Abstract

Fundamental properties of the phase-modulation ability of a nematic liquid-crystal cell are studied. Based on these phase-modulation properties of the liquid-crystal cell, a new type of speckle-shearing interferometer is proposed and studied experimentally. A liquid-crystal cell is employed as a phase shifter to implement the phase-shifting method for the conventional speckle-shearing interferometer. From the experiments used to measure the deformation of an object, the usefulness of the method is confirmed. Finally, a compensation method for phase-shift error is proposed on the basis of the statistical properties of the fully developed speckle field. In this method the speckle phase is regarded, in a statistical sense, as a standard phase object used to calibrate the measuring system. Experiments to confirm the error-compensation method are performed, and it is shown that the phase-shift error can be determined with an accuracy of as much as λ/100.

© 1991 Optical Society of America

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References

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  1. R. Jones, C. Wykes, “General parameters for the design and optimization of electronic speckle pattern interferometers,” Opt. Acta 28, 949–972 (1981).
    [CrossRef]
  2. J. H. Bruning, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, D. R. Herriott, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  3. P. Hariharan, B. F. Oreb, N. Brown, “A digital phase measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
    [CrossRef]
  4. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
    [CrossRef] [PubMed]
  5. H. K. Liu, J. A. Davis, R. A. Lilly, “Optical-data-processing properties of a liquid-crystal television spatial light modulator,” Opt. Lett. 10, 635–637 (1985).
    [CrossRef] [PubMed]
  6. M. Young, “Low-cost LCD video display for optical processing,” Appl. Opt. 25, 1024–1026 (1986).
    [CrossRef] [PubMed]
  7. F. T. S. Yu, S. Jutamulia, X. L. Huang, “Experimental application of low-cost liquid crystal TV to white-light optical signal processing,” Appl. Opt. 25, 3324–3326 (1986).
    [CrossRef] [PubMed]
  8. N. Konforti, E. Marom, S.-T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. 3, 251–253 (1988).
    [CrossRef]
  9. T. H. Barnes, T. Eiju, K. Matsuda, N. Ooyama, “Phase-only modulation using a twisted nematic liquid crystal television,” Appl. Opt. 28, 4845–4852 (1989).
    [CrossRef] [PubMed]
  10. Y. Y. Hung, “Shearography: a new optical method for strain measurement and nondestructive testing,” Opt. Eng. 21, 391–395 (1982).
    [CrossRef]
  11. J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
    [CrossRef]
  12. S. Toyooka, H. Nishida, “Automatic analysis of holographic and shearographic fringes to measure flexural strains in plates,” Opt. Eng. 28, 55–60 (1989).
    [CrossRef]
  13. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975), Chap. 2.
  14. H. Kadono, S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” Opt. Lett. 16, 883–885 (1991).
    [CrossRef] [PubMed]
  15. H. Kadono, T. Asakura, “Statistical properties of the speckle phase in the optical imaging system,” J. Opt. Soc. Am. A 2, 1787–1792 (1985).
    [CrossRef]
  16. H. Kadono, N. Takai, T. Asakura, “Statistical properties of the speckle phase in the diffraction region,” J. Opt. Soc. Am. A 3, 1080–1089 (1986).
    [CrossRef]

1991 (1)

1989 (2)

T. H. Barnes, T. Eiju, K. Matsuda, N. Ooyama, “Phase-only modulation using a twisted nematic liquid crystal television,” Appl. Opt. 28, 4845–4852 (1989).
[CrossRef] [PubMed]

S. Toyooka, H. Nishida, “Automatic analysis of holographic and shearographic fringes to measure flexural strains in plates,” Opt. Eng. 28, 55–60 (1989).
[CrossRef]

1988 (1)

N. Konforti, E. Marom, S.-T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. 3, 251–253 (1988).
[CrossRef]

1986 (4)

1985 (3)

1982 (2)

Y. Y. Hung, “Shearography: a new optical method for strain measurement and nondestructive testing,” Opt. Eng. 21, 391–395 (1982).
[CrossRef]

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
[CrossRef]

1981 (1)

R. Jones, C. Wykes, “General parameters for the design and optimization of electronic speckle pattern interferometers,” Opt. Acta 28, 949–972 (1981).
[CrossRef]

1974 (1)

Asakura, T.

Barnes, T. H.

Brangaccio, D. J.

Brown, N.

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
[CrossRef]

Bruning, J. H.

Creath, K.

Davis, J. A.

Eiju, T.

Gallagher, J. E.

Hariharan, P.

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
[CrossRef]

Herriott, D. R.

Huang, X. L.

Hung, Y. Y.

J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
[CrossRef]

Y. Y. Hung, “Shearography: a new optical method for strain measurement and nondestructive testing,” Opt. Eng. 21, 391–395 (1982).
[CrossRef]

Jones, R.

R. Jones, C. Wykes, “General parameters for the design and optimization of electronic speckle pattern interferometers,” Opt. Acta 28, 949–972 (1981).
[CrossRef]

Jutamulia, S.

Kadono, H.

Konforti, N.

N. Konforti, E. Marom, S.-T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. 3, 251–253 (1988).
[CrossRef]

Lilly, R. A.

Liu, H. K.

Marom, E.

N. Konforti, E. Marom, S.-T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. 3, 251–253 (1988).
[CrossRef]

Matsuda, K.

Nishida, H.

S. Toyooka, H. Nishida, “Automatic analysis of holographic and shearographic fringes to measure flexural strains in plates,” Opt. Eng. 28, 55–60 (1989).
[CrossRef]

Ooyama, N.

Oreb, B. F.

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
[CrossRef]

Rosenfeld, D. P.

Takai, N.

Takezaki, J.

J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
[CrossRef]

Toyooka, S.

H. Kadono, S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” Opt. Lett. 16, 883–885 (1991).
[CrossRef] [PubMed]

S. Toyooka, H. Nishida, “Automatic analysis of holographic and shearographic fringes to measure flexural strains in plates,” Opt. Eng. 28, 55–60 (1989).
[CrossRef]

White, A. D.

Wu, S.-T.

N. Konforti, E. Marom, S.-T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. 3, 251–253 (1988).
[CrossRef]

Wykes, C.

R. Jones, C. Wykes, “General parameters for the design and optimization of electronic speckle pattern interferometers,” Opt. Acta 28, 949–972 (1981).
[CrossRef]

Young, M.

Yu, F. T. S.

Appl. Opt. (5)

J. Appl. Mech. (1)

J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
[CrossRef]

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

Opt. Acta (1)

R. Jones, C. Wykes, “General parameters for the design and optimization of electronic speckle pattern interferometers,” Opt. Acta 28, 949–972 (1981).
[CrossRef]

Opt. Commun. (1)

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
[CrossRef]

Opt. Eng. (2)

Y. Y. Hung, “Shearography: a new optical method for strain measurement and nondestructive testing,” Opt. Eng. 21, 391–395 (1982).
[CrossRef]

S. Toyooka, H. Nishida, “Automatic analysis of holographic and shearographic fringes to measure flexural strains in plates,” Opt. Eng. 28, 55–60 (1989).
[CrossRef]

Opt. Lett. (3)

Other (1)

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975), Chap. 2.

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Figures (9)

Fig. 1
Fig. 1

Phase modulations of liquid-crystal cells 1 and 2 as a function of applied voltage. Applied voltage is represented in a peak-to-peak value.

Fig. 2
Fig. 2

Dependence of the phase modulation of liquid-crystal cell 1 on the incident angle of light with the applied voltage 5 Vpp.

Fig. 3
Fig. 3

Optical system of a speckle-shearing interferometer employing a liquid-crystal phase modulator. The liquid-crystal phase modulator introduces phase shifts between two sheared images of the object in order to achieve the phase-shifting technique (HM, half-mirror; L’s, lenses; SW’s, switches; PL’s, polarizers; LC, liquid-crystal cell; CCD, charge-coupled device).

Fig. 4
Fig. 4

Experimental results of the measurements of object deformation: (a) raw phase distribution, (b) median filtering with a 3 × 3 window, (c) filtering with edge detection and local average, (d) unwrapped phase distribution for (c).

Fig. 5
Fig. 5

Derivative distribution of the deformation of the object.

Fig. 6
Fig. 6

Model of phase-shift error of the phase modulator. Phasors for steps 1 and 3 suffer the same amount of shift error Δψs, i.e., symmetrical shift error.

Fig. 7
Fig. 7

Theoretical probability-density distribution of the calculated speckle phase for the phase-shift error Δψs = 2π/100, 0, −2π/100, and −2π/50 around the desired phase shift ψ = π/2.

Fig. 8
Fig. 8

Measured probability-density distributions of the speckle phase Δψs = 0.42 and Δψs = −0.21 around the desired phase shift ψ = π/2.

Fig. 9
Fig. 9

Determination of the symmetrical phase-shift error Δψs from the measured probability-density distribution of the speckle phase. The horizontal axis represents an introduced symmetrical phase-shift error around the desired phase shift ψ = π/2, and the vertical axis is the evaluated phase-shift error Δψs from ψ = π/2.

Tables (1)

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Table 1 Parameters of the Two Liquid-Crystal Cells

Equations (16)

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I i ( x , y ) = | f ( x , y ) | 2 + | f ( x + Δ x , y ) | 2 + 2 | f ( x , y ) | × | f ( x + Δ x , y ) | cos [ ϕ ( x , y ) + ψ i ] , i = 1 , 2 , 3 , ψ 1 = ψ , ψ 2 = 0 , ψ 3 = ψ ,
I i ( x , y ) = | f ( x , y ) | 2 + | f ( x + Δ x , y ) | 2 + 2 | f ( x , y ) | × | f ( x + Δ x , y ) | cos [ ϕ ( x , y ) + Δ θ ( x , y ) + ψ i ] , i = 1 , 2 , 3 ,
Δ θ ( x , y ) = ( 4 π / λ ) [ w ( x , y ) w ( x + Δ x , y ) ] ( 4 π / λ ) [ w ( x , y ) / x ] Δ x .
ϕ ( x , y ) = tan 1 { [ I 1 ( x , y ) I 3 ( x , y ) ] ( cos ψ 1 ) [ I 1 ( x , y ) + I 3 ( x , y ) 2 I 2 ( x , y ) ] sin ψ } .
γ = 2 | f ( x , y ) | | f ( x + Δ x , y ) | | f ( x , y ) | 2 + | f ( x + Δ x , y ) | 2 .
p ϕ ( ϕ ) = 1 / 2 π , π < ϕ π .
ψ 1 = ψ = ψ Δ ψ s , ψ 2 = 0 , ψ 3 = ψ = ψ + Δ ψ s .
I i ( x , y ) = | f ( x , y ) | 2 + | f ( x + Δ x , y ) | 2 + 2 | f ( x , y ) | × | f ( x + Δ x , y ) | cos [ ϕ ( x , y ) + ψ i ] , i = 1 , 2 , 3 , ψ 1 = ψ , ψ 2 = 0 , ψ 3 = ψ .
tan ϕ = I 1 I 3 I 1 + I 3 2 I 2 cos ψ 1 sin ψ .
tan ϕ = I 1 I 3 I 1 + I 3 2 I 2 cos ψ 1 sin ψ .
p ϕ ( ϕ ) = p ϕ ( ϕ ) | d ϕ / d ϕ | = 1 π | α ( α 2 1 ) cos 2 ϕ + ( α 2 + 1 ) | ,
α = sin ψ cos ψ 1 cos ψ 1 sin ψ .
p ϕ ( ϕ ) = α / π [ ( α 2 1 ) cos 2 ϕ + ( α 2 + 1 ) ] .
T = 1 2 π π π cos 2 ϕ p ϕ ( ϕ ) d ϕ .
α = T / π + [ ( T / π ) 2 + 1 ] 1 / 2 .
Δ ψ s = sin 1 1 α 2 1 + α 2 .

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