Abstract

Phase-shifting interferometry with a genetic algorithm is proposed. The correction of unknown phase-shifting error is an important task in general phase-shifting interferometry. Since phase-shifting errors generate twin image noise in a reconstructed image, we can reduce the phase-shifting errors indirectly by trying to eliminate the twin-image noise in the reconstructed image. By Zernike polynomial expansion, the reconstructed image is represented as the evenness and oddness, where the ratio of the evenness and oddness is a measure of the amount of the twin image noise. We employ the genetic algorithm for finding the fittest phase shifts of interferograms by reducing the evenness of the reconstructed image, which leads to reduction of phase-shifting errors. This phase-shifting interferometry with a genetic algorithm is confirmed experimentally.

© 2008 Optical Society of America

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References

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2008 (1)

2007 (3)

L. Waller and G. Barbastathis, “Error analysis of phase-shifting for phase and amplitude tomographic reconstruction,” in Signal Recovery and Synthesis, Technical Digest Series (OSA2007), paper CTuB5.

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]

S. T. Thurman and J. R. Fienup, “Phase error correction for digital holographic imaging,” in Signal Recovery and Synthesis, Technical Digest Series (OSA, 2007), paper SMC1.

2006 (3)

2004 (3)

2003 (1)

2001 (1)

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

2000 (3)

1999 (2)

Z. Michalewicz, Genetic Algorithms+DataStructures=Evolution Programs (Springer-Verlag, 1999).
[PubMed]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

1995 (2)

B. Y. Zel'dovich, A. V. Mamaev, and V. V. Shkunov, Speckle Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1995).

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610-3619 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (Taylor & Francis, 1993), pp. 94-140.

1991 (1)

1979 (1)

L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, 1979).

Ahlfors, L. V.

L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, 1979).

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Barbastathis, G.

L. Waller and G. Barbastathis, “Error analysis of phase-shifting for phase and amplitude tomographic reconstruction,” in Signal Recovery and Synthesis, Technical Digest Series (OSA2007), paper CTuB5.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

Cai, L. Z.

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] [PubMed]

Cai, L.-Z.

Chen, X.

Creath, K.

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610-3619 (1995).
[CrossRef] [PubMed]

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (Taylor & Francis, 1993), pp. 94-140.

Cuche, E.

Depeursinge, C.

Dong, G. Y.

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]

Fienup, J. R.

S. T. Thurman and J. R. Fienup, “Phase error correction for digital holographic imaging,” in Signal Recovery and Synthesis, Technical Digest Series (OSA, 2007), paper SMC1.

J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A 20, 609-620 (2003).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2004).

Gramaglia, M.

Hahn, J.

Han, G.-S.

He, M.-Z.

Indebetouw, G.

Kim, H.

Kim, S.-W.

Kim, T.

Lai, G.

Langoju, R.

Lee, B.

Liu, Q.

Mamaev, A. V.

B. Y. Zel'dovich, A. V. Mamaev, and V. V. Shkunov, Speckle Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1995).

Marquet, P.

Meng, X. F.

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]

Meng, X.-F.

Michalewicz, Z.

Z. Michalewicz, Genetic Algorithms+DataStructures=Evolution Programs (Springer-Verlag, 1999).
[PubMed]

Miller, J. J.

Patil, A.

Poon, T.-C.

Rastogi, P.

Schilling, B. W.

Schmit, J.

Shen, X. X.

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]

Shinoda, K.

Shkunov, V. V.

B. Y. Zel'dovich, A. V. Mamaev, and V. V. Shkunov, Speckle Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1995).

Suzuki, Y.

Thurman, S. T.

S. T. Thurman and J. R. Fienup, “Phase error correction for digital holographic imaging,” in Signal Recovery and Synthesis, Technical Digest Series (OSA, 2007), paper SMC1.

Waller, L.

L. Waller and G. Barbastathis, “Error analysis of phase-shifting for phase and amplitude tomographic reconstruction,” in Signal Recovery and Synthesis, Technical Digest Series (OSA2007), paper CTuB5.

Wang, Y. R.

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]

Wang, Y.-R.

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

Wu, M. H.

Xu, X. F.

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]

Yamaguchi, I.

I. Yamaguchi, “Phase-shifting digital holography,” in Digital Holography and Three-Dimensional Display, T. -C. Poon, ed. (Springer, 2006), pp. 145-171.
[CrossRef]

Yang, X. L.

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] [PubMed]

Yatagai, T.

Yeazell, J. A.

Zel'dovich, B. Y.

B. Y. Zel'dovich, A. V. Mamaev, and V. V. Shkunov, Speckle Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1995).

Zhang, H.

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]

Appl. Opt. (6)

Appl. Phys. Lett. (1)

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]

J. Opt. Soc. Am. A (2)

Opt. Express (2)

Opt. Lett. (2)

Other (10)

S. T. Thurman and J. R. Fienup, “Phase error correction for digital holographic imaging,” in Signal Recovery and Synthesis, Technical Digest Series (OSA, 2007), paper SMC1.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2004).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, 1979).

Z. Michalewicz, Genetic Algorithms+DataStructures=Evolution Programs (Springer-Verlag, 1999).
[PubMed]

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (Taylor & Francis, 1993), pp. 94-140.

I. Yamaguchi, “Phase-shifting digital holography,” in Digital Holography and Three-Dimensional Display, T. -C. Poon, ed. (Springer, 2006), pp. 145-171.
[CrossRef]

B. Y. Zel'dovich, A. V. Mamaev, and V. V. Shkunov, Speckle Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1995).

L. Waller and G. Barbastathis, “Error analysis of phase-shifting for phase and amplitude tomographic reconstruction,” in Signal Recovery and Synthesis, Technical Digest Series (OSA2007), paper CTuB5.

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Figures (11)

Fig. 1
Fig. 1

Optical configuration for phase-shifting interferometry in Fourier optics.

Fig. 2
Fig. 2

Relation between two phases of interferograms in the complex plane.

Fig. 3
Fig. 3

Original object and twin image in hologram with Fourier optics.

Fig. 4
Fig. 4

Numerical reconstructions of twin images in Fourier optics at (a)  z = 0.2 f , (b)  z = 0 , and (c)  z = 0.2 f .

Fig. 5
Fig. 5

Flow of the microgenetic algorithm to eliminate a twin image.

Fig. 6
Fig. 6

Numerically reconstructed images (a) before and (b) after the genetic algorithm.

Fig. 7
Fig. 7

Evolutions in the genetic algorithm of (a) the fittest chromosome and (b) the resultant evenness and oddness in reconstructed images.

Fig. 8
Fig. 8

Relationship between (a) the optimized coefficients of phase-shift bases and (b) the resultant phase shifts of interferograms.

Fig. 9
Fig. 9

Numerically reconstructed images. Parts (a) and (b) show the images at the focal plane z = 0 and defocused plane z = 0.2 f , respectively, before the optimization. Parts (c) and (d) show the images at the same positions as (a) and (b) after the optimization.

Fig. 10
Fig. 10

Relationship between (a) the optimized coefficients of phase-shift bases and (b) the resultant phase shifts of interferograms.

Fig. 11
Fig. 11

Reconstructed images by LC SLM with previous holograms. Parts (a) and (b) are before the optimization and parts (c) and (d) are after the proposed optimization.

Equations (25)

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

I i = | U O + U R , i | 2 = A O 2 + A R 2 + 2 A O A R cos ( φ α i ) ,
U O = A O exp ( j φ ) ,
U R , i = A R exp ( j α i ) .
φ ^ exp ( j φ ) = ( cos φ , sin φ ) ,
α ^ i exp ( j α i ) = ( cos α i , sin α i ) .
I i = A O 2 + A R 2 + 2 A O A R φ ^ α ^ i .
I i j ( I i I j ) / A R = 2 A O φ ^ ( α ^ i α ^ j ) = 2 A O 2 2 cos ( α i α j ) sin ( φ α i + α j 2 ) .
tan ( φ ) = I 2 I 4 I 1 I 3 ,
tan ( φ + π / 4 ) = I 1 + I 2 I 3 I 4 I 1 I 2 I 3 + I 4 ,
U O = I 13 + j I 42 ,
U O = e π / 4 ( I 12 + I 43 ) + j e π / 4 ( I 31 + I 42 ) .
U O = i 1 a i 1 I i 1 ,
U O = i 1 2 a i 1 A O [ ( cos α i cos α 1 ) cos φ + ( sin α i sin α 1 ) sin φ ] = i 1 a i 1 [ ( cos α i cos α 1 ) ( U O + U O * ) + ( sin α i sin α 1 ) ( U O U O * ) ] = i 1 a i 1 ( cos α i cos α 1 + sin α i sin α 1 ) U O + i 1 a i 1 ( cos α i cos α 1 sin α i + sin α 1 ) U O * .
U O ( λ f f X , λ f f Y ; z = 2 f ) = H ( f X , f Y ; z 0 ) F [ U O ( x , y ; z 0 ) ] .
H ( f X , f Y ; z 0 ) = A j λ f exp [ j π λ z 0 ( f X 2 + f Y 2 ) ] .
H * ( f X , f Y ; z 0 ) = H ( f X , f Y ; z 0 ) .
U twin ( x , y ; z 0 ) = F 1 [ U O * ( λ f f X , λ f f Y ; z = 2 f ) / H ( f X , f Y ; z 0 ) ] = F 1 { [ U O ( λ f f X , λ f f Y ; z = 2 f ) / H ( f X , f Y ; z 0 ) ] * } = U O * ( x , y ; z 0 ) .
U n m { ( n + 1 ) / π R n 0 ( ρ ) , m = 0 2 ( n + 1 ) / π R n m ( ρ ) cos m θ , m > 0 2 ( n + 1 ) / π R n m ( ρ ) sin m θ , m < 0 ,
R n ± m ( ρ ) = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! s ! ( ( n + m ) / 2 s ) ! ( ( n m ) / 2 s ) ! ρ n 2 s .
cost function = n m = 0 , ± 2 , ± 4 ± n ( coefficients of U n m ) n m = ± 1 , ± 3 ± n ( coefficients of U n m ) .
x i = { Re { a 21 } , Im { a 21 } , Re { a 31 } , Im { a 31 } , , Re { a N 1 } , Im { a N 1 } } .
x i = { 0 , 1 , 1 , 0 , 0 , 1 } .
U O = i 1 2 a i 1 A O [ ( cos α i cos α 1 ) cos φ + ( sin α i sin α 1 ) sin φ ] .
e = | U O c U O | = | i 1 2 a i 1 A O [ ( cos α i cos α 1 ) cos φ + ( sin α i sin α 1 ) sin φ ] c ( cos φ + j sin φ ) | .
e = | U O c U O | = norm ( ( Re { a 21 } Re { a 31 } Re { a N 1 } Im { a 21 } Im { a 31 } Im { a N 1 } ) ( cos α 2 cos α 1 cos α 3 cos α 1 cos α N cos α 1 sin α 2 sin α 1 sin α 3 sin α 1 sin α N sin α 1 ) c ( 1 0 0 1 ) ) ,

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