Abstract

We propose a robust phase-shift estimation method for statistical generalized phase-shifting digital holography using a slightly off-axis optical configuration. The phase randomness condition in the Fresnel diffraction field of an object can be sufficiently established by the linear phase factor of the oblique incident reference wave. Signed phase-shift values can be estimated with a statistical approach regardless of the statistical properties of the Fresnel diffraction field of the object. We present computer simulations and optical experiments to verify the proposed method.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  10. 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]
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    [CrossRef] [PubMed]
  12. 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]
  13. 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]
  14. X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
    [CrossRef]
  15. X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
  17. 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]
  18. H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52(26), 6572–6578 (2013).
    [CrossRef] [PubMed]
  19. N. Yoshikawa, “Phase determination method in statistical generalized phase-shifting digital holography,” Appl. Opt. 52(9), 1947–1953 (2013).
    [CrossRef] [PubMed]
  20. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

2013 (2)

2009 (2)

2008 (3)

2007 (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[CrossRef]

2006 (1)

2004 (2)

2003 (1)

1999 (2)

1997 (1)

1994 (1)

1991 (1)

1967 (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[CrossRef]

Cai, L. Z.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281(23), 5701–5705 (2008).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[CrossRef]

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]

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]

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]

Cheng, X. C.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281(23), 5701–5705 (2008).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

Cuche, E.

Depeursinge, C.

Dong, G. Y.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281(23), 5701–5705 (2008).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[CrossRef]

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]

Gao, P.

Geist, E.

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[CrossRef]

Guo, H.

Han, B.

Hao, Q.

Harder, I.

Hu, Y.

Jüptner, W.

Kemper, B.

Lai, G.

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[CrossRef]

Lindlein, N.

Liu, Q.

Mantel, K.

Marquet, P.

Meng, X. F.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281(23), 5701–5705 (2008).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[CrossRef]

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]

Osten, W.

Schnars, U.

Seebacher, S.

Shen, X. X.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281(23), 5701–5705 (2008).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[CrossRef]

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]

Sun, W. J.

von Bally, G.

Wagner, C.

Wang, Y. R.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281(23), 5701–5705 (2008).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[CrossRef]

Wang, Z.

Xu, X. F.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281(23), 5701–5705 (2008).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[CrossRef]

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]

Yamaguchi, I.

Yang, X. L.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281(23), 5701–5705 (2008).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[CrossRef]

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]

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]

Yao, B.

Yatagai, T.

Yoshikawa, N.

Zhang, H.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[CrossRef]

Zhang, T.

Zhang, Z.

Zhu, Q.

Appl. Opt. (6)

Appl. Phys. Lett. (2)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[CrossRef]

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[CrossRef]

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

Opt. Commun. (1)

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281(23), 5701–5705 (2008).
[CrossRef]

Opt. Lett. (8)

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]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
[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]

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]

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]

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. 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 (2)

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 2007), pp.547–666.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the digital hologram recording system: beam splitter (BS), beam expander (BE), mirror (M), and object (OBJ).

Fig. 2
Fig. 2

Digital holograms (left column) and their reconstructed intensity image (middle column) and phase image (right column) using the conventional method: (a)–(c) triangular aperture, (d)–(f) test chart, (g)–(i) isolated phase object, and (j)–(l) quadratic phase object.

Fig. 3
Fig. 3

Digital holograms (left column) and their reconstructed intensity image (middle column) and phase image (right column) with respect to the incident angle α in the proposed method: (a)–(c) α = 0.175 × 10−3 rad, (d)–(f) α = 0.873 × 10−3 rad, and (g)–(i) α = 1.745 × 10−3 rad.

Fig. 4
Fig. 4

Average phase error of the triangular aperture as a function of the reference wave incident angle when the hologram is corrupted by white Gaussian noise.

Fig. 5
Fig. 5

Digital holograms (left columns) and their reconstructed intensity image (middle column) and phase image (right column) using the proposed method when the incident angle is 1.745 × 10−3 rad: (a)–(c) test chart, (d)–(f) isolated phase object, and (g)–(i) quadratic phase object.

Fig. 6
Fig. 6

Average phase error of the target objects as a function of the reference wave incident angle when the signal-to-noise ratio is 20 dB.

Fig. 7
Fig. 7

Digital holograms (left column) and their reconstructed intensity image (middle column) and phase image (right column) with respect to the incident angle α: (a)–(c) α = 0 rad, (d)–(f) α = 0.349 × 10−3 rad, and (g)–(i) α = 2.269 × 10−3 rad.

Tables (1)

Tables Icon

Table 1 The estimated relative phase-shift values and phase error where the preset relative phase-shift values are (Δϕ01, Δϕ12, Δϕ20) = (1.178, −1.964, 0.785).

Equations (9)

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

I i ( x , y ) = | A | 2 + | R | 2 + 2 | A | | R | cos ( θ k x sin α ϕ i ) , ( i = 0 , 1 , 2 ) ,
Δ I p q = 4 | A | | R | sin Δ ϕ p q 2 sin ( θ k x sin α ϕ p + ϕ q 2 ) , ( p , q = 0 , 1 , 2 ) ,
E p q = | Δ I p q | 2 S = 8 | R | 2 sin 2 Δ ϕ p q 2 { | A | 2 S | A | 2 sin ( 2 θ 2 k x sin α ϕ p ϕ q ) S } ,
E pq =8|R | 2 sin 2 Δ ϕ pq 2 |A | 2 S .
Δ ϕ pq =arccos(1κ E pq ),
Δ ϕ 01 +Δ ϕ 12 +Δ ϕ 20 =2mπ,
f n (κ)= c 01 Δ ϕ 01 + c 12 Δ ϕ 12 + c 20 Δ ϕ 20 , c pq =±1,
Δ ϕ pq = c ^ pq arccos(1 κ 0 E pq ),
A(x,y)= e j ϕ 0 e jkxsinα |R|Δ { (1 e jΔ ϕ 20 )Δ I 01 +(1 e jΔ ϕ 01 )Δ I 20 },

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