Abstract

Estimation of phase derivatives is an important task in many interferometric measurements in optical metrology. This Letter introduces a method based on discrete chirp-Fourier transform for accurate and direct estimation of phase derivatives, even in the presence of noise. The method is introduced in the context of the analysis of reconstructed interference fields in digital holographic interferometry. We present simulation and experimental results demonstrating the utility of the proposed method.

© 2009 Optical Society of America

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References

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  1. C. A. Sciammarella and T. Kim, Exp. Mech. 45, 393 (2005).
    [CrossRef]
  2. Q. Kemao, S. H. Soon, and A. Asundi, Appl. Opt. 42, 6504 (2003).
    [CrossRef] [PubMed]
  3. C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
    [CrossRef]
  4. K. Qian, S. H. Soon, and A. Asundi, Opt. Lett. 28, 1657 (2003).
    [CrossRef] [PubMed]
  5. U. Schnars and W. P. O. Juptner, Meas. Sci. Technol. 13, R85 (2002).
    [CrossRef]
  6. X. Xia, IEEE Trans. Signal Process. 48, 3122 (2000).
    [CrossRef]
  7. X. Guo, H. Sun, S. Wang, and G. Liu, IEEE Trans. Signal Process. 50, 3115 (2002).
    [CrossRef]
  8. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, SIAM J. Optim. 9, 112 (1999).
    [CrossRef]
  9. J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, Opt. Lett. 30, 3018 (2005).
    [CrossRef] [PubMed]

2007 (1)

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

2005 (2)

2003 (2)

2002 (2)

U. Schnars and W. P. O. Juptner, Meas. Sci. Technol. 13, R85 (2002).
[CrossRef]

X. Guo, H. Sun, S. Wang, and G. Liu, IEEE Trans. Signal Process. 50, 3115 (2002).
[CrossRef]

2000 (1)

X. Xia, IEEE Trans. Signal Process. 48, 3122 (2000).
[CrossRef]

1999 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, SIAM J. Optim. 9, 112 (1999).
[CrossRef]

Asundi, A.

Guerrero, J. A.

Guo, X.

X. Guo, H. Sun, S. Wang, and G. Liu, IEEE Trans. Signal Process. 50, 3115 (2002).
[CrossRef]

He, X. Y.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, Meas. Sci. Technol. 13, R85 (2002).
[CrossRef]

Kemao, Q.

Kim, T.

C. A. Sciammarella and T. Kim, Exp. Mech. 45, 393 (2005).
[CrossRef]

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, SIAM J. Optim. 9, 112 (1999).
[CrossRef]

Liu, G.

X. Guo, H. Sun, S. Wang, and G. Liu, IEEE Trans. Signal Process. 50, 3115 (2002).
[CrossRef]

Marroquin, J. L.

Qian, K.

Quan, C.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

Quiroga, J. A.

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, SIAM J. Optim. 9, 112 (1999).
[CrossRef]

Rivera, M.

Schnars, U.

U. Schnars and W. P. O. Juptner, Meas. Sci. Technol. 13, R85 (2002).
[CrossRef]

Sciammarella, C. A.

C. A. Sciammarella and T. Kim, Exp. Mech. 45, 393 (2005).
[CrossRef]

Soon, S. H.

Sun, H.

X. Guo, H. Sun, S. Wang, and G. Liu, IEEE Trans. Signal Process. 50, 3115 (2002).
[CrossRef]

Sun, W.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

Tay, C. J.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

Wang, S.

X. Guo, H. Sun, S. Wang, and G. Liu, IEEE Trans. Signal Process. 50, 3115 (2002).
[CrossRef]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, SIAM J. Optim. 9, 112 (1999).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, SIAM J. Optim. 9, 112 (1999).
[CrossRef]

Xia, X.

X. Xia, IEEE Trans. Signal Process. 48, 3122 (2000).
[CrossRef]

Appl. Opt. (1)

Exp. Mech. (1)

C. A. Sciammarella and T. Kim, Exp. Mech. 45, 393 (2005).
[CrossRef]

IEEE Trans. Signal Process. (2)

X. Xia, IEEE Trans. Signal Process. 48, 3122 (2000).
[CrossRef]

X. Guo, H. Sun, S. Wang, and G. Liu, IEEE Trans. Signal Process. 50, 3115 (2002).
[CrossRef]

Meas. Sci. Technol. (1)

U. Schnars and W. P. O. Juptner, Meas. Sci. Technol. 13, R85 (2002).
[CrossRef]

Opt. Commun. (1)

C. J. Tay, C. Quan, W. Sun, and X. Y. He, Opt. Commun. 280, 327 (2007).
[CrossRef]

Opt. Lett. (2)

SIAM J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, SIAM J. Optim. 9, 112 (1999).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

(a) Plot of the real part of g ( x ) ; (b) magnitude of the DCFT of g ( x ) ; (c) estimated phase derivative of g ( x ) , at SNR of 0 dB , after optimization; (d) error in phase-derivative estimation.

Fig. 2
Fig. 2

(a) Comparison of error plots when simple windowing and overlapping windowing concepts are used, (b) 50% overlapping windowing scheme.

Fig. 3
Fig. 3

(a) Simulated fringe pattern at SNR of 10 dB ( 256 × 256 ) , (b) estimated phase derivative along the y axis using the proposed DCFT method, (c) cosine of the phase derivative along the y axis (for illustration), (d) phase obtained by the numerical integration of the derivative map shown in (b), (e) phase map after offset correction, (f) error in phase estimation.

Fig. 4
Fig. 4

(a) Recorded fringe pattern corresponding to the central loading of a circularly clamped object in a DHI experiment, (b) phase-derivative map along the y axis, (c) cosine of the phase derivative along the y axis (for illustration), (d) phase distribution obtained by numerical integration of the derivative map shown in (b), (e) phase map after offset correction.

Tables (1)

Tables Icon

Table 1 Performance Evaluation of DCFT Method ( N w = 8 ) with Respect to Noise

Equations (8)

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A ( x , y ) = b ( x , y ) exp [ j Δ ϕ ( x , y ) ] + η ( x , y ) ,
A y i = b y i exp ( j Δ ϕ y i ) + η y i ,
g ( x ) = b ( x ) exp [ j ( a 0 + a 1 x + a 2 x 2 ) ] + η ( x ) .
Δ ϕ x = a 1 + 2 a 2 x .
G ( k 1 , k 2 ) = x = 0 N s 1 g ( x ) exp [ j ( α k 1 x + β k 2 x 2 ) ] ,
G ( k 1 , k 2 ) = FFT { g ( x ) exp [ j β k 2 x 2 ] } ,
α k 1 = 2 π N s k 1 , 0 k 1 N s 1 ,
β k 2 = 2 π N s 2 k 2 , N s + 1 k 2 N s 1 ,

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