Abstract

From three interferograms with unknown phase shifts, an innovative phase retrieval approach based on the normalized difference maps is proposed. Using the subtraction operation between interferograms, two difference maps without background can be achieved. To eliminate the amplitude inequality of difference maps, normalization process is employed so that two normalized difference maps are obtained. Finally, combining two normalized difference maps and two-step phase retrieval algorithm, the measured phase with high precision can be retrieved rapidly. Comparing with the conventional two-step phase retrieval algorithm with high-pass filtering, the accuracy and processing time of the proposed approach are greatly improved. Importantly, when the phase shift is close to π, almost all two-step algorithms become invalid, but the proposed approach still performs well. That is, the proposed normalized difference maps approach is suitable for the phase retrieval with arbitrary phase shifts.

© 2014 Optical Society of America

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References

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2013

2012

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

2002

U. Schnars, W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Catal. Sci. Tech. 13, R85–R101 (2002).

1992

T. M. Kreis, W. P. O. Juptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).

1986

Belenguer, T.

Cai, L. Z.

Cai, L.-Z.

L.-Z. Cai, Y.-R. Wang, X.-F. Meng, X. Peng, “Optical experimental verification of two-step generalized phase-shifting interferometry,” Acta Phys. Sin. 58(3), 1668–1674 (2009).

Cao, S.

H. Du, H. Zhao, B. Li, J. Zhao, S. Cao, “Three frames phase-shifting shadow moiré using arbitrary unknown phase steps,” Meas. Sci. Technol. 23(10), 105201 (2012).
[CrossRef]

Carazo, J. M.

Chen, L.

Deng, J.

Du, H.

H. Du, H. Zhao, B. Li, J. Zhao, S. Cao, “Three frames phase-shifting shadow moiré using arbitrary unknown phase steps,” Meas. Sci. Technol. 23(10), 105201 (2012).
[CrossRef]

Estrada, J. C.

Guo, H.

Guo, J. P.

Han, B.

Juptner, W. P. O.

U. Schnars, W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Catal. Sci. Tech. 13, R85–R101 (2002).

T. M. Kreis, W. P. O. Juptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).

Kreis, T.

Kreis, T. M.

T. M. Kreis, W. P. O. Juptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).

Li, A. M.

Li, B.

H. Du, H. Zhao, B. Li, J. Zhao, S. Cao, “Three frames phase-shifting shadow moiré using arbitrary unknown phase steps,” Meas. Sci. Technol. 23(10), 105201 (2012).
[CrossRef]

Lu, X.

Meng, X. F.

Meng, X.-F.

L.-Z. Cai, Y.-R. Wang, X.-F. Meng, X. Peng, “Optical experimental verification of two-step generalized phase-shifting interferometry,” Acta Phys. Sin. 58(3), 1668–1674 (2009).

Peng, X.

L.-Z. Cai, Y.-R. Wang, X.-F. Meng, X. Peng, “Optical experimental verification of two-step generalized phase-shifting interferometry,” Acta Phys. Sin. 58(3), 1668–1674 (2009).

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, 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]

Quan, C.

Quiroga, J. A.

Schnars, U.

U. Schnars, W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Catal. Sci. Tech. 13, R85–R101 (2002).

Servin, M.

Servín, M.

J. Vargas, J. A. Quiroga, T. Belenguer, M. Servín, J. C. Estrada, “Two-step self-tuning phase-shifting interferometry,” Opt. Express 19(2), 638–648 (2011).
[CrossRef] [PubMed]

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

Sorzano, C. O. S.

Tay, C. J.

Vargas, J.

Wang, H.

Wang, Y. R.

Wang, Y.-R.

L.-Z. Cai, Y.-R. Wang, X.-F. Meng, X. Peng, “Optical experimental verification of two-step generalized phase-shifting interferometry,” Acta Phys. Sin. 58(3), 1668–1674 (2009).

Wang, Z.

Zhang, D.

Zhang, F.

Zhang, Z.

Zhao, H.

H. Du, H. Zhao, B. Li, J. Zhao, S. Cao, “Three frames phase-shifting shadow moiré using arbitrary unknown phase steps,” Meas. Sci. Technol. 23(10), 105201 (2012).
[CrossRef]

Zhao, J.

H. Du, H. Zhao, B. Li, J. Zhao, S. Cao, “Three frames phase-shifting shadow moiré using arbitrary unknown phase steps,” Meas. Sci. Technol. 23(10), 105201 (2012).
[CrossRef]

Zhong, L.

Acta Phys. Sin.

L.-Z. Cai, Y.-R. Wang, X.-F. Meng, X. Peng, “Optical experimental verification of two-step generalized phase-shifting interferometry,” Acta Phys. Sin. 58(3), 1668–1674 (2009).

Appl. Opt.

Catal. Sci. Tech.

U. Schnars, W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Catal. Sci. Tech. 13, R85–R101 (2002).

J. Opt. Soc. Am. A

Meas. Sci. Technol.

H. Du, H. Zhao, B. Li, J. Zhao, S. Cao, “Three frames phase-shifting shadow moiré using arbitrary unknown phase steps,” Meas. Sci. Technol. 23(10), 105201 (2012).
[CrossRef]

Opt. Commun.

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

Opt. Express

Opt. Lett.

Proc. SPIE

T. M. Kreis, W. P. O. Juptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).

Other

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC Press, 2005).

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Figures (4)

Fig. 1
Fig. 1

Simulation results. (a) one of the three simulated interferograms; (b) the theoretical phase; (c) the phase difference between the theoretical phase and the unwrapped phase obtained by GS3 approach; (d) phase difference between the theoretical phase and the unwrapped phase obtained with GS2 approach; (e) phase difference between the theoretical phase and the unwrapped phase obtained with GSm approach.

Fig. 2
Fig. 2

Relationship between the RMSE of phase retrieval with GS3, GS2 approaches and the phase shifts, respectively.

Fig. 3
Fig. 3

Experimental results. (a) one of three experimental phase-shifting interferograms; (b) the wrapped phase obtained with GS3 approach; (c) the wrapped phase obtained with GS2 approach; (d) the reference wrapped phase obtained by AIA.

Fig. 4
Fig. 4

Unwrapped phases obtained with different approaches while the phase-shift is larger than 2.5rad. (a) GS3 approach; (b) GS2 approach; (c) AIA.

Tables (1)

Tables Icon

Table 1 RMSE, Processing Time with Different Approaches

Equations (11)

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I n , k = a k + b k cos [ φ k + θ n ]
S 1 , k = I 1 , k I 2 , k = 2 b k sin θ 2 2 sin ( Φ k )
S 2 , k = I 1 , k I 3 , k = 2 b k sin θ 3 2 sin ( Φ k + Δ )
u ˜ = u / u , u = u / u
S i , k , S i , k = k = 1 K S i , k S i , k
S ˜ 1 , k = b k sin ( Φ k ) / k = 1 K b k 2 sin 2 ( Φ k )
S ˜ 2 , k = b k sin ( Φ k + Δ ) / k = 1 K b k 2 sin 2 ( Φ k + Δ )
k = 1 K b k 2 sin 2 ( Φ k ) k = 1 K b k 2 sin 2 ( Φ k + Δ ) = k = 1 K b k 2 [ sin ( Φ k ) sin ( Φ k + Δ ) ] [ sin ( Φ k ) + sin ( Φ k + Δ ) ] = 4 k = 1 K b k 2 sin ( 2 Φ k + Δ 2 ) cos ( 2 Φ k + Δ 2 ) sin ( Δ 2 ) cos ( Δ 2 ) = k = 1 K b k 2 sin ( 2 Φ k + Δ ) sin ( Δ ) 0
k = 1 K b k 2 sin 2 ( Φ k ) k = 1 K b k 2 sin 2 ( Φ k + Δ )
S ^ 1 , k = b k ' sin ( Φ k ) = b k ' cos ( Φ k + π / 2 ) = b k ' ' cos ( Φ k ' )
S ^ 2 , k = b k ' sin ( Φ k + Δ ) = b k ' cos ( Φ k + π / 2 + Δ ) = b k ' ' cos ( Φ k ' + Δ )

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