Abstract

A simple algorithm for blind extraction of phase shifts is proposed for generalized phase-shifting interferometry from only three interferograms. Based on the statistical property of the object wave, the algorithm calculates approximately the involved phase shifts as initial values. The extraction is further improved by an iterative method, considering the fact that the closer the phase shifts approach their real values, the more uniform the reconstructed reference wave will become. The feasibility of this algorithm is demonstrated by both simulation and experiment.

© 2009 Optical Society of America

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References

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2009

2008

2007

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124 (2007).
[CrossRef]

2006

2004

2003

2002

2000

S. Lai, B. King, and M. A. Neifeld, Opt. Commun. 173, 155 (2000).
[CrossRef]

1995

1983

Awatsuji, Y.

Burow, R.

Cai, L. Z.

Charrière, F.

Chen, L. J.

Colomb, T.

Creath, K.

Cuche, E.

Depeursinge, C.

Dong, G. Y.

Elssner, K. E.

Emery, Y.

Fujii, A.

Gao, P.

Grzanna, J.

Guo, C.

Guo, J. P.

Han, B.

Han, J. H.

King, B.

S. Lai, B. King, and M. A. Neifeld, Opt. Commun. 173, 155 (2000).
[CrossRef]

Kubota, T.

Kühn, J.

Lai, S.

S. Lai, B. King, and M. A. Neifeld, Opt. Commun. 173, 155 (2000).
[CrossRef]

Lei, M.

Li, A. M.

Liao, J.

Liu, Q.

Marquet, P.

Matoba, O.

Meng, X. F.

Merkel, K.

Montfort, F.

Neifeld, M. A.

S. Lai, B. King, and M. A. Neifeld, Opt. Commun. 173, 155 (2000).
[CrossRef]

Peng, X.

Schmit, J.

Schwider, J.

Shen, X. X.

Spolaczyk, R.

Wang, H.

Wang, Y. L.

Wang, Y. R.

Wang, Z.

Weible, K.

Xu, X. F.

Yang, X. L.

Yao, B. L.

Zhang, H.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124 (2007).
[CrossRef]

Zhang, L.

Zhu, Y. Y.

Appl. Opt.

Appl. Phys. Lett.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124 (2007).
[CrossRef]

Opt. Commun.

S. Lai, B. King, and M. A. Neifeld, Opt. Commun. 173, 155 (2000).
[CrossRef]

Opt. Lett.

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Figures (2)

Fig. 1
Fig. 1

Simulation results. (a), (b) Simulated phase (wrapped phase map) and amplitude distribution of the object wave; (c), (d) reconstruction errors for the amplitude and phase distribution; (e), (f) reference wave amplitude reconstructed with extracted phase shifts α 1 = 1.5712 and α 2 = 3.1405 rad, and with wrong extracted phase shifts α 1 = 1.25 and α 2 = 3.14 rad.

Fig. 2
Fig. 2

Experimental results. (a) Phase-shifting interferogram I 1 ; (b) reconstructed phase distribution φ of the tested object (unit: radian); (c) R reconstructed with extracted phase shifts α 1 = 1.68   rad and α 2 = 3.24   rad ; (d) R reconstructed with arbitrary phase shifts α 1 = 1.5   rad and α 2 = 2.85   rad .

Tables (1)

Tables Icon

Table 1 Iteration Process for Phase-Shift Extraction with Random Initial Values and the Initial Values from Eq. (4) a

Equations (11)

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I 1 ( x , y ) = | O | 2 + | R | 2 + 2 | O | | R | cos ( φ ) ,
I 2 ( x , y ) = | O | 2 + | R | 2 + 2 | O | | R | cos ( φ + α 1 ) ,
I 3 ( x , y ) = | O | 2 + | R | 2 + 2 | O | | R | cos ( φ + α 2 ) .
Q 1 ( x , y ) = I 1 I 3 = 4 | O | | R | sin ( α 2 2 ) sin ( φ + α 2 2 ) ,
Q 2 ( x , y ) = I 2 I 3 = 4 | O | | R | sin ( α 2 α 1 2 ) sin ( φ + α 2 + α 1 2 ) .
U = Q 2 Q 1 V 1 = sin ( α 2 α 1 2 ) sin ( α 2 2 ) cos α 1 2 ,
V = Q 1 Q 2 V 2 = sin ( α 2 2 ) sin ( α 2 α 1 2 ) cos α 1 2 .
α 1 = 2   arccos ( U V ) ,
α 2 = 2   arccot ( V 1 V U V 1 U V ) .
O R = Q 1 [ exp ( i α 1 ) exp ( i α 2 ) ] Q 2 [ 1 exp ( i α 2 ) ] 2 i [ sin   α 2 sin   α 1 + sin ( α 1 α 2 ) ] .
ε ( α 1 , α 2 ) = i , j | R i , j R ¯ | .

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