Abstract

A novel method is presented for one-dimensional wavefront recovery on the basis of difference measurements from two shearing interferograms with varying tilt. The method uses large shears and obtains high lateral resolution. Furthermore, the wavefront under test can be recovered exactly up to an arbitrary constant and straight line at all evaluation points with suitably chosen shears of two shearing interferograms.

© 2009 Optical Society of America

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References

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  1. W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940-952 (1947).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  15. D. Malacara and O. Harris, “Interferometric measurements of angles,” Appl. Opt. 9, 1630-1633 (1970).
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  16. H. Schreiber, “Measuring wavefront tilt using shearing interferometry,” Proc. SPIE 5965, 296-307 (2005).
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2005 (1)

H. Schreiber, “Measuring wavefront tilt using shearing interferometry,” Proc. SPIE 5965, 296-307 (2005).

2003 (1)

2002 (1)

1999 (2)

1997 (2)

T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692-2701 (1997).
[CrossRef]

W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial filigtting of lateral shearing interferometry,” Opt. Eng. 36, 905-913 (1997).
[CrossRef]

1996 (2)

1987 (1)

1983 (1)

T. Yatagai and T. Kanou, “Aspherical surface testing with shearing interferometer using fringe scanning detection method,” Proc. SPIE 429, 136 (1983).

1982 (1)

1973 (1)

1970 (1)

1964 (1)

1961 (1)

J. B. Saunders, “Measurement of wave fronts without a reference standard. Part 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239-244 (1961).

1947 (1)

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940-952 (1947).
[CrossRef]

Bates, W. J.

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940-952 (1947).
[CrossRef]

Chang, M. W.

W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial filigtting of lateral shearing interferometry,” Opt. Eng. 36, 905-913 (1997).
[CrossRef]

Eiju, T.

Elster, C.

Flynn, T. J.

Harbers, G.

Hariharan, P.

Harris, O.

Ina, H.

Kamiya, K.

Kanou, T.

T. Yatagai and T. Kanou, “Aspherical surface testing with shearing interferometer using fringe scanning detection method,” Proc. SPIE 429, 136 (1983).

Kobayashi, S.

Kunst, P. J.

Leibbrandt, G. W. R.

Malacara, D.

Marroquin, J. L.

Muñoz, J.

Nomura, T.

Okuda, S.

Oreb, B. F.

Páez, G.

Ronchi, V.

Saunders, J. B.

J. B. Saunders, “Measurement of wave fronts without a reference standard. Part 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239-244 (1961).

Schreiber, H.

H. Schreiber, “Measuring wavefront tilt using shearing interferometry,” Proc. SPIE 5965, 296-307 (2005).

Servin, M.

Shen, W.

W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial filigtting of lateral shearing interferometry,” Opt. Eng. 36, 905-913 (1997).
[CrossRef]

Strojnik, M.

Takeda, M. H.

Tashiro, H.

Wan, D. S.

W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial filigtting of lateral shearing interferometry,” Opt. Eng. 36, 905-913 (1997).
[CrossRef]

Weingärtner, I.

Wyant, J. C.

Yatagai, T.

T. Yatagai and T. Kanou, “Aspherical surface testing with shearing interferometer using fringe scanning detection method,” Proc. SPIE 429, 136 (1983).

Yoshikawa, K.

Appl. Opt. (9)

J. Opt. Soc. Am. (1)

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

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

J. B. Saunders, “Measurement of wave fronts without a reference standard. Part 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239-244 (1961).

Opt. Eng. (1)

W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial filigtting of lateral shearing interferometry,” Opt. Eng. 36, 905-913 (1997).
[CrossRef]

Proc. Phys. Soc. London (1)

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940-952 (1947).
[CrossRef]

Proc. SPIE (1)

T. Yatagai and T. Kanou, “Aspherical surface testing with shearing interferometer using fringe scanning detection method,” Proc. SPIE 429, 136 (1983).

Proc. SPIE (1)

H. Schreiber, “Measuring wavefront tilt using shearing interferometry,” Proc. SPIE 5965, 296-307 (2005).

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Figures (5)

Fig. 1
Fig. 1

Phase function and phase difference curves.

Fig. 2
Fig. 2

First group of curves obtained by the Saunders method.

Fig. 3
Fig. 3

Second group of curves obtained by the Saunders method.

Fig. 4
Fig. 4

Recovered wavefront before removing the linear trend.

Fig. 5
Fig. 5

Recovered wavefront after removing the linear trend.

Tables (3)

Tables Icon

Table 1 Smallest Aperture for Exact Wavefront Recovery

Tables Icon

Table 2 Calculating Sequence for p < s 1 · s 2

Tables Icon

Table 3 Calculating Sequence for p = s 1 · s 2

Equations (20)

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Δ f 1 ( x ) = W ( x + s 1 ) W ( x ) + E 1 s 1 ,
Δ f 2 ( x ) = W ( x + s 2 ) W ( x ) + E 2 s 2 ,
f ^ 1 ( i ) = 0 , i = 0 , 1 , , ϑ 1 1 ,
f ^ 1 ( i ) = f ^ 1 ( i ϑ 1 ) + Δ f 1 ( i - ϑ 1 ) , i = ϑ 1 , ϑ 1 + 1 , , N .
f ^ 2 ( i ) = 0 , i = 0 , 1 , , ϑ 2 1 ,
f ^ 2 ( i ) = f ^ 2 ( i ϑ 2 ) + Δ f 2 ( i - ϑ 2 ) , i = ϑ 2 , ϑ 2 + 1 , , N ,
f ( k 1 ϑ 1 ) + E 1 · k 1 ϑ 1 · δ = f ^ 1 ( k 1 ϑ 1 ) ,
f ( k 1 ϑ 1 ) + E 2 · ( k 1 ϑ 1 - r 2 ) · δ - f ( r 2 ) = f ^ 2 ( k 1 ϑ 1 ) ,
f ( k 2 ϑ 2 ) + E 2 k 2 ϑ 2 δ = f ^ 2 ( k 2 ϑ 2 ) ,
f ( k 2 ϑ 2 ) + E 1 ( k 2 ϑ 2 - r 1 ) δ - f ( r 1 ) = f ^ 1 ( k 2 ϑ 2 ) ,
E 1 k 1 ϑ 1 δ - E 2 ( k 1 ϑ 1 - r 2 ) δ + f ( r 2 ) = f ^ 1 ( k 1 ϑ 1 ) - f ^ 2 ( k 1 ϑ 1 ) = ζ 1 .
E 2 k 2 ϑ 2 - E 1 ( k 2 ϑ 2 - r 1 ) δ + f ( r 1 ) = f ^ 2 ( k 2 ϑ 2 ) - f ^ 1 ( k 2 ϑ 2 ) = ζ 2 .
Δ E = E 1 E 2 = ζ 1 ζ 2 ( k 1 ϑ 1 + k 2 ϑ 2 - r ¯ ) δ .
f ^ ( i ) = f ^ 1 ( i ) , i = 0 : ϑ 1 : N ,
f ^ ( j ) = f ^ 2 ( j ) + Δ E δ j , j = 0 : ϑ 2 : N .
f ^ ( k ) = f ^ 1 ( k ) + Δ , k = r : ϑ 1 : N ,
f ^ ( n ) = f ^ 2 ( n ) + f ( r ¯ - 1 ) + Δ E ( n - r ¯ + 1 ) δ
f ^ ( n ) = f ^ 2 ( n ) + f ( r ¯ ) + Δ E ( n - r ¯ ) δ .
f ^ ( i ) = f ^ 1 ( i ) + [ f ^ ( n ) - f ^ 1 ( n ) ] .
f ( x ) = 10.2 ( x / p ) 6 22.44 ( x / p ) 4 + 1.2 ( x / p ) 3 + 10.2 ( x / p + 0.2 ) 2 + 0.05 cos ( 80 x / p ) .

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