Abstract

A radial shearing (RS) interferogram obtained by the carrier fringe method is essentially the combination of radial and lateral shearing. The previous phase reconstruction algorithm neglects the effect of lateral shearing on the obtained RS interferogram. A mathematical formula for wavefront reconstruction from an RS interferogram is deduced. If the phase difference of the tested wavefront phase, the RS ratio s, and the laterally sheared amount x0 in the x direction and y0 in the y direction, respectively, have been determined, the tested wavefront phase can be precisely reconstructed using this formula. The result of simulation analysis and experiment shows that the formula is correct and more accurate.

© 2008 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley, 1978).
  2. A. R. Barnes and L. C. Smith, Proc. SPIE 3492, 564 (1999).
    [Crossref]
  3. P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. William, and B. M. Van Wonterghem, Proc. SPIE 3492, 1019 (1999).
    [Crossref]
  4. D. Liu, Y. Y. Yang, J. M. Weng, X. M. Zhang, B. Chen, and X. W. Qin, Opt. Commun. 275, 173 (2007).
    [Crossref]
  5. D. H. Li, H. X. Chen, and Z. P. Chen, Opt. Eng. (Bellingham) 41, 1893 (2002).
    [Crossref]
  6. M. Takeda, H. Ina, and S. Kobayashi, J. Opt. Soc. Am. 72, 156 (1982).
    [Crossref]

2007 (1)

D. Liu, Y. Y. Yang, J. M. Weng, X. M. Zhang, B. Chen, and X. W. Qin, Opt. Commun. 275, 173 (2007).
[Crossref]

2002 (1)

D. H. Li, H. X. Chen, and Z. P. Chen, Opt. Eng. (Bellingham) 41, 1893 (2002).
[Crossref]

1999 (2)

A. R. Barnes and L. C. Smith, Proc. SPIE 3492, 564 (1999).
[Crossref]

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. William, and B. M. Van Wonterghem, Proc. SPIE 3492, 1019 (1999).
[Crossref]

1982 (1)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

D. Liu, Y. Y. Yang, J. M. Weng, X. M. Zhang, B. Chen, and X. W. Qin, Opt. Commun. 275, 173 (2007).
[Crossref]

Opt. Eng. (Bellingham) (1)

D. H. Li, H. X. Chen, and Z. P. Chen, Opt. Eng. (Bellingham) 41, 1893 (2002).
[Crossref]

Proc. SPIE (2)

A. R. Barnes and L. C. Smith, Proc. SPIE 3492, 564 (1999).
[Crossref]

P. J. Wegner, M. A. Henesian, J. T. Salmon, L. G. Seppala, T. L. Weiland, W. H. William, and B. M. Van Wonterghem, Proc. SPIE 3492, 1019 (1999).
[Crossref]

Other (1)

D. Malacara, Optical Shop Testing (Wiley, 1978).

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Figures (4)

Fig. 1
Fig. 1

Schematic of a CRSI to be used to test a wavefront. F is a filter for filtering out the high frequency structure of the LCTV.

Fig. 2
Fig. 2

Schematic of the RS with lateral shearing and the coordinate system for deriving the formula suggested in this Letter.

Fig. 3
Fig. 3

Simulation analysis: (a) assumed wavefront under test, (b) expanded wavefront, (c) phase difference Δ W ( x s , y s ) , (d) cross section along the central row of the wavefront under test (dashed), the reconstructed wavefront (circles), and the reconstructed wavefront using the algorithm suggested in [5] (stars).

Fig. 4
Fig. 4

Experiment results: (a) CRS fringe pattern, (b) reconstructed phase using the algorithm in [5], (c) reconstructed phase using Eq. (10) in this Letter, (d) cross section along the central row of the reconstructed wavefront (circles) and the reconstructed wavefront using the algorithm in [5] (stars).

Equations (11)

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I ( x s , y s ) = A ( x s , y s ) exp { i k [ W ( x s , y s ) ] } + A ( s x + x 0 , s y + y 0 ) exp { i k [ W ( s x + x 0 , s y + y 0 ) ] } 2 = A 2 ( x s , y s ) + A 2 ( s x + x 0 , s y + y 0 ) + 2 A ( x s , y s ) A ( s x + x 0 , s y + y 0 ) cos { k [ W ( x s , y s ) W ( s x + x 0 , s y + y 0 ) ] } .
Δ W ( x s , y s ) = W ( x s , y s ) W ( s x + x 0 , s y ) .
Δ W ( s x , s y ) = W ( s x , s y ) W ( s 3 x + s 2 x 0 , s 3 y ) .
Δ W ( s 3 x , s 3 y ) = W ( s 3 x , s 3 y ) W ( s 5 x + s 4 x 0 , s 5 y ) ,
Δ W ( x s 2 n 1 , y s 2 n 1 ) = W ( x s 2 n 1 , y s 2 n 1 ) W ( x s 2 n + 1 + x 0 s 2 n , y s 2 n + 1 ) .
Δ W ( s x + x 0 , s y ) = W ( s x + x 0 , s y ) W ( s 3 x + s 2 x 0 + x 0 , s 3 y ) ,
Δ W ( s 3 x + s 2 x 0 + x 0 , s 3 y ) = W ( s 3 x + s 2 x 0 + x 0 , s 3 y ) W ( s 5 x + s 4 x 0 + s 2 x 0 + x 0 , s 5 y ) ,
Δ W [ x s 2 n 1 + x 0 ( k = 0 n 1 s 2 k ) , y s 2 n 1 ] = W [ x s 2 n 1 + x 0 ( k = 0 n 1 s 2 k ) , y s 2 n 1 ] W [ x s 2 n + 1 + x 0 ( k = 0 n s 2 k ) , y s 2 n + 1 ] .
Δ W ( x s , y s ) + n = 1 N Δ W [ x s 2 n 1 + x 0 ( k = 0 n 1 s 2 k ) , y s 2 n 1 ] = W ( x s , y s ) W [ x s 2 n + 1 + x 0 ( k = 0 n s 2 k ) , y s 2 n + 1 ] .
W ( x s , y s ) = Δ W ( x s , y s ) + n = 1 N Δ W [ x s 2 n 1 ± x 0 ( k = 0 n 1 s 2 k ) , y s 2 n 1 ± y 0 ( k = 0 n 1 s 2 k ) ] .
W ( x , y ) = ( 28672 y 4 + 7168 x 3 + 896 y 2 + 448 x 2 256 x ) exp ( 64 x 2 64 y 2 ) + 2.218 .

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