Abstract

Phase extraction from phase-shifting fringe patterns with unknown phase shift values is a valuable but challenging task, especially when there are only two frames of fringes. In this paper, a phase demodulation method based on the spatial-temporal fringes (STF) method is proposed, where two phase shift fringes with linear carrier are fused into one STF image, and then the measured phase can be extracted from its frequency spectrum. The algorithm is deduced by extending the traditional STF theory with at least three frames of fringes to the two frames case. In the simulations, its performance is compared with the classical Fourier Transform method, and the different carrier and phase step conditions are analyzed where the accuracy can be ensured in most cases. The algorithm is also validated by the experiment, where the reliable result can be given even if the phase shift changes within a wide range.

© 2016 Optical Society of America

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References

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2013 (1)

B. Li, L. Chen, C. Xu, and J. Li, “The simultaneous suppression of phase shift error and harmonics in the phase shifting interferometry using carrier squeezing interferometry,” Opt. Commun. 296(6), 17–24 (2013).
[Crossref]

2012 (1)

2011 (3)

2009 (4)

2008 (2)

2004 (1)

2003 (1)

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

2001 (1)

1997 (1)

1996 (1)

T. W. Ng, “The one-step phase-shifting technique for wave-front interferometry,” J. Mod. Opt. 43(10), 2129–2138 (1996).
[Crossref]

1995 (1)

P. J. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. 12(2), 354–365 (1995).
[Crossref]

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]

1987 (1)

1985 (1)

1983 (1)

1982 (1)

Almazan, S.

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

Belenguer, T.

Burow, R.

Chai, L.

Chen, L.

B. Li, L. Chen, C. Xu, and J. Li, “The simultaneous suppression of phase shift error and harmonics in the phase shifting interferometry using carrier squeezing interferometry,” Opt. Commun. 296(6), 17–24 (2013).
[Crossref]

B. Li, L. Chen, W. Tuya, S. Ma, and R. Zhu, “Carrier squeezing interferometry: suppressing phase errors from the inaccurate phase shift,” Opt. Lett. 36(6), 996–998 (2011).
[Crossref] [PubMed]

Cheng, Y. Y.

Cywiak, M.

Dai, M.

de Groot, P. J.

P. J. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. 12(2), 354–365 (1995).
[Crossref]

Deng, J.

Elssner, K. E.

Estrada, J. C.

Gao, P.

Geist, E.

Grzanna, J.

Han, B.

Harder, I.

Heppner, J.

Ina, H.

Jin, W.

Kobayashi, S.

Li, B.

B. Li, L. Chen, C. Xu, and J. Li, “The simultaneous suppression of phase shift error and harmonics in the phase shifting interferometry using carrier squeezing interferometry,” Opt. Commun. 296(6), 17–24 (2013).
[Crossref]

B. Li, L. Chen, W. Tuya, S. Ma, and R. Zhu, “Carrier squeezing interferometry: suppressing phase errors from the inaccurate phase shift,” Opt. Lett. 36(6), 996–998 (2011).
[Crossref] [PubMed]

Li, J.

B. Li, L. Chen, C. Xu, and J. Li, “The simultaneous suppression of phase shift error and harmonics in the phase shifting interferometry using carrier squeezing interferometry,” Opt. Commun. 296(6), 17–24 (2013).
[Crossref]

Lindlein, N.

Lu, X.

Ma, S.

Malacara, D.

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

Malacara-Hernandez, D.

Mantel, K.

Massig, J. H.

Merkel, K.

Ng, T. W.

T. W. Ng, “The one-step phase-shifting technique for wave-front interferometry,” J. Mod. Opt. 43(10), 2129–2138 (1996).
[Crossref]

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]

Peng, H.

Phillion, D. W.

Quiroga, J. A.

Roddier, C.

Roddier, F.

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.

Servin, M.

Servín, M.

Spolaczyk, R.

Takeda, M.

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]

Tuya, W.

Vargas, J.

Wang, H.

Wang, Y.

Wang, Z.

Wyant, J. C.

Xu, C.

B. Li, L. Chen, C. Xu, and J. Li, “The simultaneous suppression of phase shift error and harmonics in the phase shifting interferometry using carrier squeezing interferometry,” Opt. Commun. 296(6), 17–24 (2013).
[Crossref]

Xu, J.

Xu, Q.

Yao, B.

Zhang, D.

Zhang, F.

Zhong, L.

Zhu, R.

Appl. Opt. (5)

J. Mod. Opt. (1)

T. W. Ng, “The one-step phase-shifting technique for wave-front interferometry,” J. Mod. Opt. 43(10), 2129–2138 (1996).
[Crossref]

J. Opt. Soc. Am. (2)

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]

B. Li, L. Chen, C. Xu, and J. Li, “The simultaneous suppression of phase shift error and harmonics in the phase shifting interferometry using carrier squeezing interferometry,” Opt. Commun. 296(6), 17–24 (2013).
[Crossref]

Opt. Eng. (1)

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

Opt. Express (6)

Opt. Lett. (5)

Other (1)

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley-VCH, 2014).

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Figures (11)

Fig. 1
Fig. 1

The simulated surface (a) and its fringes (b).

Fig. 2
Fig. 2

The STF image rearranged by the two phase shift fringes.

Fig. 3
Fig. 3

The spectrums of STF when phase shift values are (a) ϕ = π/2 and (b) ϕ = π.

Fig. 4
Fig. 4

(a) The retrieved surface and (b) its error (ERMS value: 1.18 × 10−3 λ).

Fig. 5
Fig. 5

The phase shift fringes captured in the experiment with the phase shift values of (a) 0 and (b) π, and the fused STF image (c).

Fig. 6
Fig. 6

The spectrums of the STF in the experiment when phase-step are (a) π and (b) π/2.

Fig. 7
Fig. 7

The retrieved surface by our algorithm with phase-step of (a) π and (b) π/2. (c) The surface obtained by phase shift method. (d) The deviation between (a) and (b). (e) The deviation between (a) and (c).

Fig. 8
Fig. 8

(a) The retrieved surface by Fourier transform method and (b) its error (ERMS value: 3.66 × 10−3 λ).

Fig. 9
Fig. 9

ERMS values of the retrieved surfaces with different tilt by our method and the Fourier transform method.

Fig. 10
Fig. 10

ERMS values of the retrieved surfaces with a series of small tilt values.

Fig. 11
Fig. 11

The retrieved surface errors of our method with different phase shift value.

Equations (11)

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{ I 0 (x,y)=a(x,y)+b(x,y)cos{ 4π λ [ xtan(θ)+w(x,y) ] } I 1 (x,y)=a(x,y)+b(x,y)cos{ 4π λ [ xtan(θ)+w(x,y) ]+ϕ }
I i (x,y)=a(x,y)+b(x,y)cos[ 2π f 0 x+α( x,y )+ϕ ],(i=0,1)
I'(x',y)=a( x' 2 ,y)+b( x' 2 ,y)cos[ 2π f 0 'x'+α( x' 2 ,y )+ ε ϕ ]
ε ϕ ={ 0 , while x' is odd ϕ , while x' is even
I'(x',y)=a( x' 2 ,y)+ 1 2 b( x' 2 ,y)exp{ j[ 2π f 0 'x'+α( x' 2 ,y ) ] }exp( j ε ϕ ) + 1 2 b( x' 2 ,y)exp{ j[ 2π f 0 'x'+α( x' 2 ,y ) ] }exp( j ε ϕ )
1exp( ±jϕ ) 2 δ( f+ f c )+ 1+exp( ±jϕ ) 2 δ( f )+ 1exp( ±jϕ ) 2 δ( f f c )
[ I'(x',y) ]=A( f )+ 1 4π B( f )S( f )
S( f )= 1 4 π 2 Θ( f )[ 1exp( jϕ ) 2 δ( f+ f c f 0 ' )+ 1+exp( jϕ ) 2 δ( f f 0 ' ) + 1exp( jϕ ) 2 δ( f f c f 0 ' ) ] + 1 4 π 2 Θ ( f )[ 1+exp( jϕ ) 2 δ( f+ f c + f 0 ' )+ 1exp( jϕ ) 2 δ( f+ f 0 ' ) + 1+exp( jϕ ) 2 δ( f f c + f 0 ' ) ]
[ I'(x',y) ] f c f 0 ' = 1 4π [ b( x' 2 ,y) ] 1 4 π 2 Θ ( f )[ 1+exp( jϕ ) 2 δ( f f c + f 0 ' ) ]
I"(x',y)= 1 4π [ 1+exp( jϕ ) 2 ]b( x' 2 ,y)exp{ j[ 2π( f c f 0 ' )x'α( x' 2 ,y ) ] }
α( x' 2 ,y )= tan 1 { im[ I"(x',y) ] real[ I"(x',y) ] }+2π( f c f 0 ' )x'

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