Abstract

A modified form of a phase tracking method to demodulate a single fringe pattern is presented. Phase values from local areas of the interferogram are recovered by means of a spatial synchronous technique instead of solving the set of nonlinear equations obtained from the implementation of the ordinary algorithm. This results in a significant speed improvement of the method. Additionally, the robustness against noise is maintained, and the sensitivity to contrast variations is decremented with respect to the phase tracking technique.

© 2007 Optical Society of America

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References

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  1. K. Creath and J. C. Wyant, Optical Shop Testing (Wiley, 1992), Chap. 16.
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, "Digital wavefront measuring interferometer for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974).
    [CrossRef] [PubMed]
  3. D. W. Robinson, "Phase unwrapping methods," in Interferogram Analysis, D.W.Robinson and G.T.Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 195.
  4. M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  5. M. Servin, J. L. Marroquín, and F. J. Cuevas, "Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique," Appl. Opt. 36, 4540-4548 (1997).
    [CrossRef] [PubMed]
  6. M. Servin, J. L. Marroquín, and J. A. Quiroga, "Regularized quadrature and phase tracking from a single closed-fringe interferogram," J. Opt. Soc. Am. A 21, 411-419 (2004).
    [CrossRef]
  7. J. A. Quiroga, J. A. Gómez-Pedrero, and A. García-Botella, "Algorithm for fringe pattern normalization," Opt. Commun. 197, 43-51 (2001).
    [CrossRef]
  8. K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

2004 (1)

2001 (1)

J. A. Quiroga, J. A. Gómez-Pedrero, and A. García-Botella, "Algorithm for fringe pattern normalization," Opt. Commun. 197, 43-51 (2001).
[CrossRef]

1997 (1)

1984 (1)

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

1982 (1)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Creath, K.

K. Creath and J. C. Wyant, Optical Shop Testing (Wiley, 1992), Chap. 16.

Cuevas, F. J.

Gallagher, J. E.

García-Botella, A.

J. A. Quiroga, J. A. Gómez-Pedrero, and A. García-Botella, "Algorithm for fringe pattern normalization," Opt. Commun. 197, 43-51 (2001).
[CrossRef]

Gómez-Pedrero, J. A.

J. A. Quiroga, J. A. Gómez-Pedrero, and A. García-Botella, "Algorithm for fringe pattern normalization," Opt. Commun. 197, 43-51 (2001).
[CrossRef]

Herriott, D. R.

Ina, H.

Kobayashi, S.

Marroquín, J. L.

Quiroga, J. A.

M. Servin, J. L. Marroquín, and J. A. Quiroga, "Regularized quadrature and phase tracking from a single closed-fringe interferogram," J. Opt. Soc. Am. A 21, 411-419 (2004).
[CrossRef]

J. A. Quiroga, J. A. Gómez-Pedrero, and A. García-Botella, "Algorithm for fringe pattern normalization," Opt. Commun. 197, 43-51 (2001).
[CrossRef]

Robinson, D. W.

D. W. Robinson, "Phase unwrapping methods," in Interferogram Analysis, D.W.Robinson and G.T.Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 195.

Rosenfeld, D. P.

Servin, M.

Takeda, M.

White, A. D.

Womack, K. H.

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Wyant, J. C.

K. Creath and J. C. Wyant, Optical Shop Testing (Wiley, 1992), Chap. 16.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

J. A. Quiroga, J. A. Gómez-Pedrero, and A. García-Botella, "Algorithm for fringe pattern normalization," Opt. Commun. 197, 43-51 (2001).
[CrossRef]

Opt. Eng. (1)

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Other (2)

K. Creath and J. C. Wyant, Optical Shop Testing (Wiley, 1992), Chap. 16.

D. W. Robinson, "Phase unwrapping methods," in Interferogram Analysis, D.W.Robinson and G.T.Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 195.

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Figures (5)

Fig. 1
Fig. 1

Interferogram with (a) closed fringes, (b) cosine of a selected plane and the result of the multiplication of the original interferogram with (c) the cosine and (d) the sine of the chosen plane.

Fig. 2
Fig. 2

Low-pass filtered intensities corresponding to the multiplication of the interferogram with (a) the cosine and (b) the sine. (c) Normalized modulation and (d) its binarized version with a threshold value of 0.5.

Fig. 3
Fig. 3

Localization of (a) square window and (b) the phase found. (c) Overlapping of the second window and (d) the phase found covering the two windows.

Fig. 4
Fig. 4

(a) Partial estimation of the phase with spatial frequencies ( φ x and φ y ) higher than 0.12 π . (b) Phase recovered over the complete field.

Fig. 5
Fig. 5

(a) Noisy fringe pattern with variations in contrast. (b) Estimated phase, (c) its wrapped version, and (d) wrapped noiseless phase.

Equations (13)

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I ( x , y ) = I b ( x , y ) + I m ( x , y ) cos [ ϕ ( x , y ) ] ,
I N ( x , y ) = cos [ ϕ ( x , y ) ] .
I C ( x , y ) = I N ( x , y ) cos [ φ ( x , y ) ] = 1 2 { cos [ ϕ ( x , y ) φ ( x , y ) ] + cos [ ϕ ( x , y ) + φ ( x , y ) ] } ,
I S ( x , y ) = I N ( x , y ) sin [ φ ( x , y ) ] = 1 2 { sin [ ϕ ( x , y ) φ ( x , y ) ] + sin [ ϕ ( x , y ) + φ ( x , y ) ] } .
ψ W ( x , y ) = arctan [ I ˜ S ( x , y ) I ˜ C ( x , y ) ] .
ϕ W ( x , y ) = arctan { sin [ ψ W ( x , y ) + φ ( x , y ) ] cos [ ψ W ( x , y ) + φ ( x , y ) ] } .
I M ( x , y ) = [ I ˜ S ( x , y ) ] 2 + [ I ˜ C ( x , y ) ] 2 .
W = ( ξ , η ) N x , y [ ϕ ( ξ , η ) φ ( ξ , η ) ] 2 p ( ξ , η ) ,
φ ( ξ , η ) = φ 0 + φ x ξ + φ y η .
ϕ ( x , y ) = { ϕ 1 ( x , y ) i f p 1 ( x , y ) = 1   and   p 2 ( x , y ) = 0 ϕ 2 ( x , y ) i f p 1 ( x , y ) = 0   and   p 2 ( x , y ) = 1 ϕ 1 ( x , y ) + ϕ 2 ( x , y ) + k 2 i f p 1 ( x , y ) = 1   and   p 2 ( x , y ) = 1 } ,
k = [ ϕ 1 ( x , y ) ϕ 2 ( x , y ) ] p 1 ( x , y ) p 2 ( x , y ) p 1 ( x , y ) p 2 ( x , y ) .
p 1 ( x , y ) = { 1 if   p 1 ( x , y ) = 1 or   p 2 ( x , y ) = 1 0 otherwise } .
| φ y | t or   | φ x | t .

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