Abstract

An experimental setup for optical phase extraction from 2-D interferograms using a one-shot phase-shifting technique able to achieve four interferograms with 90° phase shifts in between is presented. The system uses a common-path interferometer consisting of two windows in the input plane and a phase grating in Fourier plane as its pupil. Each window has a birefringent wave plate attached in order to achieve nearly circular polarization of opposite rotations one respect to the other after being illuminated with a 45° linear polarized beam. In the output, interference of the fields associated with replicated windows (diffraction orders) is achieved by a proper choice of the windows spacing with respect to the grating period. The phase shifts to achieve four interferograms simultaneously to perform phase-shifting interferometry can be obtained by placing linear polarizers on each diffraction orders before detection at an appropriate angle. Some experimental results are shown.

© 2008 Optical Society of America

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References

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  1. J. E. Greivenkamp and J. H. Bruning, "C.14 Phase shifting interferometry," in Optical Shop Testing, D. Malacara, ed., (John Wiley and Sons, 1992), pp. 501-598.
  2. J. Schwider, "Advanced evaluation techniques in interferometry," in Progress in Optics, E. Wolf, ed., (North-Holland, 1990), pp. 271-359.
  3. K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, E. Wolf, ed., (North-Holland, 1998), 26 pp. 349-393.
  4. M. P. Kothiyal and C. Delisle, "Shearing interferometer for phase shifting interferometry with polarization phase shifter," Appl. Opt. 44, 4439-4442 (1985).
    [CrossRef]
  5. B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping," Appl. Opt. 38, 5944-5947 (1999).
    [CrossRef]
  6. B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Spatial Phase-stepped Interferometry using a holographic optical element," Opt. Eng. 38, 2069-2074 (1999).
    [CrossRef]
  7. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, "Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer," Appl. Opt. 44, 6861-6868 (2005).
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    [CrossRef]
  11. C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, "Optical Tomography of transparent objects with Phase-Shifting Interferometry and Stepping Wise Shifted Ronchi Ruling," J. Opt. Soc. Am. A 23, 298-305 (2006).
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    [CrossRef] [PubMed]
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  15. P. W. Remijan, Processing stereo photographs by Optical Subtraction, PhD Thesis, University of Rochester (1978).
  16. S. R. Dashiell and A. W. Lohmann, "Image subtraction by Polarization-Shifted Periodic Carrier," Opt. Commun. 8, 100-102 (1973).
    [CrossRef]
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2006 (2)

2005 (1)

2004 (1)

1999 (2)

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping," Appl. Opt. 38, 5944-5947 (1999).
[CrossRef]

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Spatial Phase-stepped Interferometry using a holographic optical element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

1986 (1)

1985 (1)

M. P. Kothiyal and C. Delisle, "Shearing interferometer for phase shifting interferometry with polarization phase shifter," Appl. Opt. 44, 4439-4442 (1985).
[CrossRef]

1973 (1)

S. R. Dashiell and A. W. Lohmann, "Image subtraction by Polarization-Shifted Periodic Carrier," Opt. Commun. 8, 100-102 (1973).
[CrossRef]

1971 (1)

Almi, U.

Arrizón, V.

Barrientos-García, B.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping," Appl. Opt. 38, 5944-5947 (1999).
[CrossRef]

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Spatial Phase-stepped Interferometry using a holographic optical element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

Brock, N.

Burow, R.

Dashiell, S. R.

S. R. Dashiell and A. W. Lohmann, "Image subtraction by Polarization-Shifted Periodic Carrier," Opt. Commun. 8, 100-102 (1973).
[CrossRef]

Delisle, C.

M. P. Kothiyal and C. Delisle, "Shearing interferometer for phase shifting interferometry with polarization phase shifter," Appl. Opt. 44, 4439-4442 (1985).
[CrossRef]

Elssner, K.-E.

Grzanna, J.

Hayes, J.

Kothiyal, M. P.

M. P. Kothiyal and C. Delisle, "Shearing interferometer for phase shifting interferometry with polarization phase shifter," Appl. Opt. 44, 4439-4442 (1985).
[CrossRef]

Lohmann, A. W.

S. R. Dashiell and A. W. Lohmann, "Image subtraction by Polarization-Shifted Periodic Carrier," Opt. Commun. 8, 100-102 (1973).
[CrossRef]

Meneses-Fabian, C.

Millerd, J.

Moore, A. J.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Spatial Phase-stepped Interferometry using a holographic optical element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping," Appl. Opt. 38, 5944-5947 (1999).
[CrossRef]

North-Morris, M.

Novak, M.

Pérez-López, C.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping," Appl. Opt. 38, 5944-5947 (1999).
[CrossRef]

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Spatial Phase-stepped Interferometry using a holographic optical element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

Rodríguez-Zurita, G.

Sánchez-De-La-Llave, D.

Schwider, J.

Spolaczyk, R.

Tschudi, T.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping," Appl. Opt. 38, 5944-5947 (1999).
[CrossRef]

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Spatial Phase-stepped Interferometry using a holographic optical element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

Wang, L.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Spatial Phase-stepped Interferometry using a holographic optical element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping," Appl. Opt. 38, 5944-5947 (1999).
[CrossRef]

Weinberger, H.

Wyant, J.

Appl. Opt. (5)

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

Opt. Commun. (2)

C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, "Common-path phase-shifting interferometer with binary grating," Opt. Commun. 264, 13-17 (2006).
[CrossRef]

S. R. Dashiell and A. W. Lohmann, "Image subtraction by Polarization-Shifted Periodic Carrier," Opt. Commun. 8, 100-102 (1973).
[CrossRef]

Opt. Eng. (1)

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Spatial Phase-stepped Interferometry using a holographic optical element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

Opt. Lett. (1)

Other (7)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), pp. 243-246.

P. W. Remijan, Processing stereo photographs by Optical Subtraction, PhD Thesis, University of Rochester (1978).

J. E. Greivenkamp and J. H. Bruning, "C.14 Phase shifting interferometry," in Optical Shop Testing, D. Malacara, ed., (John Wiley and Sons, 1992), pp. 501-598.

J. Schwider, "Advanced evaluation techniques in interferometry," in Progress in Optics, E. Wolf, ed., (North-Holland, 1990), pp. 271-359.

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, E. Wolf, ed., (North-Holland, 1998), 26 pp. 349-393.

J. E Miller,  et al., "Methods and apparatus for splitting, imaging, and measuring wavefronts in Interferometry," U. S. Patent 6552808B2 (2002).

J. E. Miller,  et al., "Methods and apparatus for splitting, imaging, and measuring wavefronts in Interferometry," U. S. Patent 20030053071A1 (2003).

Supplementary Material (2)

» Media 1: MOV (2012 KB)     
» Media 2: MOV (268 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

One-shot phase-shifting grating interferometer with modulation of polarization. A, B: windows. Side view: ψi : polarizing angles (with i=1,2,3,4 to obtain phase-shifts ξi =0°, 90°, 180°, 270° respectively). Involved interference order superpositions are indicated from -2 to +2.

Fig. 2.
Fig. 2.

(a) Phase shift ξ(ψ, α′) as a function of ψ for several values of α′. Insert: α′ for ideal retardation and experimental retardation. (b) Amplitude A(ψ,α′) as a function of ψ for several values of α′.

Fig. 3.
Fig. 3.

Relative positions of diffraction orders from windows A and B. The case of order +1 out of phase by π with respect to the others is shown. Two interferograms with inverse contrast result. X 0=x 0.

Fig. 4.
Fig. 4.

Image plane from a system as depicted in Fig. 1. Neither plate retardation plates nor polarizing filters were used. Two opposite fringe contrast can be seen. Compare with Fig. 3.

Fig. 5.
Fig. 5.

Upper row: phase dot. Four 90° phase-shifted interferograms and unwrapped phase. Lower row: phase step. Four 90° phase-shifted interferograms and unwrapped phase.

Fig. 6.
Fig. 6.

Unwrapped calculated phases along typical raster lines of each object of Fig. 5. Scale factor: 0.405 rad.

Fig. 7.
Fig. 7.

Simplify polarizer filters array

Fig. 8.
Fig. 8.

Typical four 90° phase-shifted interferograms from oil flowing (Media 1)

Fig. 9.
Fig. 9.

typical unwrapped phase from interferograms of oil flowing (Media 2).

Equations (22)

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J = J ψ l J L J = J ψ l J R
J ψ l = ( cos 2 ψ sin ψ cos ψ sin ψ cos ψ sin 2 ψ ) .
J T 2 = J + J 2
= 1 + cos ϕ · cos 2 ψ + sin 2 ψ · cos α + sin 2 ψ · cos [ α ϕ ( x , y ) ]
+ sin 2 ψ · cos [ 2 α ϕ ( x , y ) ]
= 1 + sin 2 ψ · cos α + A ( ψ , α ) cos [ ξ ( ψ , α ) ϕ ( x , y ) ]
ξ ( ψ , α ) = tan 1 [ sin 2 ψ · sin α + sin 2 ψ · sin 2 α cos 2 ψ + sin 2 ψ · cos 2 α + sin 2 ψ · cos α ]
A ( ψ , α ) = ( cos 4 ψ + sin 4 ψ + ( 1 + 1 2 cos 2 α ) · sin 2 2 ψ + 2 sin 2 ψ · cos α ) 1 2 .
ξ ( ψ , π 2 ) = 2 ψ , A 2 ( ψ , π 2 ) = 1 ,
J T 2 = 1 + cos [ 2 ψ ϕ ( x , y ) ] ,
J i 2 = 1 + cos [ 2 ψ i ϕ ( x , y ) ] ,
tan ϕ = J 1 2 J 3 2 J 2 2 J 4 2
cos 2 ψ + sin 2 ψ · cos 2 α + sin 2 ψ · cos α = 0 ,
{ ( 1 cos 2 α ) 2 + 4 cos 2 α } cos 4 ψ + { 2 ( 1 cos 2 α ) · cos 2 α 4 cos 2 α } cos 2 ψ + cos 2 2 α = 0
cos 2 ( ψ ) =
{ 2 cos 2 α ( 1 cos 2 α ) cos 2 α } ± { ( 1 cos 2 α ) cos 2 α 2 cos 2 α } 2 { ( 1 cos 2 α ) 2 + 4 cos 2 α } cos 2 2 α ( 1 cos 2 α ) 2 + 4 cos 2 α
ψ 2 = ψ a , ψ 4 = ψ b + π ,
ψ 3 = n π + arctan ( sec ( α ) ) .
O ( x , y ) = J L ( x + 1 2 x 0 , y ) · w ( x + 1 2 x 0 , y ) + J R ( x 1 2 x 0 , y ) · w ( x 1 2 x 0 , y )
G ( μ , v ) = G P ( μ ) * n = δ ( μ n X 0 )
G ~ ( x , y ) = X 0 · n = G ~ P ( n · X 0 ) · δ ( x n · X 0 , y ) .
α = π 2 λ a λ = 1.519 rad

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