Abstract

A new method of estimating reference phase shifts in phase-shifting interferometry is proposed. The reference phase shifts are determined from a matrix that represents the interframe intensity correlation (IIC) of phase-shifted interferograms. The root-mean-square error of intensity measurement is automatically obtained from the smallest eigenvalue of the IIC matrix. The proposed method requires only four interferograms, unlike others, and can extract phase shifts reliably even from interferograms without well-defined fringes, such as speckle patterns. In typical conditions, reference phase shifts and wave-front phases can be determined with an accuracy of λ/6310 and λ/150, respectively. The validity of the method is tested by comparing it with other methods in experiments and simulations.

© 2005 Optical Society of America

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2003 (2)

2001 (2)

2000 (1)

1999 (2)

H. van Brug, “Phase-step calibration for phase-stepped interferometry,” Appl. Opt. 38, 3549–3555 (1999).
[Crossref]

H. Huang, M. Itoh, T. Yatagai, “Phase retrieval of phase-shifting interferometry with iterative least-squares fitting algorithm: experiments,” Opt. Rev. 6, 196–203 (1999).
[Crossref]

1998 (3)

1997 (3)

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

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 36, 3101–3106 (1997).
[Crossref]

Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36, 271–276 (1997).
[Crossref] [PubMed]

1996 (1)

1995 (1)

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).
[Crossref]

1994 (1)

1991 (1)

K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[Crossref]

1985 (2)

1983 (1)

1974 (1)

1966 (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Bokor, J.

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Cai, L. Z.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Chen, X.

Cheng, Y. Y.

Creath, K.

Elssner, K.-E.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988), pp. 408–455.

Gallagher, J. E.

Goldberg, K. A.

Gramaglia, M.

Grzanna, J.

Han, G.-S.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 36, 3101–3106 (1997).
[Crossref]

G.-S. Han, S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–7325 (1994).
[Crossref] [PubMed]

Herriott, D. R.

Huang, H.

H. Huang, M. Itoh, T. Yatagai, “Phase retrieval of phase-shifting interferometry with iterative least-squares fitting algorithm: experiments,” Opt. Rev. 6, 196–203 (1999).
[Crossref]

Huntley, J. M.

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
[Crossref]

Itoh, M.

H. Huang, M. Itoh, T. Yatagai, “Phase retrieval of phase-shifting interferometry with iterative least-squares fitting algorithm: experiments,” Opt. Rev. 6, 196–203 (1999).
[Crossref]

Kang, M.-G.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 36, 3101–3106 (1997).
[Crossref]

Kim, B. C.

Kim, S. W.

Kim, S.-W.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 36, 3101–3106 (1997).
[Crossref]

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).
[Crossref]

G.-S. Han, S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–7325 (1994).
[Crossref] [PubMed]

Kong, I.-B.

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).
[Crossref]

Larkin, K.

Liu, Q.

Marroquin, J. L.

Merkel, K.

Ochoa, N. A.

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
[Crossref]

Okada, K.

K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[Crossref]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988), pp. 408–455.

Rodriguez Vera, R.

Rosenfeld, D. P.

Sato, A.

K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[Crossref]

Schwider, J.

Servin, M.

Spolaczyk, R.

Strang, G.

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich, 1988), pp. 201–202.

Surrel, Y.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988), pp. 408–455.

Tsujiuchi, J.

K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[Crossref]

van Brug, H.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988), pp. 408–455.

White, A. D.

Wyant, J. C.

Yamaguchi, I.

Yang, X. L.

Yatagai, T.

H. Huang, M. Itoh, T. Yatagai, “Phase retrieval of phase-shifting interferometry with iterative least-squares fitting algorithm: experiments,” Opt. Rev. 6, 196–203 (1999).
[Crossref]

Yeazell, J. A.

Zhang, T.

Appl. Opt. (9)

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

Metrologia (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Opt. Commun. (1)

K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[Crossref]

Opt. Eng. (3)

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).
[Crossref]

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 36, 3101–3106 (1997).
[Crossref]

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Opt. Rev. (1)

H. Huang, M. Itoh, T. Yatagai, “Phase retrieval of phase-shifting interferometry with iterative least-squares fitting algorithm: experiments,” Opt. Rev. 6, 196–203 (1999).
[Crossref]

Other (3)

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich, 1988), pp. 201–202.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1988), pp. 408–455.

Ref. 22, pp. 408–412.

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Figures (7)

Fig. 1
Fig. 1

IIC: (a) phase-shifted interferograms, (b) intensity matrix I, (c) IIC matrix C(1), (d) differential correlation matrix C.

Fig. 2
Fig. 2

Synthesized model for testing the method (256 × 256 pixels). (a) A nominal wave-front phase. (b) An interferogram with 20% additive noise.

Fig. 3
Fig. 3

Noise level estimation. (a) σ is the square root of the smallest eigenvalue of temporal correlation matrix, and σ0 is the nominal value. Four results per each noise level are plotted. The length of the error bar is twice the standard deviation of 128 estimations. (b) σ is the square root of the minimized variance with the iterative least-squares method.

Fig. 4
Fig. 4

Phase-shift estimation error. Each point represents a phase-shifted frame, and 16 sets are plotted. The length of the bar is twice the standard deviation of 128 estimations. One typical set is chosen and connected by lines. The noise level is 5%. (a) M = 4. The phasor diagram of the selected set is displayed in the upper right corner. Root-mean-square error of the whole data is 0.796 × 10−3 rad. (b) M = 5. Root-mean-square error of the whole data is 0.996 × 10−3 rad.

Fig. 5
Fig. 5

Wave-front phase error. (a) Nominal wave-front phase, which is the horizontal cross section of Fig. 2(a) at the center. Horizontal axis is the pixel index. (b) Wave-front phase error of the set chosen in Fig. 4(a). Phase error of the proposed method is compared with the one that is calculated neglecting phase-shift error. The bias and harmonic error in the wave-front phase is manifested as the effects of the phase-shift error.

Fig. 6
Fig. 6

Convergence paths. The paths are those of phase shifts in iteration procedures for the sets chosen in Fig. 4. Initial values far from exact ones converge to true values. (a) M = 4, (b) M = 5.

Fig. 7
Fig. 7

Actual experiment. (a) Mach–Zehnder interferometer. BS, beam splitter; BE, beam expander. (b) Part of fringe acquired by CCD1. (c) Part of speckle pattern acquired by CCD2.

Tables (1)

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Table 1 Comparison with the Fourier-Transform Method

Equations (50)

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I ( t , r ) = a ( r ) + b ( r ) cos [ ψ ( r ) - θ ( t ) ] + Δ I ( t , r ) ,
i m n = a n + b n cos ( ψ n - θ m ) + Δ i m n , { m = 1 , 2 , , M n = 1 , 2 , , N .
i m n = ½ ( u m z n * + u m * z n + 2 a n ) + Δ i m n .
I = ½ UA + + Δ I ,
U = [ u 1 u 1 * 1 u M u M * 1 ] ,
A = [ z 1 z 1 * 2 a 1 z N z N * 2 a N ] .
E = m = 1 M n = 1 N [ a n + b n cos ( ψ n - θ m ) - i m n ] 2 .
j m n i m n - i 1 n = ½ [ ( u m - 1 ) z n * + ( u m * - 1 ) z n ] + Δ j m n ,
J = DI = ½ VB + + Δ J ,
D = [ - 1 1 0 0 0 0 - 1 0 0 1 ] ,
V [ u 2 - 1 u 2 * - 1 u M - 1 u M * - 1 ] ,
B [ z 1 z 1 * z N z N * ] .
I = ½ U k A k + ,
A k = 2 I + U k ( U k + U k ) - 1 ,
I = ½ U k + 1 A k + ,
U k + 1 = 2 IA k + ( A k A k + ) - 1 ,
U k + 1 = Proj ( U k + 1 ) .
Proj [ μ 11 μ 12 μ 13 μ 21 μ 22 μ 23 μ M 1 μ M 2 μ M 3 ] = [ 1 1 1 exp ( i θ 2 ) exp ( - i θ 2 ) 1 exp ( i θ M ) exp ( - i θ M ) 1 ] ,
exp ( i θ m ) = u m u m u 1 u 1 ,
U k + 1 = Proj [ 2 IA k + ( A k A k + ) - 1 ] .
U k + 1 = Proj [ 2 IA k + ( A k A k + ) - 1 ] = Proj [ I I + U k ( U k + I I + U k ) - 1 U k + U k ] .
C ( θ k , θ l ) ~ d r I ( θ k , r ) I ( θ l , r ) .
C ( 1 ) 1 N I I + ,
C ( 1 ) = 1 N ( 1 4 UA + AU + + Δ I Δ I + + 1 2 UA + Δ I + + 1 2 Δ IAU + )
( Δ IA ) k 1 = n = 1 N Δ i k n z n = N Δ i k z ¯ N Δ i k ¯ z ¯ N × 0 × z ¯ = 0 ,
Δ I Δ I + Δ I Δ I + = N σ 0 2 1 M ,
C ( 1 ) = 1 4 N UA + AU + + σ 0 2 1 M .
C ( 1 ) v = μ v ,
1 4 N ( UA + AU + ) v ( μ - σ 0 2 ) v .
C ( n s ) 1 4 N UA + AU +
C ( n s ) 1 4 [ u 1 u 1 * 1 u M u M * 1 ] [ z n 2 ¯ z n * 2 ¯ 2 z n * a n ¯ z n 2 ¯ z n 2 ¯ 2 z n a n ¯ 2 z n a n ¯ 2 z n * a n ¯ 4 a n 2 ¯ ] × [ u 1 * u M * u 1 u M 1 1 ] ,
z n 2 ¯ = 1 N n = 1 N z n 2 .
C D C ( n s ) D T .
C 1 4 [ u 2 - 1 u 2 * - 1 u M - 1 u M * - 1 ] [ z n 2 ¯ z n * 2 ¯ z n 2 ¯ z n 2 ¯ ] × [ u 2 * - 1 u M * - 1 u 2 - 1 u M - 1 ] .
C 1 4 N VB + BV + ,
C VHV + ,
C = V k HV k + .
H k = ( V k + V k ) - 1 V k + CV k ( V k + V k ) - 1 .
C = V k + 1 H k V k + .
V k + 1 = CV k ( V k + CV k ) - 1 V k + V k ,
V = Proj ( V )
exp ( i θ m + 1 ) = ( V ) m 1 + 1 ( V ) m 1 + 1 m = 1 , 2 , , M - 1.
V k + 1 = Proj [ CV k ( V k + CV k ) - 1 V k + V k ] ,
f = f ( θ 2 , , θ M , z n 2 ¯ , z n 2 ¯ ) = k = 1 , l = 1 M , M c k l - c k l ( m o ) 2 ,
[ z n 2 ¯ z n * 2 ¯ z n 2 ¯ z n 2 ¯ ] = [ b n 2 ¯ b n 2 exp ( i ψ n ) ¯ b n 2 exp ( i ψ n ) ¯ b n 2 ¯ ] [ b 2 0 0 b 2 ] ,
c k l ½ b 2 ¯ [ cos ( θ k + 1 - θ l + 1 ) - cos θ k + 1 - cos θ l + 1 + 1 ] .
θ m 2 π ( m - 1 ) / M ,
z n 2 ¯ = b 2 ½ c max ,
z n 2 ¯ 0 + i 0.
f ( 1 ) = k = 2 , n = 1 M , N ( Δ i k n - Δ i 1 n ) 2 , f ( 2 ) = k = 1 , n = 1 M , N Δ i k n 2 ,

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