Abstract

A blind phase shift estimation algorithm that allows simultaneous calculation of phases and phase shifts from three or more interferograms is presented. In phase-shifting interferometry, the phase shift errors introduce specific correlations between the calculated background intensity distribution and the fringe component. These correlations can be measured with a cross-power spectrum. By minimization of an objective function based on this cross-power spectrum, the actual phase shifts are estimated and used for phase recovery. The validity of this algorithm is verified by both the numerical simulation and the experiment results.

© 2006 Optical Society of America

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  1. K. Okada, A. Sato, and J. Tsujiuchi, 'Simultaneous calculation of phase distribution and scanning phase shifting interferometry,' Opt. Commun. 84, 118-124 (1991).
    [CrossRef]
  2. G.-S. Ham and S.-W. Kim, 'Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,' Appl. Opt. 33, 7321-7325 (1994).
    [CrossRef]
  3. I.-B. Kong and S.-W. Kim, 'General algorithm of phase-shifting interferometry by iterative least-squares fitting,' Opt. Eng. 34, 183-187 (1995).
    [CrossRef]
  4. C. T. Farrell and M. A. Player, 'Phase step measurement and variable step algorithms in phase-shifting interferometry,' Meas. Sci. Technol. 3, 953-958 (1992).
    [CrossRef]
  5. C. Wei, M. Chen, and Z. Wang, 'General phase-stepping algorithm with automatic calibration of phase steps,' Opt. Eng. 38, 1357-1360 (1999).
    [CrossRef]
  6. X. Chen, M. Gramaglia, and J. A. Yeazell, 'Phase-shifting interferometry with uncalibrated phase shifts,' Appl. Opt. 39, 585-591 (2000).
    [CrossRef]
  7. 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] [PubMed]
  8. L. Z. Cai, Q. Liu, and X. L. Yang, 'Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,' Opt. Commun. 233, 21-26 (2004).
    [CrossRef]
  9. Z. Wang and B. Han, 'Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,' Opt. Lett. 29, 1671-1673 (2004).
    [CrossRef] [PubMed]
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2004

2003

2001

2000

1999

C. Wei, M. Chen, and Z. Wang, 'General phase-stepping algorithm with automatic calibration of phase steps,' Opt. Eng. 38, 1357-1360 (1999).
[CrossRef]

1995

I.-B. Kong and S.-W. Kim, 'General algorithm of phase-shifting interferometry by iterative least-squares fitting,' Opt. Eng. 34, 183-187 (1995).
[CrossRef]

1994

1992

C. T. Farrell and M. A. Player, 'Phase step measurement and variable step algorithms in phase-shifting interferometry,' Meas. Sci. Technol. 3, 953-958 (1992).
[CrossRef]

1991

K. Okada, A. Sato, and J. Tsujiuchi, 'Simultaneous calculation of phase distribution and scanning phase shifting interferometry,' Opt. Commun. 84, 118-124 (1991).
[CrossRef]

1986

1984

J. E. Greivenkamp, 'Generalized data reduction for heterodyne interferometry,' Opt. Eng. 23, 350-352 (1984).

1982

Bachor, H.-A.

Bokor, J.

Bone, D. J.

Cai, L. Z.

L. Z. Cai, Q. Liu, and X. L. Yang, 'Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,' Opt. Commun. 233, 21-26 (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] [PubMed]

Chen, M.

C. Wei, M. Chen, and Z. Wang, 'General phase-stepping algorithm with automatic calibration of phase steps,' Opt. Eng. 38, 1357-1360 (1999).
[CrossRef]

Chen, X.

Cornejo-Rodríguez, A.

Farrell, C. T.

C. T. Farrell and M. A. Player, 'Phase step measurement and variable step algorithms in phase-shifting interferometry,' Meas. Sci. Technol. 3, 953-958 (1992).
[CrossRef]

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, 1989).

Goldberg, K. A.

Gramaglia, M.

Greivenkamp, J. E.

J. E. Greivenkamp, 'Generalized data reduction for heterodyne interferometry,' Opt. Eng. 23, 350-352 (1984).

Ham, G.-S.

Han, B.

Ina, H.

Kim, S.-W.

I.-B. Kong and S.-W. Kim, 'General algorithm of phase-shifting interferometry by iterative least-squares fitting,' Opt. Eng. 34, 183-187 (1995).
[CrossRef]

G.-S. Ham and S.-W. Kim, 'Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,' Appl. Opt. 33, 7321-7325 (1994).
[CrossRef]

Kobayashi, S.

Kong, I.-B.

I.-B. Kong and S.-W. Kim, 'General algorithm of phase-shifting interferometry by iterative least-squares fitting,' Opt. Eng. 34, 183-187 (1995).
[CrossRef]

Larkin, K. G.

Liu, Q.

L. Z. Cai, Q. Liu, and X. L. Yang, 'Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,' Opt. Commun. 233, 21-26 (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] [PubMed]

Luna, E.

Nava-Vega, A.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, 'Simultaneous calculation of phase distribution and scanning phase shifting interferometry,' Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Player, M. A.

C. T. Farrell and M. A. Player, 'Phase step measurement and variable step algorithms in phase-shifting interferometry,' Meas. Sci. Technol. 3, 953-958 (1992).
[CrossRef]

Salas, L.

Sandeman, R. J.

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, 'Simultaneous calculation of phase distribution and scanning phase shifting interferometry,' Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Takeda, M.

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, 'Simultaneous calculation of phase distribution and scanning phase shifting interferometry,' Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Wang, Z.

Z. Wang and B. Han, 'Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,' Opt. Lett. 29, 1671-1673 (2004).
[CrossRef] [PubMed]

C. Wei, M. Chen, and Z. Wang, 'General phase-stepping algorithm with automatic calibration of phase steps,' Opt. Eng. 38, 1357-1360 (1999).
[CrossRef]

Wei, C.

C. Wei, M. Chen, and Z. Wang, 'General phase-stepping algorithm with automatic calibration of phase steps,' Opt. Eng. 38, 1357-1360 (1999).
[CrossRef]

Yang, X. L.

L. Z. Cai, Q. Liu, and X. L. Yang, 'Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,' Opt. Commun. 233, 21-26 (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] [PubMed]

Yeazell, J. A.

Appl. Opt.

J. Opt. Soc. Am.

Meas. Sci. Technol.

C. T. Farrell and M. A. Player, 'Phase step measurement and variable step algorithms in phase-shifting interferometry,' Meas. Sci. Technol. 3, 953-958 (1992).
[CrossRef]

Opt. Commun.

L. Z. Cai, Q. Liu, and X. L. Yang, 'Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,' Opt. Commun. 233, 21-26 (2004).
[CrossRef]

K. Okada, A. Sato, and J. Tsujiuchi, 'Simultaneous calculation of phase distribution and scanning phase shifting interferometry,' Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Opt. Eng.

J. E. Greivenkamp, 'Generalized data reduction for heterodyne interferometry,' Opt. Eng. 23, 350-352 (1984).

C. Wei, M. Chen, and Z. Wang, 'General phase-stepping algorithm with automatic calibration of phase steps,' Opt. Eng. 38, 1357-1360 (1999).
[CrossRef]

I.-B. Kong and S.-W. Kim, 'General algorithm of phase-shifting interferometry by iterative least-squares fitting,' Opt. Eng. 34, 183-187 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Other

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, 1989).

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Figures (6)

Fig. 1
Fig. 1

Results of numerical simulations in the noise-free case. (a) An interferogram of a smooth phase map. (b) Reconstructed smooth phase map. (c) An interferogram of a rough phase map. (d) Reconstructed rough phase map.

Fig. 2
Fig. 2

Numerical simulation results for measuring a smooth phase map in the presence of noises. (a) An interferogram corrupted by Gaussian noise (the noise SD is 0.05). (b) The phase map reconstructed from (a). (c) An interferogram corrupted by salt-and-pepper noise. (d) The phase map reconstructed from (c).

Fig. 3
Fig. 3

Profiles of the objective functions with the actual phase shifts being (a) { δ 1 , δ 2 } = { 0.52 π , 1.26 π } and (b) { δ 1 , δ 2 } = { 0.4070 , 6.2099 } . The left-hand panels display the functions as images, and the right-hand ones are their three-dimensional views.

Fig. 4
Fig. 4

Simulation results when the background and the phase map have discontinuities. (a) From left to right, one of the fringe patterns, the recovered phase map, F { a a ¯ } , and F { g g ¯ } . (b) and (c) Parallel rows with the tilt of the reference mirror increasing.

Fig. 5
Fig. 5

Experimental result. (a) An interferogram. (b) Reconstructed phase map.

Fig. 6
Fig. 6

Comparison among the nominal phase shifts, the values estimated by using the suggested algorithm, and those by using the iterative least-squares algorithm.

Tables (3)

Tables Icon

Table 1 Statistics of the Estimated Phase Shifts When Measuring a Smooth Phase Map

Tables Icon

Table 2 Statistics of the Estimated Phase Shifts When Measuring a Rough Phase Map

Tables Icon

Table 3 Random Phase Shifts and Their Estimates

Equations (19)

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I k ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + δ k ] ,
I k ( x , y ) = c 0 ( x , y ) + c 1 ( x , y ) cos δ k + c 2 ( x , y ) sin δ k ,
c = A 1 b ,
A = k = 0 K 1 [ 1 cos δ k sin δ k cos δ k cos 2 δ k cos δ k sin δ k sin δ k cos δ k sin δ k sin 2 δ k ] ,
b = k = 0 K 1 [ I k ( x , y ) I k ( x , y ) cos δ k I k ( x , y ) sin δ k ] T .
ϕ ( x , y ) = arctan [ c 2 ( x , y ) c 1 ( x , y ) ] .
δ ̂ k = δ k + ε k .
I ̂ k ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + δ k + ε k ] .
g ( x , y ) = b ( x , y ) exp [ i ϕ ( x , y ) ] = c 1 ( x , y ) i c 2 ( x , y ) ,
I ̂ k ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + δ k + ε k ] = a ( x , y ) + 1 2 b ( x , y ) exp [ i ϕ ( x , y ) ] exp [ i ( δ k + ε k ) ] + 1 2 b ( x , y ) exp [ i ϕ ( x , y ) ] exp [ i ( δ k + ε k ) ] = a ( x , y ) + 1 2 g ( x , y ) exp [ i ( δ k + ε k ) ] + 1 2 g * ( x , y ) exp [ i ( δ k + ε k ) ] ,
I ̂ k ( x , y ) = I k ( x , y ) + n = 1 + 1 n ! d n I k ( x , y ) d δ k n ε k n = I k ( x , y ) + 1 2 g ( x , y ) exp ( i δ k ) n = 1 + ( i ε k ) n n ! + 1 2 g * ( x , y ) exp ( i δ k ) n = 1 + ( i ε k ) n n ! .
ξ k = exp ( i δ k ) n = 1 + ( i ε k ) n n ! ,
I ̂ k ( x , y ) = I k ( x , y ) + 1 2 g ( x , y ) ξ k + 1 2 g * ( x , y ) ξ k * .
b ̂ = b + Δ b ,
Δ b = 1 2 g ( x , y ) k = 0 K 1 [ ξ k ξ k cos δ k ξ k sin δ k ] T + 1 2 g * ( x , y ) k = 0 K 1 [ ξ k * ξ k * cos δ k ξ k * sin δ k ] T .
c ̂ = A 1 ( b + Δ b ) = c + A 1 Δ b .
P ( u , v ) = F { a ̂ ( x , y ) a ̂ ¯ } F * { g ̂ ( x , y ) g ̂ ¯ } ,
β ( δ ) = u = π π v = π π P ( u , v ) .
P ( u , v ) = F { a ̂ a ̂ ¯ } F * ( g ̂ g ̂ ¯ ) = F { a a ¯ } F * { g g ¯ } + ( e + i f ) F { a a ¯ } F * { g g ¯ } + ( e * + i f * ) F { a a ¯ } F * { g * g ¯ * } + d ( 1 + e + i f ) F { g g ¯ } F * { g g ¯ } + d * ( 1 + e + i f ) F { g * g ¯ * } F * { g g ¯ } + d ( e * + i f * ) F { g g ¯ } F * { g * g ¯ * } + d * ( e * + i f * ) F { g * g ¯ * } F * { g * g ¯ * } .

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