Abstract

We present a study of the variations of a speckle pattern passing through a grating that can be displaced. This study is described theoretically by a simple model based on the scalar diffraction theory in the Fresnel zone. The intensity correlation of the modified speckle as a function of the grating displacement is obtained and compared with experimental results. The possibilities of metrological applications in optical encoders are suggested.

© 2006 Optical Society of America

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References

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  1. Y. Arai and T. Kurata, "A high-resolution encoder by multiplication of Moire fringes," Eng. Technol. 9, 1-7 (1987).
  2. K. Engelhardt and P. Seitz, "High-resolution optical position encoder with large mounting tolerances," Appl. Opt. 36, 2912-2916 (1997).
    [CrossRef] [PubMed]
  3. Y. Wei-Hung, W. Bletscher, and M. Mansuripur, "High resolution optical shaft encoder for motor speed control based on an optical disk pick-up," Rev. Sci. Instrum. 69, 3068-3071 (1998).
    [CrossRef]
  4. D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
    [CrossRef]
  5. Y. Jourlin, J. Jay, and O. Parriaux, "Compact diffractive interferometric displacement sensor in reflection," Precis. Eng. 26, 1-6 (2002).
    [CrossRef]
  6. J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena (Springer-Verlag, 1975), p. 36.
    [CrossRef]
  7. R. E. Luna, E. R. Méndez, J. Q. Lu, and Z.-H. Gu, "Enhanced backscattering due to total internal reflection at a dielectric-air interface," J. Mod. Opt. 42, 257-269 (1995).
    [CrossRef]
  8. F. Perez Quintián, M. A. Rebollo, and N. G. Gaggioli, "Diffusion of light transmitted from rough surfaces," J. Mod. Opt. 44, 447-460 (1997).
    [CrossRef]
  9. A. Papoulis, Systems and Transforms with Applications in Optics (Krieger, 1986), pp. 315-320.

2002 (1)

Y. Jourlin, J. Jay, and O. Parriaux, "Compact diffractive interferometric displacement sensor in reflection," Precis. Eng. 26, 1-6 (2002).
[CrossRef]

2000 (1)

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
[CrossRef]

1998 (1)

Y. Wei-Hung, W. Bletscher, and M. Mansuripur, "High resolution optical shaft encoder for motor speed control based on an optical disk pick-up," Rev. Sci. Instrum. 69, 3068-3071 (1998).
[CrossRef]

1997 (2)

F. Perez Quintián, M. A. Rebollo, and N. G. Gaggioli, "Diffusion of light transmitted from rough surfaces," J. Mod. Opt. 44, 447-460 (1997).
[CrossRef]

K. Engelhardt and P. Seitz, "High-resolution optical position encoder with large mounting tolerances," Appl. Opt. 36, 2912-2916 (1997).
[CrossRef] [PubMed]

1995 (1)

R. E. Luna, E. R. Méndez, J. Q. Lu, and Z.-H. Gu, "Enhanced backscattering due to total internal reflection at a dielectric-air interface," J. Mod. Opt. 42, 257-269 (1995).
[CrossRef]

1987 (1)

Y. Arai and T. Kurata, "A high-resolution encoder by multiplication of Moire fringes," Eng. Technol. 9, 1-7 (1987).

Alonso, J.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
[CrossRef]

Arai, Y.

Y. Arai and T. Kurata, "A high-resolution encoder by multiplication of Moire fringes," Eng. Technol. 9, 1-7 (1987).

Bernabeu, E.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
[CrossRef]

Bletscher, W.

Y. Wei-Hung, W. Bletscher, and M. Mansuripur, "High resolution optical shaft encoder for motor speed control based on an optical disk pick-up," Rev. Sci. Instrum. 69, 3068-3071 (1998).
[CrossRef]

Crespo, D.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
[CrossRef]

Engelhardt, K.

Gaggioli, N. G.

F. Perez Quintián, M. A. Rebollo, and N. G. Gaggioli, "Diffusion of light transmitted from rough surfaces," J. Mod. Opt. 44, 447-460 (1997).
[CrossRef]

Goodman, J. W.

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena (Springer-Verlag, 1975), p. 36.
[CrossRef]

Gu, Z.-H.

R. E. Luna, E. R. Méndez, J. Q. Lu, and Z.-H. Gu, "Enhanced backscattering due to total internal reflection at a dielectric-air interface," J. Mod. Opt. 42, 257-269 (1995).
[CrossRef]

Jay, J.

Y. Jourlin, J. Jay, and O. Parriaux, "Compact diffractive interferometric displacement sensor in reflection," Precis. Eng. 26, 1-6 (2002).
[CrossRef]

Jourlin, Y.

Y. Jourlin, J. Jay, and O. Parriaux, "Compact diffractive interferometric displacement sensor in reflection," Precis. Eng. 26, 1-6 (2002).
[CrossRef]

Kurata, T.

Y. Arai and T. Kurata, "A high-resolution encoder by multiplication of Moire fringes," Eng. Technol. 9, 1-7 (1987).

Lu, J. Q.

R. E. Luna, E. R. Méndez, J. Q. Lu, and Z.-H. Gu, "Enhanced backscattering due to total internal reflection at a dielectric-air interface," J. Mod. Opt. 42, 257-269 (1995).
[CrossRef]

Luna, R. E.

R. E. Luna, E. R. Méndez, J. Q. Lu, and Z.-H. Gu, "Enhanced backscattering due to total internal reflection at a dielectric-air interface," J. Mod. Opt. 42, 257-269 (1995).
[CrossRef]

Mansuripur, M.

Y. Wei-Hung, W. Bletscher, and M. Mansuripur, "High resolution optical shaft encoder for motor speed control based on an optical disk pick-up," Rev. Sci. Instrum. 69, 3068-3071 (1998).
[CrossRef]

Méndez, E. R.

R. E. Luna, E. R. Méndez, J. Q. Lu, and Z.-H. Gu, "Enhanced backscattering due to total internal reflection at a dielectric-air interface," J. Mod. Opt. 42, 257-269 (1995).
[CrossRef]

Morlanes, T.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (Krieger, 1986), pp. 315-320.

Parriaux, O.

Y. Jourlin, J. Jay, and O. Parriaux, "Compact diffractive interferometric displacement sensor in reflection," Precis. Eng. 26, 1-6 (2002).
[CrossRef]

Quintián, F. Perez

F. Perez Quintián, M. A. Rebollo, and N. G. Gaggioli, "Diffusion of light transmitted from rough surfaces," J. Mod. Opt. 44, 447-460 (1997).
[CrossRef]

Rebollo, M. A.

F. Perez Quintián, M. A. Rebollo, and N. G. Gaggioli, "Diffusion of light transmitted from rough surfaces," J. Mod. Opt. 44, 447-460 (1997).
[CrossRef]

Seitz, P.

Wei-Hung, Y.

Y. Wei-Hung, W. Bletscher, and M. Mansuripur, "High resolution optical shaft encoder for motor speed control based on an optical disk pick-up," Rev. Sci. Instrum. 69, 3068-3071 (1998).
[CrossRef]

Appl. Opt. (1)

Eng. Technol. (1)

Y. Arai and T. Kurata, "A high-resolution encoder by multiplication of Moire fringes," Eng. Technol. 9, 1-7 (1987).

J. Mod. Opt. (2)

R. E. Luna, E. R. Méndez, J. Q. Lu, and Z.-H. Gu, "Enhanced backscattering due to total internal reflection at a dielectric-air interface," J. Mod. Opt. 42, 257-269 (1995).
[CrossRef]

F. Perez Quintián, M. A. Rebollo, and N. G. Gaggioli, "Diffusion of light transmitted from rough surfaces," J. Mod. Opt. 44, 447-460 (1997).
[CrossRef]

Opt. Eng. (1)

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
[CrossRef]

Precis. Eng. (1)

Y. Jourlin, J. Jay, and O. Parriaux, "Compact diffractive interferometric displacement sensor in reflection," Precis. Eng. 26, 1-6 (2002).
[CrossRef]

Rev. Sci. Instrum. (1)

Y. Wei-Hung, W. Bletscher, and M. Mansuripur, "High resolution optical shaft encoder for motor speed control based on an optical disk pick-up," Rev. Sci. Instrum. 69, 3068-3071 (1998).
[CrossRef]

Other (2)

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena (Springer-Verlag, 1975), p. 36.
[CrossRef]

A. Papoulis, Systems and Transforms with Applications in Optics (Krieger, 1986), pp. 315-320.

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Figures (3)

Fig. 1
Fig. 1

Scheme of the proposed system to generate an incremental signal. The notation chosen for the fields at different planes is shown.

Fig. 2
Fig. 2

Typical speckle pattern at the observation plane.

Fig. 3
Fig. 3

Intensity correlation coefficient obtained from relation (10) introducing the grating and diffuser parameters (dashed curve) and the one obtained from the experiment (solid curve).

Tables (1)

Tables Icon

Table 1 Measured Intensities of the Diffraction Orders for an Incident Beam at 0°

Equations (15)

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γ I A I B = I A I B I A I B [ I A 2 I A 2 I B 2 I B 2 ] 1 / 2 ,
γ I A I B = | E A * E B | 2 I A I B .
t B ( x ) = m = a m exp ( i m q x ) ,
t A ( x ) = n = a n exp [ i n q ( x Δ x ) ] ,
E 3 A * ( x ) E 3 B ( x ) m , n = a n * a m exp ( i n q Δ x ) × E 1 * ( x 1 ) E 1 ( x 1 ) ×  exp [ i k 2 z 2 ( x x 1 ) 2 ] × exp [ i k 2 z 2 ( x x 1 ) 2 ] exp ( i n q x 1 ) × exp ( i m q x 1 ) d x 1 d x 1 ,
E 1 * ( x 1 ) E 1 ( x 1 ) = E 0 * ( x 0 ) E 0 ( x 0 ) × exp [ i k 2 z 1 ( x 1 x 0 ) 2 ] × exp [ i k 2 z 1 ( x 1 x 0 ) 2 ] d x 0 d x 0 .
E 3 A * ( x ) E 3 B ( x ) m , n = a n * a m exp ( i n q Δ x ) × exp [ i q 2 z 1 z 2 ( n 2 m 2 ) 2 k ( z 1 + z 2 ) ] × exp [ i q z 1 ( n m ) z 1 + z 2 x ] × E 0 * ( x 0 ) E 0 ( x 0 ) × exp [ i k 2 ( z 1 + z 2 ) ( x x 0 ) 2 ] × exp [ i k 2 ( z 1 + z 2 ) ( x x 0 ) 2 ] × exp [ i q z 2 z 1 + z 2 × ( n x 0 m x 0 ) ] d x 0 d x 0 .
E 0 * ( x 0 ) E 0 ( x 0 ) = | P ( x 0 ) | 2 exp ( τ 0 2 T 0 2 ) = exp ( x 0 2 b 0 2 ) exp ( τ 0     2 T 0     2 ) ,
E 3 A * ( x ) E 3 B ( x ) m , n = a n * a m exp ( i n q Δ x ) × exp [ i q 2 z 1 z 2 ( n 2 m 2 ) k ( z 1 + z 2 ) ] × exp [ i q z 1 ( n m ) z 1 + z 2 x ] × exp ( τ 0     2 T 0     2 ) × exp { i k ( z 1 + z 2 ) [ x q k z 2 m ] τ 0 } × exp { b 0 2 k 2 4 ( z 1 + z 2 ) 2 × [ τ 0 + q k z 2 ( n m ) ] 2 } × exp [ i k τ 0 2 2 ( z 1 + z 2 ) ] d τ 0 .
E 3 A * ( x ) E 3 B ( x ) m = | a m | 2 exp ( i m q Δ x ) ×  exp [ k 2 4 T 0     2 ( z 1 + z 2 ) 2 ( x m q z 2 k ) 2 ] .
E 3 A * E 3 B m = | a m | 2 exp ( i m q Δ x ) × exp [ m 2 T 0     2 q 2 z 2     2 4 ( z 1 + z 2 ) 2 ] ,
E 3A * E 3B | a 0 | 2 + | a 1 | 2 exp ( i q Δ x ) + | a 1 | 2 exp ( i q Δ x ) ,
| a 1 | 2 = | a 1 | 2 exp [ T 0     2 q 2 z 2       2 4 ( z 1 + z 2 ) 2 ] ,
| a 1 | 2 = | a 1 | 2 exp [ T 0     2 q 2 z 2       2 4 ( z 1 + z 2 ) 2 ] .
γ I A I B | a 0 | 4 + | a 1 | 4 + | a 1 | 4 + 2 | a 0 | 2 ( | a 1 | 2 + | a 1 | 2 ) × cos ( q Δ x ) + 2 | a 1 | 2 | a 1 | 2 cos ( 2 q Δ x ) .

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