Abstract

We propose a novel phase shifting interferometry from two normalized interferograms with random tilt phase-shift. The determination of tilt phase-shift is performed by extracting the tilted phase-shift plane from the phase difference of two normalized interferograms, and with the calculated tilt phase-shift value the phase distribution can be retrieved from the two normalized frames. By analyzing the distribution of phase difference and utilizing special points fitting method, the tilted phase-shift plane is extracted in three different cases, which relate to different magnitudes of tilts. Proposed method has been applied to simulations and experiments successfully and the satisfactory results manifest that proposed method is of high accuracy and high speed compared with the three step iterative method. Additionally, both open and closed fringe can be analyzed with proposed method. What’s more, it cannot only eliminate the small tilt-shift error caused by slight vibration in phase-shifting interferometry, but also detect the large tilt phase-shift in phase-tilting interferometry. Thus, it will relaxes the requirements on the accuracy of phase shifter, and the costly phase shifter may even be useless by applying proposed method in high amplitude vibrated circumstance to achieve high-precision analysis.

© 2015 Optical Society of America

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References

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  1. R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express 22(4), 4738–4750 (2014).
    [Crossref] [PubMed]
  2. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor and Francis, 2005).
  3. L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. 48(20), 3948–3960 (2009).
    [Crossref] [PubMed]
  4. 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]
  5. 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]
  6. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
    [Crossref] [PubMed]
  7. 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]
  8. 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]
  9. 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]
  10. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
    [Crossref] [PubMed]
  11. J. Xu, Q. Xu, and L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt. 47(3), 480–485 (2008).
    [Crossref] [PubMed]
  12. J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
    [Crossref]
  13. Q. Liu, Y. Wang, F. Ji, and J. He, “A three-step least-squares iterative method for tilt phase-shift interferometry,” Opt. Express 21(24), 29505–29515 (2013).
    [Crossref] [PubMed]
  14. F. Zeng, Q. Tan, H. Gu, and G. Jin, “Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method,” Opt. Express 21(14), 17234–17248 (2013).
    [Crossref] [PubMed]
  15. O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express 13(10), 3743–3753 (2005).
    [Crossref] [PubMed]
  16. J. Li, R. Zhu, L. Chen, and Y. He, “Phase-tilting interferometry for optical testing,” Opt. Lett. 38(15), 2838–2841 (2013).
    [Crossref] [PubMed]
  17. J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett. 36(17), 3485–3487 (2011).
    [Crossref] [PubMed]
  18. J. Xu, Y. Li, H. Wang, L. Chai, and Q. Xu, “Phase-shift extraction for phase-shifting interferometry by histogram of phase difference,” Opt. Express 18(23), 24368–24378 (2010).
    [Crossref] [PubMed]
  19. R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
    [Crossref]
  20. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
    [Crossref] [PubMed]
  21. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34(8), 1210–1212 (2009).
    [Crossref] [PubMed]
  22. Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
    [Crossref] [PubMed]
  23. J. Xu, Q. Xu, L. Chai, Y. Li, and H. Wang, “Direct phase extraction from interferograms with random phase shifts,” Opt. Express 18(20), 20620–20627 (2010).
    [Crossref] [PubMed]

2015 (1)

2014 (1)

2013 (4)

2011 (2)

2010 (2)

2009 (3)

2008 (2)

J. Xu, Q. Xu, and L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt. 47(3), 480–485 (2008).
[Crossref] [PubMed]

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[Crossref]

2006 (1)

2005 (1)

2004 (2)

2003 (1)

2000 (1)

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]

Aguilar, L. A.

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Cai, L. Z.

Carazo, J. M.

Chai, L.

Chen, L.

Chen, M.

Deck, L. L.

Dong, G. Y.

Estrada, J. C.

Gu, H.

Guerrero-Sanchez, F.

R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express 22(4), 4738–4750 (2014).
[Crossref] [PubMed]

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Guo, H.

Guo, J. P.

Han, B.

Hao, Q.

He, J.

He, Y.

Hu, Y.

Ji, F.

Jin, G.

Jin, W.

Juarez-Salazar, R.

R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express 22(4), 4738–4750 (2014).
[Crossref] [PubMed]

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Li, A. M.

Li, J.

Li, Y.

Liu, F.

Liu, Q.

Meneses-Fabian, C.

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Meng, X. F.

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, X.

Quiroga, J. A.

Rangel-Huerta, A.

Robledo-Sanchez, C.

R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express 22(4), 4738–4750 (2014).
[Crossref] [PubMed]

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

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]

Shen, X. X.

Soloviev, O.

Sorzano, C. O. S.

Tan, Q.

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]

Vargas, J.

Vdovin, G.

Wang, H.

Wang, Y.

Wang, Y. R.

Wang, Z.

Wei, C.

Wu, F.

Wu, Y.

Xu, J.

Xu, Q.

Xu, X. F.

Yang, X. L.

Zeng, F.

Zhu, Q.

Zhu, R.

Appl. Opt. (3)

J. Opt. A, Pure Appl. Opt. (1)

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[Crossref]

Opt. Commun. (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]

Opt. Express (8)

R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express 22(4), 4738–4750 (2014).
[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]

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]

Q. Liu, Y. Wang, F. Ji, and J. He, “A three-step least-squares iterative method for tilt phase-shift interferometry,” Opt. Express 21(24), 29505–29515 (2013).
[Crossref] [PubMed]

F. Zeng, Q. Tan, H. Gu, and G. Jin, “Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method,” Opt. Express 21(14), 17234–17248 (2013).
[Crossref] [PubMed]

O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express 13(10), 3743–3753 (2005).
[Crossref] [PubMed]

J. Xu, Y. Li, H. Wang, L. Chai, and Q. Xu, “Phase-shift extraction for phase-shifting interferometry by histogram of phase difference,” Opt. Express 18(23), 24368–24378 (2010).
[Crossref] [PubMed]

J. Xu, Q. Xu, L. Chai, Y. Li, and H. Wang, “Direct phase extraction from interferograms with random phase shifts,” Opt. Express 18(20), 20620–20627 (2010).
[Crossref] [PubMed]

Opt. Lasers Eng. (1)

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Opt. Lett. (8)

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref] [PubMed]

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34(8), 1210–1212 (2009).
[Crossref] [PubMed]

Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
[Crossref] [PubMed]

J. Li, R. Zhu, L. Chen, and Y. He, “Phase-tilting interferometry for optical testing,” Opt. Lett. 38(15), 2838–2841 (2013).
[Crossref] [PubMed]

J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett. 36(17), 3485–3487 (2011).
[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]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[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]

Other (1)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor and Francis, 2005).

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Figures (8)

Fig. 1
Fig. 1 Extraction of the tilted phase-shift plane: (a) the distribution of phase difference; (b) the binary image of (a); (c) the x- directional component of phase difference; (d) the selected points and the x- directional fitted line; (e) the y- directional component of phase difference; (f) the selected points and the y- directional fitted line.
Fig. 2
Fig. 2 Extraction of the tilted phase-shift plane: (a) the distribution of phase difference in situation (1); (b) the selected points and the fitted phase-shift plane in situation (1); (c) the distribution of phase difference in situation (2); (d) the selected points and the fitted phase-shift plane in situation (2); (e) the distribution of phase difference in situation (3); (f) the selected points and the fitted phase-shift plane in situation (3).
Fig. 3
Fig. 3 Extraction of the tilted phase-shift plane: (a) the distribution of phase difference; (b) the platform of (a); (c) the binary image of (b); (d) the Hough transform process; (e) the peak points in the Hough domain.
Fig. 4
Fig. 4 Simulation results1: (a) and (b) the two normalized interferograms; (c) the surface distribution of the phase to be measured; (d) the phase difference of the two normalized interferograms; (e) the histogram of phase shift; (f) and (g) the retrieved surface and the residual surface error of Xu’s method; (h) the extracted tilted phase-shift plane; (i) and (j) the retrieved surface and the residual surface error of proposed method.
Fig. 5
Fig. 5 Simulation results2: (a)-(c) the calculated tilt gradients, i.e. a, b and translational phase shift, i.e. c of the 5 interferograms with three different methods; (d) the computing time and residual surface error (rms) of three different methods .
Fig. 6
Fig. 6 The average phase shift extraction error for different numbers of frames that used for normalization.
Fig. 7
Fig. 7 The relation between the residual surface error and SNR.
Fig. 8
Fig. 8 Experimental results: (a) and (b) the two selected interferograms with tilt phase shift error; (c) and (d) the two normalized interferograms; (e) the reference surface calculated without tilt phase-shift error; (f) the surface calculated by Zygo’s 13-step PSA when tilt phase shift error is introduced; (g) the residual surface error between (e) and (f); (h) the phase difference of the two normalized interferograms; (i) the retrieved surface of proposed method; (j) the residual surface error between (e) and (i).

Tables (1)

Tables Icon

Tabel 1 Comparison of the tilt phase shift coefficients according to different magnitudes of tilts

Equations (16)

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I k (x,y)=A(x,y)+B(x,y)cos[ϕ(x,y)+ δ k (x,y)] k=1,2,3M .
A(x,y)= I max + I min 2 ,B(x,y)= I max I min 2 .
I k ¯ (x,y)= I k (x,y)A(x,y) B(x,y) .
ϕ k [0,π] (x,y)=arccos( I k ¯ (x,y)) k=1,2,...M .
ϕ k [π,π] = tan 1 [ cot(Δ δ k ) I k ¯ sin(Δ δ k ) I k1 ¯ ].
δ kx =ax+ c x
δ ky =by+ c y
c= c x + c y
δ k1 (x,y)=ax+by+c
δ k2 (x,y)=axbyc
ax+by+c+(n1)2π=0,n=1,2,3...
a=2πcos(ϑ)/Δρ
b=2πsin(ϑ)/Δρ
c=bρsin(ϑ)aρcos(ϑ) (ϑ= 90 θ)
A(x,y)=145exp(1.8( x 2 + y 2 )), B(x,y)=100exp(0.2( x 2 + y 2 ));(-1x,y1)
e={ [ | (a'a) |+| (b'b) | ]×200+| (c'c) | }/3

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