Abstract

A procedure for obtaining real-time holographic moirélike patterns and measuring small angles is proposed. Two rotated sinusoidal phase gratings are superposed, and the result represents a promising technique for making small-angle measurements in metrological applications. The experiments are performed with a diffusion-only recording mechanism in the photorefractive crystal Bi12TiO20 illuminated by λ = 0.633 µm light from a He–Ne laser.

© 2000 Optical Society of America

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References

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  1. J. W. Dally, W. F. Riley, Experimental Stress Analysis, 2nd ed. (McGraw-Hill, Tokyo, 1978).
  2. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]
  3. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  4. A. A. Kamshilin, M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53, 23–26 (1985).
    [CrossRef]
  5. S. I. Stepanov, “Applications of photorefractive crystals,” Rep. Prog. Phys. 57, 39–116 (1994).
    [CrossRef]
  6. A. Marrakchi, R. V. Johnson, A. R. Tanguay, “Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime),” J. Opt. Soc. Am. B 3, 321–336 (1986).
    [CrossRef]
  7. J. F. Nye, Physical Properties of Crystals (Oxford U. Press, New York, 1993).
  8. P. A. M. dos Santos, “Photorefractive intrinsic parameters measured in Bi12TiO20 crystal at λ = 0.633 µm,” Opt. Commun. 80, 225–228 (1991).
    [CrossRef]

1994 (1)

S. I. Stepanov, “Applications of photorefractive crystals,” Rep. Prog. Phys. 57, 39–116 (1994).
[CrossRef]

1991 (1)

P. A. M. dos Santos, “Photorefractive intrinsic parameters measured in Bi12TiO20 crystal at λ = 0.633 µm,” Opt. Commun. 80, 225–228 (1991).
[CrossRef]

1986 (1)

1985 (1)

A. A. Kamshilin, M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53, 23–26 (1985).
[CrossRef]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Dally, J. W.

J. W. Dally, W. F. Riley, Experimental Stress Analysis, 2nd ed. (McGraw-Hill, Tokyo, 1978).

dos Santos, P. A. M.

P. A. M. dos Santos, “Photorefractive intrinsic parameters measured in Bi12TiO20 crystal at λ = 0.633 µm,” Opt. Commun. 80, 225–228 (1991).
[CrossRef]

Johnson, R. V.

Kamshilin, A. A.

A. A. Kamshilin, M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53, 23–26 (1985).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Marrakchi, A.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U. Press, New York, 1993).

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Petrov, M. P.

A. A. Kamshilin, M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53, 23–26 (1985).
[CrossRef]

Riley, W. F.

J. W. Dally, W. F. Riley, Experimental Stress Analysis, 2nd ed. (McGraw-Hill, Tokyo, 1978).

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Stepanov, S. I.

S. I. Stepanov, “Applications of photorefractive crystals,” Rep. Prog. Phys. 57, 39–116 (1994).
[CrossRef]

Tanguay, A. R.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

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

Opt. Commun. (2)

P. A. M. dos Santos, “Photorefractive intrinsic parameters measured in Bi12TiO20 crystal at λ = 0.633 µm,” Opt. Commun. 80, 225–228 (1991).
[CrossRef]

A. A. Kamshilin, M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53, 23–26 (1985).
[CrossRef]

Rep. Prog. Phys. (1)

S. I. Stepanov, “Applications of photorefractive crystals,” Rep. Prog. Phys. 57, 39–116 (1994).
[CrossRef]

Other (3)

J. F. Nye, Physical Properties of Crystals (Oxford U. Press, New York, 1993).

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

J. W. Dally, W. F. Riley, Experimental Stress Analysis, 2nd ed. (McGraw-Hill, Tokyo, 1978).

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Figures (8)

Fig. 1
Fig. 1

Sketch of the geometrical approach to showing the relation between rotated gratings and the moiré pattern that they produce.

Fig. 2
Fig. 2

Sketch of the experimental setup. Abbreviations are defined in text.

Fig. 3
Fig. 3

BTO sample and characteristics of its crystal orientation and light-beam illumination.

Fig. 4
Fig. 4

Dynamic moiré patterns produced by two sinusoidal phase gratings superposed but rotated by small angle α.

Fig. 5
Fig. 5

Experimental curve of the dependence of moiré fringe spacing δ on grating rotation angle α.

Fig. 6
Fig. 6

Mechanical arrangement for rotating the photorefractive sample to calibrate the standard micrometer head by the moiré pattern produced.

Fig. 7
Fig. 7

Sketch of the mechanical test system of Fig. 6. Here α is the small rotation produced by micrometer head displacement δx.

Fig. 8
Fig. 8

Experimental curve of the moiré fringe spacing for one step of the micrometer head (5 mm), which means one complete lap of the unit.

Equations (12)

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Λsin α=δsinϕ-α
tan α=sin ϕδ/Λ+cos ϕ.
α=Λ/δ.
Ix=I01+m cos kgx.
Δnˆx=12r41n03Escx1000-10000.
η=χˆ2,
Rzz=-iuˆ·χˆwˆSz,  Szz=-iwˆ·χˆuˆRz,
χˆ=1ε0π2λ cosθB εˆ1,
χˆ=χ01000-10000,
χ0=12λ cosθBr41n03Escx.
u·χˆw=χ0cos γ sin γ01000-10000cos βsin β0=χ0 cosγ+β.
η=πn3r41Esc2λ cosθB m sin ρlρ2,

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