient information to reconstruct the figure of both surfaces with a maximum likelihood reconstruction method. The method is demonstrated for the measurement of a 1.6m flat mirror to 2nm rms, using a smaller reference mirror that had significant figure error.

© 2010 Optical Society of America

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  1. J. Yellowhair, P. Su, M. Novak, and J. H. Burge, “Fabrication and testing of large flats,” Proc. SPIE 6671, 667107 (2007).
  2. P. Su, J. H. Burge, R. A. Sprowl, and J. M. Sasian, “Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing,” Proc. SPIE 6342, 63421X (2006).
  3. C. Kim and J. Wyant, “Subaperture test of a large flat on a fast aspheric surface,” J. Opt. Soc. Am. 71, 15-87 (1981).
    [CrossRef]
  4. J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19-27 (1982).
  5. W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. 8, 468-470 (1983).
    [CrossRef]
  6. S. C. Jensen, W. W. Chow, and G. N. Lawrence, “Subaperture testing approaches: a comparison,” Appl. Opt. 23, 740-745(1984).
    [CrossRef]
  7. C. Kim, “Polynomial fit of subaperture interferograms,” Appl. Opt. 21, 4521-4525 (1982).
    [CrossRef]
  8. T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118-127 (1986).
  9. M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurements of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33, 608-613 (1994).
  10. QED Technologies Inc., “Software Patent: Simultaneous self-calibrated subaperture stitching for surface figure measurement (interferometer), ” European patent EP1,324,006 (1 July 2003).
  11. S. Chen, S. Li, and Y. Dai, “Iterative algorithm for subaperture stitching interferometry for general surfaces,” J. Opt. Soc. Am. A 22, 1929-1936 (2005).
    [CrossRef]
  12. H. Barrell and R. Marriner, “Liquid surface interferometry,” Nature 162, 529-530 (1948).
  13. I. Powell and E. Goulet, “Absolute figure measurements with a liquid-flat reference,” Appl. Opt. 37, 2579-2588 (1998).
    [CrossRef]
  14. G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077-1084 (1967).
    [CrossRef]
  15. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379-383 (1984).
  16. R. E. Parks, “Removal of test optics errors,” Proc. SPIE 153, 56-63 (1978).
  17. C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015-1021 (1996).
    [CrossRef]
  18. C. Ai and J. C. Wyant, “Absolute testing of flats using even and odd functions,” Appl. Opt. 32, 4698-4705 (1993).
    [CrossRef]
  19. R. E. Parks, L. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951-5965(1998).
    [CrossRef]
  20. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856-5865 (2006).
    [CrossRef]
  21. P. Su, J. H. Burge and J. M. Sasian, “Shear test of the off-axis surface with an axis-symmetric parent,” Proc. SPIE 6671, 66710R (2007).
  22. B. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer-Verlag, 1983).
  23. H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A 24, 391-414 (2007).
    [CrossRef]
  24. W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in C (Cambridge Univ. Press, 1992).
  25. P. Su, “Absolute measurements of large mirrors,” Ph.D.dissertation (University of Arizona, 2008).
  26. See the stitching software information at www.mb-optics.com.
  27. C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).

2007 (3)

J. Yellowhair, P. Su, M. Novak, and J. H. Burge, “Fabrication and testing of large flats,” Proc. SPIE 6671, 667107 (2007).

P. Su, J. H. Burge and J. M. Sasian, “Shear test of the off-axis surface with an axis-symmetric parent,” Proc. SPIE 6671, 66710R (2007).

H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A 24, 391-414 (2007).
[CrossRef]

2006 (3)

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).

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

P. Su, J. H. Burge, R. A. Sprowl, and J. M. Sasian, “Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing,” Proc. SPIE 6342, 63421X (2006).

2005 (1)

1998 (2)

1996 (1)

1994 (1)

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurements of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33, 608-613 (1994).

1993 (1)

1986 (1)

T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118-127 (1986).

1984 (2)

1983 (1)

1982 (2)

J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19-27 (1982).

C. Kim, “Polynomial fit of subaperture interferograms,” Appl. Opt. 21, 4521-4525 (1982).
[CrossRef]

1981 (1)

1978 (1)

R. E. Parks, “Removal of test optics errors,” Proc. SPIE 153, 56-63 (1978).

1967 (1)

1948 (1)

H. Barrell and R. Marriner, “Liquid surface interferometry,” Nature 162, 529-530 (1948).

Ai, C.

Barrell, H.

H. Barrell and R. Marriner, “Liquid surface interferometry,” Nature 162, 529-530 (1948).

Barrett, H. H.

Bray, M.

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).

Burge, J. H.

P. Su, J. H. Burge and J. M. Sasian, “Shear test of the off-axis surface with an axis-symmetric parent,” Proc. SPIE 6671, 66710R (2007).

J. Yellowhair, P. Su, M. Novak, and J. H. Burge, “Fabrication and testing of large flats,” Proc. SPIE 6671, 667107 (2007).

P. Su, J. H. Burge, R. A. Sprowl, and J. M. Sasian, “Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing,” Proc. SPIE 6342, 63421X (2006).

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).

Chen, S.

Chow, W. W.

Dai, Y.

Dainty, C.

Evans, C. J.

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in C (Cambridge Univ. Press, 1992).

Frieden, B.

B. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer-Verlag, 1983).

Fritz, B. S.

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

Goulet, E.

Griesmann, U.

Jensen, S. C.

Kestner, R. N.

Kim, C.

Kwon, O. Y.

J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19-27 (1982).

Lara, D.

Lawrence, G. N.

Li, S.

Marriner, R.

H. Barrell and R. Marriner, “Liquid surface interferometry,” Nature 162, 529-530 (1948).

Novak, M.

J. Yellowhair, P. Su, M. Novak, and J. H. Burge, “Fabrication and testing of large flats,” Proc. SPIE 6671, 667107 (2007).

Okada, K.

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurements of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33, 608-613 (1994).

Otsubo, M.

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurements of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33, 608-613 (1994).

Parks, R. E.

Powell, I.

Press, W.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in C (Cambridge Univ. Press, 1992).

Sasian, J. M.

P. Su, J. H. Burge and J. M. Sasian, “Shear test of the off-axis surface with an axis-symmetric parent,” Proc. SPIE 6671, 66710R (2007).

P. Su, J. H. Burge, R. A. Sprowl, and J. M. Sasian, “Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing,” Proc. SPIE 6342, 63421X (2006).

Schulz, G.

Schwider, J.

Shao, L.

Sprowl, R. A.

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).

P. Su, J. H. Burge, R. A. Sprowl, and J. M. Sasian, “Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing,” Proc. SPIE 6342, 63421X (2006).

Stuhlinger, T. W.

T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118-127 (1986).

Su, P.

J. Yellowhair, P. Su, M. Novak, and J. H. Burge, “Fabrication and testing of large flats,” Proc. SPIE 6671, 667107 (2007).

P. Su, J. H. Burge and J. M. Sasian, “Shear test of the off-axis surface with an axis-symmetric parent,” Proc. SPIE 6671, 66710R (2007).

P. Su, J. H. Burge, R. A. Sprowl, and J. M. Sasian, “Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing,” Proc. SPIE 6342, 63421X (2006).

P. Su, “Absolute measurements of large mirrors,” Ph.D.dissertation (University of Arizona, 2008).

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in C (Cambridge Univ. Press, 1992).

Thunen, J. G.

J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19-27 (1982).

Tsujiuchi, J.

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurements of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33, 608-613 (1994).

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in C (Cambridge Univ. Press, 1992).

Wyant, J.

Wyant, J. C.

Yellowhair, J.

J. Yellowhair, P. Su, M. Novak, and J. H. Burge, “Fabrication and testing of large flats,” Proc. SPIE 6671, 667107 (2007).

Zhao, C.

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).

Appl. Opt. (8)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nature (1)

H. Barrell and R. Marriner, “Liquid surface interferometry,” Nature 162, 529-530 (1948).

Opt. Eng. (2)

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurements of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33, 608-613 (1994).

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

Opt. Lett. (1)

Proc. SPIE (7)

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).

R. E. Parks, “Removal of test optics errors,” Proc. SPIE 153, 56-63 (1978).

P. Su, J. H. Burge and J. M. Sasian, “Shear test of the off-axis surface with an axis-symmetric parent,” Proc. SPIE 6671, 66710R (2007).

T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118-127 (1986).

J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19-27 (1982).

J. Yellowhair, P. Su, M. Novak, and J. H. Burge, “Fabrication and testing of large flats,” Proc. SPIE 6671, 667107 (2007).

P. Su, J. H. Burge, R. A. Sprowl, and J. M. Sasian, “Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing,” Proc. SPIE 6342, 63421X (2006).

Other (5)

QED Technologies Inc., “Software Patent: Simultaneous self-calibrated subaperture stitching for surface figure measurement (interferometer), ” European patent EP1,324,006 (1 July 2003).

B. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer-Verlag, 1983).

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in C (Cambridge Univ. Press, 1992).

P. Su, “Absolute measurements of large mirrors,” Ph.D.dissertation (University of Arizona, 2008).

See the stitching software information at www.mb-optics.com.

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Figures (14)

Fig. 1
Fig. 1

Fizeau test of a 1.6 m flat using a diameter 1 m reference flat and a commercial vibration insensitive Fizeau interferometer.

Fig. 2
Fig. 2

Geometric setup of the subaperture test.

Fig. 3
Fig. 3

(a) Measurement results of the 1.6 m flat rms = 6 nm , before it was put into a cell; (b)  rms = 21 nm , after it was put into a cell; (c)  rms = 6 nm , after it was put into a cell and astigmatisms were removed mathematically.

Fig. 4
Fig. 4

Final surface measurement result of 24 nm rms for the 1.6 m flat calculated from a combination of the result, shown in Fig. 3b, and power determined using the pentaprism test.

Fig. 5
Fig. 5

(a) Surface measurement result of the reference flat rms = 42 nm , p - v = 184 nm . (b) Reference flat finite element analysis (FEA) result : rms = 29 nm , p - v = 129 nm .

Fig. 6
Fig. 6

Zernike coefficients from two independent measurements of the reference flat, showing the repeatability of 1.8 nm rms.

Fig. 7
Fig. 7

(a) Measurement result of the Parks method was 37 nm rms; (b) measurement result of the 6 rotation method was 39 nm . Both methods can only measure the non-axis-symmetric form of errors in the surface.

Fig. 8
Fig. 8

Numerically generated covariance matrix C. Red (on dashed line) means that the estimates of the alignment terms have relatively large uncertainties compared to the estimates of the surface coefficients. Along the main diagonal of the matrix C, the first 1–30 terms correspond to the coefficients of the 1.6 m flat, the 35–109 correspond to the coefficients of the reference flat, and the 31–34 and the rest of the terms are related with the alignment terms in each subaperture test.

Fig. 9
Fig. 9

Crosstalk errors increase as more Zernike terms are used to describe the surfaces.

Fig. 10
Fig. 10

Surface irregularities in the form of higher order aberrations couple into the estimates of the low order surface coefficents with a magnitude less than 20%.

Fig. 11
Fig. 11

Typical subaperture map of the least squares fitting residual, rms = 5.5 nm . The aperture shape is not exactly circular due to the geometry of the subaperture test.

Fig. 12
Fig. 12

(a)  1.6 m flat by MLR, 6 nm rms after power and astigmatism removed mathematically. (b)  1.6 m flat by MBSI, 7 nm rms after power and astigmatism removed mathematically.

Fig. 13
Fig. 13

Estimated Zernike coefficients of 1.6 m flat from MLR method and MBSI, difference is 1.38 nm rms.

Fig. 14
Fig. 14

Difference map between MLR and MBSI, difference is 5 nm rms.

Tables (2)

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Table 1 Subaperture Measurement Scheme

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Table 2 X Displacement Scale Factors of Zernike Standard Polynomial Z5–Z14

Equations (21)

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D i j = D i j a + residuals = surface A + surface B + alignments + residuals , surface A = p = 5 m A p Z p ( ρ a , θ a + ϕ a i ) , surface B = q = 5 n B q Z q ( ρ b , θ b + ϕ b i ) , alignments = P i Z 1 ( ρ a , θ a + ϕ a i ) + T x i Z 2 ( ρ a , θ a + ϕ a i ) + T y i Z 3 ( ρ a , θ a + ϕ a i ) + D e i Z 4 ( ρ a , θ a + ϕ a i ) ,
L ( A p , B q | D i j ) = ( 2 π σ ) u v exp [ 1 2 σ 2 i = 1 u j = 1 v ( D i j D a i j ) 2 ] ,
i = 1 u j = 1 v ( D i j D a i j ) 2 = i = 1 u j = 1 v ( D i j surface A surface B alignments ) 2 = m inimum
x = [ A 5 , A 6 A m , B 5 , B 6 , B n , P 1 , T x 1 , T y 1 , D e 1 , P u , T x u , T y u , D e u ] T ,
ϕ i = [ D i 1 , D i 2 , , D i v ] T .
y i = ( U i ) 1 ϕ i ,
T = U i S i S i ,
y = [ y 1 , y 2 , , y u ] T ,
Z C i t = ( U i ) 1 Z i t .
M i = [ Z C i 1 , Z C i 2 , Z C i 4 , Z C i 5 A , Z C i m A , Z C i 5 B , Z C i n B ] ,
M = [ M 1 , M 2 , , M u ] T ,
y = M x ,
x = ( M T M ) 1 M T y .
σ 2 ( x p ) = C p p · σ 2 ( y q ) , C = ( M T M ) 1 ,
σ 2 ( y q ) = C Y q q · σ k 2 , C Y = ( U i T U i ) 1 ,
C Y q q = 1 ; σ ( y q ) = σ k .
σ a = 0.0034 σ k = 0.3 nm rms ,
σ b = 0.0078 σ k = 0.4 nm rms .
rms = Δ θ · m f · coefficient,
( σ r t ) 2 = ( σ k ) 2 + ( σ r a ) 2 + ( σ r b ) 2 ,
rms = σ k 2 + σ r 2 + σ r r 2 = 1.6 nm rms

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