Abstract

A two-step self-tuning phase-shifting method is presented. The phase-step between the two interferograms is not known when the experiment is performed. Our demodulating method finds, in a robust way, this unknown phase-step. Once the phase-step is estimated we proceed to phase demodulate the interferograms. Moreover our method only requires the fringe patterns to have a constant unknown phase-shift between them. As a consequence, this technique can be used to demodulate open and closed-fringed patterns without phase-sign ambiguity. The method may be regarded as a self-tuning quadrature filter, which determines the phase-shift between the two fringe patterns and finally estimates the demodulated phase map. The proposed technique has been tested with simulated and real interferograms obtaining satisfactory results.

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References

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  1. D. Malacara, M. Servín, and Z. Malacara, “Interferogram analisis for optical testing”, Cambridge University Press, (2004), Marcel Dekker, Inc, (1998)
  2. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
    [CrossRef] [PubMed]
  3. F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).
  4. S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42(12), 3524–3531 (2003).
    [CrossRef]
  5. Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008).
    [CrossRef]
  6. X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007).
    [CrossRef]
  7. 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]
  8. M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009).
    [CrossRef] [PubMed]
  9. J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
    [CrossRef]
  10. 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]

2009

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
[CrossRef] [PubMed]

M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009).
[CrossRef] [PubMed]

2008

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]

Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008).
[CrossRef]

2007

X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007).
[CrossRef]

2004

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]

2003

S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42(12), 3524–3531 (2003).
[CrossRef]

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

1988

F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

Almazán-Cuéllar, S.

S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42(12), 3524–3531 (2003).
[CrossRef]

Cai, L. Z.

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]

X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007).
[CrossRef]

Cheng, X. C.

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]

Dong, G. Y.

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]

X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007).
[CrossRef]

Estrada, J. C.

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
[CrossRef] [PubMed]

M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009).
[CrossRef] [PubMed]

Han, B.

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]

Kerr, D.

F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

Liu, L.

Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008).
[CrossRef]

Luana, Z.

Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008).
[CrossRef]

Malacara-Hernandez, D.

S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42(12), 3524–3531 (2003).
[CrossRef]

Mendoza-Santoyo, F.

F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

Meng, X. F.

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]

X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007).
[CrossRef]

Quiroga, J. A.

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
[CrossRef] [PubMed]

M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009).
[CrossRef] [PubMed]

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

Servin, M.

M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009).
[CrossRef] [PubMed]

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
[CrossRef] [PubMed]

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.

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]

X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007).
[CrossRef]

Sun, J.

Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008).
[CrossRef]

Sun, W. J.

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]

Tyrer, J. R.

F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

Wang, Y. R.

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]

Wang, Z.

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]

Wanga, Y. R.

X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007).
[CrossRef]

Xu, X. F.

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]

X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007).
[CrossRef]

Zhang, H.

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]

X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007).
[CrossRef]

Zhu, Y.

Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008).
[CrossRef]

Appl. Opt.

F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

Opt. Commun.

X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007).
[CrossRef]

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

Opt. Eng.

S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42(12), 3524–3531 (2003).
[CrossRef]

Opt. Express

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
[CrossRef] [PubMed]

M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009).
[CrossRef] [PubMed]

Opt. Lett.

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. 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]

Optik (Stuttg.)

Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008).
[CrossRef]

Other

D. Malacara, M. Servín, and Z. Malacara, “Interferogram analisis for optical testing”, Cambridge University Press, (2004), Marcel Dekker, Inc, (1998)

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Figures (9)

Fig. 1
Fig. 1

Scheme of the detuning error that appears when ω f ω 0 . (a) In this case ω f coincides with ω 0 and there is non detuning error. (b) In this case ω f ω 0 and there is non-detuning

Fig. 2
Fig. 2

Two fringe patterns used in the first simulation

Fig. 3
Fig. 3

Plot between the standard deviation of the modulation map σ [ | g ˜ ( 0 ) | 2 ] and the different temporal frequencies ω f obtained in the first simulation

Fig. 4
Fig. 4

Obtained wrapped phase in the first simulation

Fig. 5
Fig. 5

Reconstructed phase (a) and computed error between actual and computed phases (b)

Fig. 6
Fig. 6

(a), (b) Real Interferograms (c), (d) Resultant interferograms after the normalization process.

Fig. 7
Fig. 7

Computed wrapped phase obtained with the two-step proposed method (b) Wrapped phase obtained by the least-squares demodulation method using 10 interferograms

Fig. 8
Fig. 8

Reconstructed phases with the proposed method (a) and with the least-squares method (b).

Fig. 9
Fig. 9

Difference between the reconstructed phases shown in Fig. 12 using the proposed method and the least-squares method

Tables (1)

Tables Icon

Table 1 Obtained temporal frequencies ω ^ 0 , absolute difference between the actual and computed temporal frequencies | ω 0 ω ^ 0 | , rms errors obtained between the actual and computed phase maps (with the phase maps retrieved by interferograms without noise and from the computed temporal frequencies) and processing times for different signal to noise ratios

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

g ( x , y , t ) = a ( x , y ) + b ( x , y ) cos [ Φ ( x , y ) + ω 0 t ]
g ( t ) = a + b 2 [ e i [ Φ ( x , y ) + ω 0 t ] + e i [ Φ ( x , y ) + ω 0 t ] ]
G ( ω ) = a δ ( ω ) + b 2 [ δ ( ω + ω 0 ) e i Φ + δ ( ω ω 0 ) e i Φ ]
G ˜ ( ω ) = b 2 H ( ω 0 ) δ ( ω + ω 0 ) e i Φ
g ˜ ( t ) = b 2 H ( ω 0 ) e i Φ e i ω o t
Φ = tan 1 ( Im { g ˜ ( 0 ) } Re { g ˜ ( 0 ) } )
| g ˜ ( 0 ) | = b 2 | H ( ω 0 ) |
g ( x , y , t ) = cos [ Φ ( x , y ) + ω 0 t ]
G ˜ ( ω ) = 1 2 [ H ( ω 0 ) δ ( ω + ω 0 ) e i Φ + H ( ω 0 ) δ ( ω ω 0 ) e i Φ ]
g ˜ ( t ) = 1 2 [ H ( ω 0 ) e i Φ e i ω o t + H ( ω 0 ) e i Φ e i ω o t ]
| g ˜ ( 0 ) | 2 = 1 4 [ A ( ω 0 ) + B ( ω 0 ) + ( C ( ω 0 ) cos [ 2 Φ ] D ( ω 0 ) sin [ 2 Φ ] ) ]
σ [ | g ˜ ( 0 ) | 2 ] = 1 4 σ [ C ( ω 0 ) cos [ 2 Φ ] D ( ω 0 ) sin [ 2 Φ ] ]
σ [ | g ˜ ( x , y , 0 ) | 2 ] = 1 M x y ( | g ˜ ( x , y , 0 ) | 2 g ˜ m ) 2
H ( ω ) = sin ( ω ω f 2 ) e i ω / 2
h ( t ) = 1 2 π H ( ω ) e i ω t d ω = 1 2 i [ e i ω f / 2 δ ( t + 1 ) e i ω f / 2 δ ( t ) ]

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