Abstract

A simple estimation method of the relative phase shift for generalized phase-shifting digital holography based on a statistical method is proposed. This method consists of a selection procedure of an optimum cost function and a simple root-finding procedure. The value and sign of the relative phase shift are determined using the coefficient and the solution of the optimum cost function. The complex field of an object wave is obtained using the estimated relative phase shift. The proposed method lifts the typical restriction on the range of the phase shift due to the phase ambiguity problem. Computer simulations and optical experiments are performed to verify the proposed method.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
    [CrossRef]
  2. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef]
  3. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [CrossRef]
  4. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 547–666.
  5. 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, 1808–1810 (2003).
    [CrossRef]
  6. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29, 183–185 (2004).
    [CrossRef]
  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, 1966–1968 (2006).
    [CrossRef]
  8. 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, 121124 (2007).
    [CrossRef]
  9. 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, 776–778 (2008).
    [CrossRef]
  10. 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, 5701–5705 (2008).
    [CrossRef]
  11. P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34, 3553–3555(2009).
    [CrossRef]
  12. B. Kemper and G. V. Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47, A52–A61 (2008).
    [CrossRef]
  13. C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820 (1999).
    [CrossRef]
  14. J. W. Goodman, “Introduction to Fourier Optics,” 2nd ed. (McGraw-Hill, 1996).

2009

2008

2007

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, 121124 (2007).
[CrossRef]

2006

2004

2003

1999

1997

1994

1967

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

Bally, G. V.

Bruning, J. H.

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

Cai, L. Z.

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, 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, 776–778 (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, 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, 1966–1968 (2006).
[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, 183–185 (2004).
[CrossRef]

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, 1808–1810 (2003).
[CrossRef]

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, 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, 776–778 (2008).
[CrossRef]

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, 776–778 (2008).
[CrossRef]

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, 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, 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, 1966–1968 (2006).
[CrossRef]

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, 77–79 (1967).
[CrossRef]

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

Harder, I.

Jüptner, W.

Kemper, B.

Lawrence, R. W.

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

Lindlein, N.

Liu, Q.

Mantel, K.

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, 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, 776–778 (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, 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, 1966–1968 (2006).
[CrossRef]

Osten, W.

Schnars, U.

Schreiber, H.

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

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, 776–778 (2008).
[CrossRef]

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, 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, 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, 1966–1968 (2006).
[CrossRef]

Sun, W. J.

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, 776–778 (2008).
[CrossRef]

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, 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, 121124 (2007).
[CrossRef]

Xu, 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, 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, 776–778 (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, 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, 1966–1968 (2006).
[CrossRef]

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, 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, 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, 183–185 (2004).
[CrossRef]

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, 1808–1810 (2003).
[CrossRef]

Yao, B.

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, 776–778 (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, 121124 (2007).
[CrossRef]

Zhang, T.

Appl. Opt.

Appl. Phys. Lett.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[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, 121124 (2007).
[CrossRef]

Opt. Commun.

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, 5701–5705 (2008).
[CrossRef]

Opt. Lett.

Other

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

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Optical configuration: BE, beam expander; BS, beam splitter; M, mirror; PS, phase shifter; and OBJ, object.

Fig. 2.
Fig. 2.

Variation of cost functions within the available range of κ when the optimum cost function is (a) f1(κ) and (b) f3(κ), where f1(κ)2π is also illustrated because the principal value is considered.

Fig. 3.
Fig. 3.

Computer simulations: (a) phase object, (b) typical phase-shifted digital hologram, (c) and (d) phase map reconstructed from the hologram obtained by using the proposed phase-shifting method and the Fresnel transform alone, respectively.

Fig. 4.
Fig. 4.

Determination of κ0 using the bisection method.

Fig. 5.
Fig. 5.

Phase estimation error as a function of the signal-to-noise ratio.

Fig. 6.
Fig. 6.

Typical phase-shifted digital holograms.

Fig. 7.
Fig. 7.

Numerically reconstructed (a) intensity and (b) phase; and (c) and (d) magnified reconstructed images of (a) and (b), respectively.

Equations (11)

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

Ii(x,y)=|A|2+|R|2+2|A||R|cos(θϕi),(i=0,1,2),
ΔIpq=Iq(x,y)Ip(x,y)=4|A||R|sinΔϕpq2sin(θϕp+ϕq2),(p,q=0,1,2),
Epq=|ΔIpq|2=8|R|2sin2Δϕpq2{|A|2|A|2sin(2θϕpϕq)},
Epq=8|R|2sin2Δϕpq2|A|2.
Δϕpq=arccos(1κEpq).
Δϕ01+Δϕ12+Δϕ20=2mπ,
fn(κ)=c01Δϕ01+c12Δϕ12+c20Δϕ20,cpq=±1,
f1=Δϕ01+Δϕ12+Δϕ20,f2=Δϕ01+Δϕ12Δϕ20,f3=Δϕ01Δϕ12+Δϕ20,f4=Δϕ01Δϕ12Δϕ20,f5=Δϕ01Δϕ12Δϕ20(=f1),f6=Δϕ01Δϕ12+Δϕ20(=f2),f7=Δϕ01+Δϕ12Δϕ20(=f3),f8=Δϕ01+Δϕ12+Δϕ20(=f4).
0<κ2Emax.
Δϕpq=c^pqarccos(1κ0Epq),
A(x,y)=ejϕ0|R|Δ{(1ejΔϕ20)ΔI01+(1ejΔϕ01)ΔI20},

Metrics