Abstract

The most basic task in subaperture stitching test work is to confirm the precision of the system. We propose a method with which to calculate the system's precision. Statistical theory, especially regressive analysis, is employed. To discuss the statistical characteristics of all the random variables is the objective of the precision analyses. The results show that the number of sample points and the connatural error of the interferometer are the most important factors in the analyses.

© 2006 Optical Society of America

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References

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  1. J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, "Specification of optical components using the power spectral density function," in Optical Manufacturing and Testing, V.J.Doherty and H.P.Stahl, eds., Proc. SPIE 2536, 38-50 (1995).
  2. R. J. Noll, "Effect of mid- and high-spatial frequencies on the optical performance," Opt. Eng. 18, 137-142 (1979).
  3. M. Otsubo, K. Okada, and J. Tsujiuchi, "Measurement of large plane surface shapes by connecting small-aperture interferograms," Opt. Eng. 33, 608-613 (1994).
    [CrossRef]
  4. M. Bray, "Stitching interferometer for large plano optics using a standard interferometer," in Optical Manufacturing and Testing II, H.P.Stahl, ed., Proc. SPIE 3134, 39-50 (1997).
  5. M. Bray, "Stitching interferometry and absolute surface shape metrology: similarities," in Optical Manufacturing and Testing IV, H.P.Stahl, ed., Proc. SPIE 4451, 375-383 (2001).
  6. T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparre and B.Singh, eds., Proc. SPIE 4449, 265-275 (2001).

1994 (1)

M. Otsubo, K. Okada, and J. Tsujiuchi, "Measurement of large plane surface shapes by connecting small-aperture interferograms," Opt. Eng. 33, 608-613 (1994).
[CrossRef]

1979 (1)

R. J. Noll, "Effect of mid- and high-spatial frequencies on the optical performance," Opt. Eng. 18, 137-142 (1979).

Aikens, D. M.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, "Specification of optical components using the power spectral density function," in Optical Manufacturing and Testing, V.J.Doherty and H.P.Stahl, eds., Proc. SPIE 2536, 38-50 (1995).

Bray, M.

M. Bray, "Stitching interferometer for large plano optics using a standard interferometer," in Optical Manufacturing and Testing II, H.P.Stahl, ed., Proc. SPIE 3134, 39-50 (1997).

M. Bray, "Stitching interferometry and absolute surface shape metrology: similarities," in Optical Manufacturing and Testing IV, H.P.Stahl, ed., Proc. SPIE 4451, 375-383 (2001).

English, R. E.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, "Specification of optical components using the power spectral density function," in Optical Manufacturing and Testing, V.J.Doherty and H.P.Stahl, eds., Proc. SPIE 2536, 38-50 (1995).

Hänsel, T.

T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparre and B.Singh, eds., Proc. SPIE 4449, 265-275 (2001).

Lawson, J. K.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, "Specification of optical components using the power spectral density function," in Optical Manufacturing and Testing, V.J.Doherty and H.P.Stahl, eds., Proc. SPIE 2536, 38-50 (1995).

Manes, K. R.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, "Specification of optical components using the power spectral density function," in Optical Manufacturing and Testing, V.J.Doherty and H.P.Stahl, eds., Proc. SPIE 2536, 38-50 (1995).

Nickel, A.

T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparre and B.Singh, eds., Proc. SPIE 4449, 265-275 (2001).

Noll, R. J.

R. J. Noll, "Effect of mid- and high-spatial frequencies on the optical performance," Opt. Eng. 18, 137-142 (1979).

Okada, K.

M. Otsubo, K. Okada, and J. Tsujiuchi, "Measurement of large plane surface shapes by connecting small-aperture interferograms," Opt. Eng. 33, 608-613 (1994).
[CrossRef]

Otsubo, M.

M. Otsubo, K. Okada, and J. Tsujiuchi, "Measurement of large plane surface shapes by connecting small-aperture interferograms," Opt. Eng. 33, 608-613 (1994).
[CrossRef]

Schindler, A.

T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparre and B.Singh, eds., Proc. SPIE 4449, 265-275 (2001).

Trenholme, J. B.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, "Specification of optical components using the power spectral density function," in Optical Manufacturing and Testing, V.J.Doherty and H.P.Stahl, eds., Proc. SPIE 2536, 38-50 (1995).

Tsujiuchi, J.

M. Otsubo, K. Okada, and J. Tsujiuchi, "Measurement of large plane surface shapes by connecting small-aperture interferograms," Opt. Eng. 33, 608-613 (1994).
[CrossRef]

Wolfe, C. R.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, "Specification of optical components using the power spectral density function," in Optical Manufacturing and Testing, V.J.Doherty and H.P.Stahl, eds., Proc. SPIE 2536, 38-50 (1995).

Opt. Eng. (2)

R. J. Noll, "Effect of mid- and high-spatial frequencies on the optical performance," Opt. Eng. 18, 137-142 (1979).

M. Otsubo, K. Okada, and J. Tsujiuchi, "Measurement of large plane surface shapes by connecting small-aperture interferograms," Opt. Eng. 33, 608-613 (1994).
[CrossRef]

Other (4)

M. Bray, "Stitching interferometer for large plano optics using a standard interferometer," in Optical Manufacturing and Testing II, H.P.Stahl, ed., Proc. SPIE 3134, 39-50 (1997).

M. Bray, "Stitching interferometry and absolute surface shape metrology: similarities," in Optical Manufacturing and Testing IV, H.P.Stahl, ed., Proc. SPIE 4451, 375-383 (2001).

T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparre and B.Singh, eds., Proc. SPIE 4449, 265-275 (2001).

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, "Specification of optical components using the power spectral density function," in Optical Manufacturing and Testing, V.J.Doherty and H.P.Stahl, eds., Proc. SPIE 2536, 38-50 (1995).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the stitching interferometer.

Fig. 2
Fig. 2

(Color online) Result of testing the first subaperture (W1).

Fig. 3
Fig. 3

(Color online) Result of testing the second subaperture (W2).

Fig. 4
Fig. 4

(Color online) Result of stitching.

Fig. 5
Fig. 5

(Color online) Result of testing the whole aperture.

Fig. 6
Fig. 6

Comparison of stitching and whole-aperture results.

Equations (18)

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Φ ̂ W 2 = Φ W 2 + a + b x + c y ,
Φ 1 Φ 1 + a + b x 1 + c y 1 = v 1 , Φ 2 Φ 2 + a + b x 2 + c y 2 = v 2 , Φ 3 Φ 3 + a + b x 3 + c y 3 = v 3 ,
i v i     2 min .
i v i     2 a = 0 , i v i     2 b = 0 , i v i     2 c = 0.
a [ x ] + b [ x x ] + c [ x y ] [ x l ] = 0 ,
a [ y ] + b [ x y ] + c [ y y ] [ y l ] = 0 ,
a + b [ x z ] + c [ y z ] [ l ] = 0 ,
U = ( a b c ) , H = [ N [ x ] [ y ] [ x ] [ x x ] [ x y ] [ y ] [ x y ] [ y y ] ] , L = ( [ x l ] [ y l ] [ l ] ) ,
U = H 1 L .
Φ W 2 = Φ W 2 + A + B x + C y ,
E [ a b c ] = [ A B C ] .
E { [ a A b B c C ] [ a A , b B , c C ] } = 2 σ 2 [ h 00 h 01 h 02 h 10 h 11 h 12 h 20 h 21 h 22 ] = 2 σ 2 H 1 .
E [ Φ ̂ W 2 Φ W 2 ] = 0
D [ Φ ̂ W 2 Φ W 2 ] = D [ ( a A ) + x ( b B ) + y ( c C ) ]
= 2 σ 2 { 1 N + h 11 ( x x ¯ ) 2 + ( h 12 + h 21 ) ( x x ¯ ) ( y y ¯ ) + h 22 ( y y ¯ ) 2 } ,
x ¯ = 1 N i = 1 N x i , y ¯ = 1 N i = 1 N y i .
H = [ N x max m ( m + 1 ) 2 y max m ( m + 1 ) 2 x max m ( m + 1 ) 2 x max              2 m ( 2 m + 1 ) 6 x max y max ( m + 1 ) 2 4 y max m ( m + 1 ) 2 x max y max ( m + 1 ) 2 4 y max               2 m ( 2 m + 1 ) 6 ] .
lim N H = N [ 1 x max 2 y max 2 x max 2 x max              2 3 x max y max 4 x max y max 4 y max 2 y max             2 3 ] .

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