Abstract

We propose a non-iterative approach to extract the unknown phase shift in phase shifting interferometry without the assumption of equal distribution of measured phase in [0,2π]. According to the histogram of the phase difference between two adjacent frames, the phase shift can be accurately extracted by finding the bin of histogram with the highest frequency. The main factors that influence the accuracy of the proposed method are analyzed and discussed, such as the random noise, the quantization bit of CCD, the number of fringe patterns used and the bin width of histogram. Numerical simulations and optical experiments are also implemented to verify the effectiveness of this method.

© 2010 OSA

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  23. X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  28. J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
    [CrossRef]
  29. Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
    [CrossRef] [PubMed]

2010 (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[CrossRef]

2009 (4)

2008 (2)

2005 (1)

2004 (2)

2003 (1)

J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[CrossRef]

2002 (1)

J. Hayes, “Dynamic interferometry handles vibration,” Laser Focus World 38, 109–116 (2002).

2000 (2)

1998 (2)

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

1997 (2)

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997).
[CrossRef]

1995 (4)

1994 (1)

1993 (1)

1992 (1)

1991 (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[CrossRef]

1990 (1)

1987 (1)

1985 (1)

Apostol, D.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

Brock, N.

Cai, L. Z.

Chen, X.

Cheng, X. C.

Cheng, Y.-Y.

Damian, V.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

Deck, L. L.

Dobroiu, A.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

Dong, G. Y.

Eiju, T.

Farrant, D. I.

Frejlich, J.

Freschi, A. A.

Gao, P.

Geist, E.

Gramaglia, M.

Guo, J. P.

Han, B.

Han, G.-S.

Hao, Q.

Harder, I.

Hariharan, P.

Hayes, J.

Hibino, K.

Hu, Y.

Ishii, Y.

Kim, S.-W.

B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[CrossRef]

G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33(31), 7321–7325 (1994).
[CrossRef] [PubMed]

Kong, B.

B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[CrossRef]

Larkin, K. G.

Li, A. M.

Li, D. L.

X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[CrossRef]

Lindlein, N.

Liu, Q.

Logofatu, P. C.

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

Mantel, K.

Meneses-Fabian, C.

Meng, X. F.

Millerd, J.

Nascov, V.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

North-Morris, M.

Novak, M.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[CrossRef]

Onodera, R.

Oreb, B. F.

Peng, X.

Quiroga, J. A.

J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[CrossRef]

Robledo-Sánchez, C.

Rodriguez-Zurita, G.

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[CrossRef]

Schwider, J.

Servín, M.

J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[CrossRef]

Shen, X. X.

Sun, W. J.

Surrel, Y.

Toto-Arellano, N.-I.

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[CrossRef]

Vázquez-Castillo, J. F.

Wang, Y. R.

Wang, Z.

Wizinowich, P. L.

Wyant, J.

Wyant, J. C.

Xu, X. F.

Yang, X. L.

Yao, B.

Yeazell, J. A.

Zhang, H.

Zhu, Q.

Appl. Opt. (8)

Chin. Phys. Lett. (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[CrossRef]

J. Opt. Soc. Am. A (4)

Laser Focus World (1)

J. Hayes, “Dynamic interferometry handles vibration,” Laser Focus World 38, 109–116 (2002).

Meas. Sci. Technol. (3)

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

Opt. Commun. (2)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[CrossRef]

J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[CrossRef]

Opt. Eng. (1)

B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[CrossRef]

Opt. Express (1)

Opt. Lett. (8)

A. A. Freschi and J. Frejlich, “Adjustable phase control in stabilized interferometry,” Opt. Lett. 20(6), 635–637 (1995).
[CrossRef] [PubMed]

Y. Ishii and R. Onodera, “Phase-extraction algorithm in laser-diode phase-shifting interferometry,” Opt. Lett. 20(18), 1883–1885 (1995).
[CrossRef] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[CrossRef] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[CrossRef] [PubMed]

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34(8), 1210–1212 (2009).
[CrossRef] [PubMed]

P. Gao, B. Yao, N. Lindlein, J. Schwider, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009).
[CrossRef] [PubMed]

Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

(a) The relation between two phases and their phase difference in one dimension and (b) the histogram of the absolute value of the phase difference.

Fig. 2
Fig. 2

The relation between the phase-shift extraction error and SNR. (a) The average phase-shift extraction errors for different SNRs and (b) the phase-shift extraction errors of 50 frames when SNR = 60dB.

Fig. 3
Fig. 3

The relation between the phase-shift extraction error and the bit of CCD. (a) The average phase-shift extraction errors for different bits of CCD and (b) the phase-shift extraction errors of 30 frames when the bit of CCD is 8.

Fig. 4
Fig. 4

The relation between the phase-shift extraction error and the number of fringe patterns used. (a) The average phase-shift extraction errors for different numbers of frames and (b) the phase-shift extraction errors of 50 frames when N = 50.

Fig. 5
Fig. 5

The relation between the average phase-shift extraction error and the bin width.

Fig. 6
Fig. 6

Comparison between our method and Xu’s method. (a) interferogram,(b) the real phase shift between adjacent frames, (c) the histogram of phase difference,(d) the residual phase error of our method,(e) the residual phase error of Xu’s method and (f) the phase-shift extraction errors of our method and Xu’s method.

Fig. 7
Fig. 7

Optical experiment results:(a)-(b) two typical adjacent interferograms,(c)-(d) the phase recovered by the proposed method and ZYGO’s standard method respectively and (e) the difference between (c) and (d).

Equations (9)

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I ( x , y , n ) = A ( x , y ) + B ( x , y ) cos [ φ ( x , y ) + θ ( n ) ]
I max ( x , y ) = A ( x , y ) + B ( x , y )
I min ( x , y ) = A ( x , y ) B ( x , y )
A ( x , y ) = [ I max ( x , y ) + I min ( x , y ) ] / 2
B ( x , y ) = [ I max ( x , y ) I min ( x , y ) ] / 2
Φ 0 , π ( n ) = a r c cos [ I ( n ) A B ]
Φ π , π ( n ) = tan 1 [ cot ( Δ θ ( n ) ) I ( n ) A sin ( Δ θ ( n ) ) ( I ( n 1 ) A ) ]
Δ Φ 0 , π ( n ) = Φ 0 , π ( n ) Φ 0 , π ( n 1 )
| Δ Φ 0 , π ( x , y , n ) | = Δ θ ( n ) + η ( x , y , n )

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