Abstract

A general spatial phase-shifting (GSPS) interferometry method is proposed to achieve phase retrieval from one-frame spatial carrier frequency interferogram. By optimizing the internal signal retrieving function of the spatial phase-shifting (SPS) method, the accuracy, anti-noise ability and speed of phase retrieval can be significantly improved, meanwhile the corresponding local calculation property is reserved. Especially, in the case that the ratio of the spatial carrier to the phase variation rate are small, the proposed method reveals obvious advantage in the accuracy improvement relative to the conventional SPS methods, so the more details of measured sample can be effectively reserved through introducing smaller spatial carrier frequency, and this will facilitate its application in interference microscopy. The principle analysis, numerical simulation and experimental result are employed to verify the performance of the proposed GSPS method.

© 2017 Optical Society of America

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References

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2014 (2)

2013 (1)

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

2012 (3)

2011 (3)

2010 (1)

2009 (1)

2008 (2)

2007 (2)

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogram with linear carrier,” Opt. Commun. 271(1), 59–64 (2007).
[Crossref]

H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principle and applications,” Appl. Opt. 46(7), 1057–1065 (2007).
[Crossref] [PubMed]

2005 (1)

2004 (2)

2001 (1)

1999 (1)

1997 (1)

1995 (2)

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Interferogram Analysis Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[Crossref]

M. Servin and F. J. Cuevas, “A Novel Technique for Spatial Phase-shifting Interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[Crossref]

1987 (1)

1986 (1)

1985 (1)

1983 (2)

1982 (1)

Abdulhalim, I.

Awatsuji, Y.

Bachor, H. A.

Barnes, T. H.

Bone, D. J.

Bryanston-Cross, P. J.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Interferogram Analysis Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[Crossref]

Carazo, J. M.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Chan, P. H.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Interferogram Analysis Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[Crossref]

Chen, M.

Cuevas, F. J.

M. Servin and F. J. Cuevas, “A Novel Technique for Spatial Phase-shifting Interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[Crossref]

Debnath, S. K.

Ding, H.

Do, M.

Du, Y.

Estrada, J. C.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. M. Padilla, M. Servin, and J. C. Estrada, “Synchronous phase-demodulation and harmonic rejection of 9-step pixelated dynamic interferograms,” Opt. Express 20(11), 11734–11739 (2012).
[Crossref] [PubMed]

Feng, G.

Ferrari, J. A.

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogram with linear carrier,” Opt. Commun. 271(1), 59–64 (2007).
[Crossref]

Frins, E. M.

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogram with linear carrier,” Opt. Commun. 271(1), 59–64 (2007).
[Crossref]

Gao, W.

Girshovitz, P.

Guo, H.

Han, B.

Heppner, J.

Huyen, N. T. T.

Ichihashi, Y.

Ina, H.

Ito, T.

Kakue, T.

Kemao, Q.

Kobayashi, S.

Kubota, T.

Kujawinska, M.

M. Kujawinska and J. Wojciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” in Proceedings of SPIE (1991), pp. 61–67.
[Crossref]

Li, H.

Liu, J. B.

Loi, H. S.

Macy, W. W.

Massig, J. H.

Masuda, N.

Matoba, O.

Mutoh, K.

Nishio, K.

Nugent, K. A.

Padilla, J. M.

Park, Y.

Parker, S. C.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Interferogram Analysis Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[Crossref]

Patel, S.

Peng, H.

Pham, H.

Popescu, G.

Roddier, C.

Roddier, F.

Ronney, P. D.

Safrani, A.

Sandeman, R. J.

Servin, M.

Shaked, N. T.

Shimobaba, T.

Sobh, N.

Sorzano, C. O. S.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Tahara, T.

Takada, N.

Takeda, M.

Tan, S. M.

Ura, S.

Vargas, J.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Y. Du, G. Feng, H. Li, J. Vargas, and S. Zhou, “Spatial carrier phase-shifting algorithm based on principal component analysis method,” Opt. Express 20(15), 16471–16479 (2012).
[Crossref]

Wang, H.

Wang, Z.

Watkins, L. R.

Weng, J.

Wojciak, J.

M. Kujawinska and J. Wojciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” in Proceedings of SPIE (1991), pp. 61–67.
[Crossref]

Xu, J.

Xu, Q.

Yang, Q.

Yonesaka, R.

Zhong, J.

Zhou, S.

Appl. Opt. (11)

W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983).
[Crossref] [PubMed]

K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24(18), 3101–3105 (1985).
[Crossref] [PubMed]

D. J. Bone, H. A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25(10), 1653–1660 (1986).
[Crossref] [PubMed]

C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26(9), 1668–1673 (1987).
[Crossref] [PubMed]

J. B. Liu and P. D. Ronney, “Modified Fourier transform method for interferogram fringe pattern analysis,” Appl. Opt. 36(25), 6231–6241 (1997).
[Crossref] [PubMed]

J. H. Massig and J. Heppner, “Fringe-pattern analysis with high accuracy by use of the fourier-transform method: theory and experimental tests,” Appl. Opt. 40(13), 2081–2088 (2001).
[Crossref] [PubMed]

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983).
[Crossref] [PubMed]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004).
[Crossref] [PubMed]

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47(29), 5408–5419 (2008).
[Crossref] [PubMed]

H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principle and applications,” Appl. Opt. 46(7), 1057–1065 (2007).
[Crossref] [PubMed]

J. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. 47(29), 5446–5453 (2008).
[Crossref] [PubMed]

Biomed. Opt. Express (2)

J. Mod. Opt. (1)

M. Servin and F. J. Cuevas, “A Novel Technique for Spatial Phase-shifting Interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogram with linear carrier,” Opt. Commun. 271(1), 59–64 (2007).
[Crossref]

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Opt. Express (4)

Opt. Lasers Eng. (1)

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Interferogram Analysis Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[Crossref]

Opt. Lett. (7)

Other (2)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, Second Edition (CRC, 2005).

M. Kujawinska and J. Wojciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” in Proceedings of SPIE (1991), pp. 61–67.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Frequency spectrum distribution of one-frame carrier interferogram, in which the filtering function cannot totally eliminate the 0 and −1 peaks, so the retrieved signal will contain the WCE induced by the 0 peak and −1 peak).

Fig. 2
Fig. 2

(a) One-frame simulated carrier interference pattern; (b) the theoretical phase distribution of (a); the plane view of the achieved phase by using (c) GSPS-AIA method with RMSE of 0.029 rad, (d) SPS-AIA method with RMSE of 0.048 rad; (e) SPS-PCA method with RMSE of 0.048 rad; the difference between the theoretical phase and the phase achieved with (f) GSPS-AIA method; (g) SPS-AIA method; (h) SPS-PCA method; The plane view of error distribution with (i) GSPS-AIA method; (j) SPS-AIA method;(k) SPS-PCA method.

Fig. 3
Fig. 3

(a) Variation of RMSE of the retrieved phase (a) in different spatial carrier; (b) in different SNR.

Fig. 4
Fig. 4

Experimental setup of the Mach-Zhnder type phase-shifting system.

Fig. 5
Fig. 5

(a) One frame experimental carrier interferogram of a subovate transparent dot in guide plate with the resolution about 770nm; (b) the reference phase achieved from 140-frame temporal phase-shifting interferograms with AIA method; the differences between the reference phase and the phase achieved with different methods (c) GSPS-AIA; (d) SPS-AIA; (e) SPS-PCA; (f) the difference between the reference phase and the retrieved phases of the 60th row

Tables (2)

Tables Icon

Table 1 RMSE, PVE and calculation time achieved with different methods

Tables Icon

Table 2 RMSE, PVE and calculation time achieved with different methods

Equations (17)

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I(x,y)=A(x,y)+B(x,y)cos[ϕ(x,y)+ f 0x x+ f 0y y]+n(x,y).
{ I i =A(x,y)+B(x,y)cos[ϕ(x,y)+ f 0x x+ f 0y y+ δ i ]+ n i (x,y)+ d i (x,y) =A(x,y)+B(x,y)cos[ϕ'(x,y)+ δ i ]+ o i (x,y) i=1...N
{ B(x,y)cosϕ'(x,y)= i c i I i =c(x,y)I(x,y) B(x,y)sinϕ'(x,y)= i c i ' I i =c'(x,y)I(x,y) i=1......N .
{ B(x,y)cosϕ'(x,y)= I 1 I 3 =c(x,y)I(x,y) B(x,y)sinϕ'(x,y)= I 4 I 2 =c'(x,y)I(x,y) .
{ c=[δ(x,y)δ(x2 x 0 ,y)]/2 c'=[δ(x3 x 0 ,y)δ(x x 0 ,y)]/2 .
h(x,y)=c(x,y)+ic'(x,y).
B(x,y)exp[iϕ'(x,y)]=h(x,y)I(x,y).
{ ξ[B(x)expiϕ'(x)]=H( f x )ξ[I(x)] H( f x )=ξ[h(x)]= 1exp(i4π x 0 f x )iexp(i2π x 0 f x )+iexp(i6π x 0 f x ) 2 .
{ H( T f /4)=H(0)=0 H( T f /4)=2 .
I( f x , f y )=ξ[I(x,y)]=A( f x , f y )+ξ[B(x,y)cosϕ'(x,y)] I'( f x , f y )=G( f x , f y )I( f x , f y )=ξ[B(x,y)cosϕ'(x,y)]. I'(x,y)= ξ 1 [I'( f x , f y )]=B(x,y)cosϕ'(x,y)
{ I i '=B(x,y)cos[ϕ(x,y)+ f 0x x+ f 0y y+ δ i ]+ o i (x,y) =B(x,y)cos[ϕ'(x,y)+ δ i ]+ o i (x,y) i=1......N .
L f =( i=1 i=N cos 2 δ i i=1 i=N cos δ i sin δ i i=1 i=N cos δ i sin δ i i=1 i=N sin 2 δ i ) Λ f ={ i=1 i=N I i '(x,y)cos δ i , i=1 i=N I i '(x,y)sin δ i } X f ={B(x,y)cosϕ'(x,y),B(x,y)sinϕ'(x,y)}.
{ X f = L f 1 Λ f ϕ'(x,y)=arctan( X f (2)/ X f (1)) .
L f '=( j=1 j=M cos 2 ϕ ' j j=1 j=M sinϕ ' j cosϕ ' j j=1 j=M sinϕ ' j cosϕ ' j j=1 j=M sin 2 ϕ ' j ). Λ f '={ j=1 j=M I j 'cosϕ ' j , j=1 j=M I j 'sinϕ ' j } X f '={bcos δ i ,bsin δ i }
{ X f '= L f ' 1 Λ f ' δ i =arctan( X f '(2)/ X f '(1)) .
{ δ x =arccos( x,y I 1 ' I 2 ' x,y I 1 ' I 1 ' ) δ y =arccos( x,y I 1 ' I 3 ' x,y I 1 ' I 1 ' ) .
I(x)=A(x)+ x 0 /2 x 0 /2 B(x')cos( ϕ 0 + f x x'+ f x x) dx' x 0 . =A(x)+ 2sin( f x x 0 /2) f x x 0 B(x)cos( ϕ 0 + f x x)

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