Abstract

In the case that the phase distribution of interferogram is nonuniform and the background/modulation amplitude change rapidly, the current self-calibration algorithms with better performance like principal components analysis (PCA) and advanced iterative algorithm (AIA) cannot work well. In this study, from three or more phase-shifting interferograms with unknown phase-shifts, we propose a spatial dual-orthogonal (SDO) phase-shifting algorithm with high accuracy through using the spatial orthogonal property of interference fringe, in which a new sequence of fringe patterns with uniform phase distribution can be constructed by pre-recomposing original interferograms to determine their corresponding optimum combination coefficients, which are directly related with the phase shifts. Both simulation and experimental results show that using the proposed SDO algorithm, we can achieve accurate phase from the phase-shifting interferograms with nonuniform phase distribution, non-constant background and arbitrary phase shifts. Specially, it is found that the accuracy of phase retrieval with the proposed SDO algorithm is insensitive to the variation of fringe pattern, and this will supply a guarantee for high accuracy phase measurement and application.

© 2017 Optical Society of America

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References

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  1. M. Servín, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley-VCH, 2014).
  2. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982).
    [Crossref] [PubMed]
  3. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. 7(4), 542–551 (1990).
    [Crossref]
  4. 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]
  5. K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988).
    [Crossref]
  6. X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
    [Crossref] [PubMed]
  7. P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009).
    [Crossref] [PubMed]
  8. C.-S. Guo, B. Sha, Y.-Y. Xie, and X.-J. Zhang, “Zero difference algorithm for phase shift extraction in blind phase-shifting holography,” Opt. Lett. 39(4), 813–816 (2014).
    [Crossref] [PubMed]
  9. 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]
  10. H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express 19(8), 7807–7815 (2011).
    [Crossref] [PubMed]
  11. J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011).
    [Crossref] [PubMed]
  12. F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015).
    [Crossref] [PubMed]
  13. J. Deng, H. Wang, F. Zhang, D. Zhang, L. Zhong, and X. Lu, “Two-step phase demodulation algorithm based on the extreme value of interference,” Opt. Lett. 37(22), 4669–4671 (2012).
    [Crossref] [PubMed]
  14. H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24(1), 25–33 (2007).
    [Crossref] [PubMed]
  15. J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step demodulation based on the Gram-Schmidt orthonormalization method,” Opt. Lett. 37(3), 443–445 (2012).
    [Crossref] [PubMed]
  16. 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]
  17. H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
    [Crossref] [PubMed]
  18. J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express 19(21), 20483–20492 (2011).
    [Crossref] [PubMed]
  19. J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015).
    [Crossref] [PubMed]
  20. K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016).
    [Crossref] [PubMed]
  21. J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
    [Crossref] [PubMed]
  22. J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Pys. B B 115(3), 355–364 (2014).
    [Crossref]
  23. 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]
  24. J. Vargas and C. O. S. Sorzano, “Quadrature Component Analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
    [Crossref]
  25. Q. Weijuan, Y. Yingjie, C. O. Choo, and A. Asundi, “Digital holographic microscopy with physical phase compensation,” Opt. Lett. 34(8), 1276–1278 (2009).
    [Crossref] [PubMed]
  26. M. Roy, J. Schmit, and P. Hariharan, “White-light interference microscopy: minimization of spurious diffraction effects by geometric phase-shifting,” Opt. Express 17(6), 4495–4499 (2009).
    [Crossref] [PubMed]
  27. Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19(2), 1016–1026 (2011).
    [Crossref] [PubMed]
  28. J. C. Estrada, M. Servin, and J. A. Quiroga, “A self-tuning phase-shifting algorithm for interferometry,” Opt. Express 18(3), 2632–2638 (2010).
    [Crossref] [PubMed]
  29. Y. Kikuchi, D. Barada, T. Kiire, and T. Yatagai, “Doppler phase-shifting digital holography and its application to surface shape measurement,” Opt. Lett. 35(10), 1548–1550 (2010).
    [Crossref] [PubMed]

2016 (1)

2015 (2)

2014 (3)

2013 (2)

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. Vargas and C. O. S. Sorzano, “Quadrature Component Analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

2012 (3)

2011 (5)

2010 (2)

2009 (4)

2007 (1)

2006 (1)

2004 (1)

1990 (1)

1988 (1)

K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

1982 (1)

Asundi, A.

Barada, D.

Belenguer, T.

Cai, L. Z.

Carazo, J. M.

J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Pys. B B 115(3), 355–364 (2014).
[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]

J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step demodulation based on the Gram-Schmidt orthonormalization method,” Opt. Lett. 37(3), 443–445 (2012).
[Crossref] [PubMed]

Chai, L.

Chen, M.

Chen, W.

Choo, C. O.

Creath, K.

K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Deng, J.

Ding, H.

Dong, G. Y.

Du, Y.

Estrada, J. C.

Feng, G.

Freischlad, K.

Gao, P.

Geist, E.

Gillette, M. U.

Guo, C.-S.

Guo, H.

Han, B.

Hao, J.

Harder, I.

Hariharan, P.

Ishikawa, K.

Jin, W.

Kiire, T.

Kikuchi, Y.

Koliopoulos, C. L.

Li, C.

Li, H.

Lindlein, N.

Liu, F.

Lu, X.

Luo, C.

Lv, X.

Ma, S.

Mantel, K.

Meng, X. F.

Millet, L.

Mir, M.

Morgan, C. J.

Oikawa, Y.

Popescu, G.

Qin, J.

Quiroga, J. A.

Rogers, J.

Roy, M.

Schmit, J.

Servin, M.

Sha, B.

Shen, X. X.

Sorzano, C. O. S.

J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Pys. B B 115(3), 355–364 (2014).
[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]

J. Vargas and C. O. S. Sorzano, “Quadrature Component Analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step demodulation based on the Gram-Schmidt orthonormalization method,” Opt. Lett. 37(3), 443–445 (2012).
[Crossref] [PubMed]

Unarunotai, S.

Vargas, J.

Wang, H.

Wang, K.

Wang, Z.

Weijuan, Q.

Wu, D.

Wu, F.

Wu, Y.

Xie, Y.-Y.

Xu, J.

Xu, Q.

Xu, X. F.

Yao, B.

Yatabe, K.

Yatagai, T.

Yingjie, Y.

Yu, Y.

Zhang, D.

Zhang, F.

Zhang, X.-J.

Zhong, L.

Zhou, S.

Appl. Pys. B B (1)

J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Pys. B B 115(3), 355–364 (2014).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (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]

Opt. Express (11)

M. Roy, J. Schmit, and P. Hariharan, “White-light interference microscopy: minimization of spurious diffraction effects by geometric phase-shifting,” Opt. Express 17(6), 4495–4499 (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]

J. C. Estrada, M. Servin, and J. A. Quiroga, “A self-tuning phase-shifting algorithm for interferometry,” Opt. Express 18(3), 2632–2638 (2010).
[Crossref] [PubMed]

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19(2), 1016–1026 (2011).
[Crossref] [PubMed]

H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express 19(8), 7807–7815 (2011).
[Crossref] [PubMed]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016).
[Crossref] [PubMed]

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015).
[Crossref] [PubMed]

J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015).
[Crossref] [PubMed]

J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express 19(21), 20483–20492 (2011).
[Crossref] [PubMed]

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]

Opt. Lasers Eng. (1)

J. Vargas and C. O. S. Sorzano, “Quadrature Component Analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

Opt. Lett. (11)

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982).
[Crossref] [PubMed]

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, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
[Crossref] [PubMed]

Q. Weijuan, Y. Yingjie, C. O. Choo, and A. Asundi, “Digital holographic microscopy with physical phase compensation,” Opt. Lett. 34(8), 1276–1278 (2009).
[Crossref] [PubMed]

P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009).
[Crossref] [PubMed]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
[Crossref] [PubMed]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011).
[Crossref] [PubMed]

Y. Kikuchi, D. Barada, T. Kiire, and T. Yatagai, “Doppler phase-shifting digital holography and its application to surface shape measurement,” Opt. Lett. 35(10), 1548–1550 (2010).
[Crossref] [PubMed]

J. Deng, H. Wang, F. Zhang, D. Zhang, L. Zhong, and X. Lu, “Two-step phase demodulation algorithm based on the extreme value of interference,” Opt. Lett. 37(22), 4669–4671 (2012).
[Crossref] [PubMed]

C.-S. Guo, B. Sha, Y.-Y. Xie, and X.-J. Zhang, “Zero difference algorithm for phase shift extraction in blind phase-shifting holography,” Opt. Lett. 39(4), 813–816 (2014).
[Crossref] [PubMed]

J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step demodulation based on the Gram-Schmidt orthonormalization method,” Opt. Lett. 37(3), 443–445 (2012).
[Crossref] [PubMed]

Prog. Opt. (1)

K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Other (1)

M. Servín, 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 (4)

Fig. 1
Fig. 1 Simulation results of a Gauss phase distribution sample with quadratic background and a sloping surface. (a1)(a2) three-frame simulated phase-shifting interference patterns; (b1)(b2) theoretical phase of (a1) and (a2), respectively ; the phases achieved with different algorithms (c1)(c2) PCA; (d1)(d2) SDO; (e1)(e2) AIA.
Fig. 2
Fig. 2 Phase distributions achieved with the proposed SDO algorithm in different SNR (a) 30db; (c) 50db; (b) variation curves of RMSE with the SNR; the achieved fringe patterns in different K values (d) 0.3; (f) 3.3 ; (e) variation curves of RMSE with the K value.
Fig. 3
Fig. 3 RMSE variation curve of SDO algorithm with the value of filtering window σ.
Fig. 4
Fig. 4 (a) (b) One-frame interferogram with size of 1300 × 1300 pixels and the intercepted area marked with red square and size of 480 × 520 pixels, respectively; (c) the reference phase distribution achieved with Fourier Transform algorithm; the error distributions achieved with different algorithms (d) PCA; (e) AIA (f) SDO; (g) variation curves (the 250th row) of the phase difference between the reference phase and the phases achieved with different algorithms.

Tables (1)

Tables Icon

Table 1 PVE, RMSE and Processing time of phase retrieval achieved with different algorithms

Equations (32)

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

I n (x,y)=A(x,y)+B(x,y)cos[ϕ(x,y)+ θ n ]
x,y B 2 (x,y) cos 2 ϕ(x,y) x,y B 2 (x,y) sin 2 ϕ(x,y)
x,y B 2 (x,y)cosϕ(x,y)sinϕ(x,y)<< x,y B 2 (x,y) sin 2 ϕ(x,y)
x,y B(x,y)cos ϕ(x,y)=0
x,y B(x,y)sinϕ(x,y) =0
I ˜ n ( f x )=H( f x ) I n ( f x ) =H( f x )(A( f x )+ξ{B(x)cos[ϕ(x)+ δ n ]}) =H( f x )ξ{B(x)cos[ϕ(x)+ δ n ]}
H(x)= ξ 1 {H( f x )}
I ˜ n (x)=H(x)B(x)cos[ϕ(x)+ δ n ]= τ H(τ)B(xτ)cos[ϕ(xτ)+ δ n ] = τ H(τ)B(xτ)cos[ϕ(x)+f(x,τ)+ δ n ] =cos[ϕ(x)+ δ n ] τ H(τ)B(xτ)cosf(x,τ) sin[ϕ(x)+ δ n ] τ H(τ)B(xτ)sinf(x,τ) = T(x) cos[ϕ(x)+ δ n +η(x)]= T(x) cos[ϕ'(x)+ δ n ]
f(x,τ)=ϕ(xτ)ϕ(x)
T(x)= [ τ H(τ)B(xτ)sinf(x,τ) ] 2 + [ τ H(τ)B(xτ)cosf(x,τ) ] 2
η(x)=arccos( τ H(τ)B(xτ)cosf(x,τ) T(x) ).
ϕ'(x)=ϕ(x)+η(x).
I ˜ n ( f x ) f x =0 = x,y T(x,y) cos [ϕ'(x,y)+ δ n ]=0,n=1......N.
{ A mn = x I ˜ m (x) I ˜ n (x) m=1...N,n=1...N
A= I p I p T .
PA P T =P I p I p T P T =P I p (P I p ) T =O.
O=( λ 1 0 0 0 0 λ 2 0 0 0 0 ...... 0 0 0 0 λ N ).
I p =( I 1 I 2 ...... I N ).
P=( P 1 (1)...... P 1 (N) P 2 (2)...... P 2 (N) ...... P N (2)...... P N (N) ).
[ n N P 1 (n) I n (x) ] 2 = C 1 x,y [ T(x,y) cosϕ(x,y)] 2 = λ 1 .
[ n N P 2 (n) I n (x) ] 2 = C 2 x,y [ T(x,y) sinϕ(x,y)] 2 = λ 2 .
x,y [ T(x,y) cosϕ(x,y)] 2 x,y [ T(x,y) sinϕ(x,y)] 2 .
P ˜ 1 = P 1 λ 1 , P ˜ 2 = P 2 λ 2 .
U 1 (x)=B(x)cosϕ(x)= n N P ˜ 1 (n) I n (x) .
U 2 (x)=B(x)sinϕ(x)= m N P ˜ 2 (m) I m (x) .
ϕ(x)=arctan( U 2 (x) U 1 (x) ).
T 1 = x,y B (x,y) 2 cos 2 ϕ(x,y) x,y B (x,y) 2 sin 2 ϕ(x,y) .
T 2 = x,y B (x,y) 2 cosϕ(x,y)sinϕ(x,y) x,y B (x,y) 2 .
K 1 =H(x,y)B(x,y)cosϕ(x,y).
K 2 =H(x,y)B(x,y)sinϕ(x,y).
T 1 '= x,y K 1 2 x,y K 2 2 .
T 2 '= x,y K 1 K 2 x,y K 1 2 + K 2 2 .

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