Abstract

To extract phase distributions, which evolve in time using phase-shifting interferometry, the simultaneous capture of several interferograms with a prescribed shift has to be done. Previous interferometric systems aimed to fulfill such a task were reported to get only four interferograms. It is pointed out that more than four suitable interferograms can be obtained with an interferometer that uses two windows in the object plane, a phase grid as a pupil, and modulation of polarization for each diffraction orders in the image plane. Experimental results for five, seven, and nine interferograms are given.

© 2008 Optical Society of America

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References

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  1. B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, Opt. Eng. 38, 2069 (1999).
    [CrossRef]
  2. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, Appl. Opt. 44, 6861 (2005).
    [CrossRef] [PubMed]
  3. G. Rodriguez-Zurita, C. Meneses-Fabian, N. I. Toto-Arellano, J. Vázquez-Castillo, and C. Robledo-Sánchez, Opt. Express 16, 7806 (2008).
    [CrossRef] [PubMed]
  4. D. Malacara, M. Servin, and Z. Malacara, in Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
  5. N. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, Opt. Express 16, 19330 (2008).
    [CrossRef]

2008 (2)

2005 (1)

1999 (1)

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, Opt. Eng. 38, 2069 (1999).
[CrossRef]

1998 (1)

D. Malacara, M. Servin, and Z. Malacara, in Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Barrientos-García, B.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, Opt. Eng. 38, 2069 (1999).
[CrossRef]

Brock, N.

Hayes, J.

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, in Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, in Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Meneses-Fabian, C.

Millerd, J.

Moore, A. J.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, Opt. Eng. 38, 2069 (1999).
[CrossRef]

North-Morris, M.

Novak, M.

Pérez-López, C.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, Opt. Eng. 38, 2069 (1999).
[CrossRef]

Robledo-Sánchez, C.

Rodriguez-Zurita, G.

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, in Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Toto-Arellano, N.

Toto-Arellano, N. I.

Tschudi, T.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, Opt. Eng. 38, 2069 (1999).
[CrossRef]

Vázquez-Castillo, J.

Vázquez-Castillo, J. F.

Wang, L.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, Opt. Eng. 38, 2069 (1999).
[CrossRef]

Wyant, J.

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Figures (4)

Fig. 1
Fig. 1

Setup. A, B, windows; G 2 , phase grid; t f , transmitted image; Q j , retarders; P, linear polarizer; x q = x ( q + 1 2 ) X 0 , y r = y ( r + 1 2 ) X 0 , coordinates about diffraction order q r . Polarization filter arrays at the right showing different angle ψ n for each transmission axis of polarizing filter P i .

Fig. 2
Fig. 2

(a) One-dimensional spectra of identical phase gratings to be crossed to construct a grid. (b) Corresponding image plane for the phase grid. (c) Two shifted Fourier spectra are superimposed according to the windows displacement A B .

Fig. 3
Fig. 3

Polarizing filter arrays for several cases. (a) Twelve interference patterns detected with a polarizing filter at ψ = 35 ° covering all of them. (b) N = 4 , symmetrical five. (c) N = 6 , symmetrical seven. (d) N = 8 , symmetrical nine.

Fig. 4
Fig. 4

Flow of oil drops on glass. Phase-shifted interferograms and unwrapped phases. Upper two rows, two examples of five 90° phase shifts. Center rows, seven 60° phase shifts. Lower rows, nine 45° phase shifts. Reference square is for scale dimensions.

Equations (10)

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G 2 ( μ , ζ ) = q = J q ( 2 π A g ) e i 2 π q X 0 μ r = J r ( 2 π A g ) e i 2 π r X 0 ζ ,
G ̃ 2 ( x , y ) = q = q = r = r = J q ( 2 π A g ) J r ( 2 π A g ) δ ( x q X 0 , y r X 0 ) ,
t 2 ( x , y ) = J L w ( x x 0 2 , y y 0 2 ) + J R w ( x + x 0 2 , y + y 0 2 ) ,
J L = ( 1 e i α ) , J R = ( 1 e i α ) ,
t f ( x , y ) = 1 2 I t 2 ( x , y ) I 1 { G ̃ 2 ( μ μ 0 , ζ ζ 0 ) } ,
q = r = { J L J q J r + J R J q + 1 J r + 1 e i 2 π ( μ 0 + ζ 0 ) x 0 e [ i ϕ ( x ( q + 1 2 ) x 0 , y ( r + 1 2 ) x 0 ) ] } e i 2 π ( q μ 0 + r ζ 0 ) x 0 ,
A ( ψ , α ) ( J q J r ) 2 + ( J q + 1 J r + 1 ) 2 + 2 J q J r J q + 1 J r + 1 cos [ ξ ( ψ , α ) ϕ ( x q , y r ) ] ,
J ψ L = ( cos ψ sin ψ sin ψ cos ψ ) , J L = J ψ L J L , J R = J ψ L J R ,
A ( ψ , α ) = 1 + sin ( 2 ψ ) cos ( α ) ,
ξ ( ψ , α ) = ArcTan [ sin ( α ) cot ( 2 ψ ) 1 + tan ( ψ ) cos ( α ) + cos ( α ) ] .

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