Abstract

The term generalized grating imaging is used to describe the process of image formation of a grating using only another grating as imaging system. The moiré and the Lau effects could be regarded as particular cases of such a process. Here we deal with the less-studied case of images formed at finite distances from the gratings, using an extended monochromatic light source. Some experimental results are shown for the images obtained in this last case, and they are compared with theoretical predictions.

© 2002 Optical Society of America

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References

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  1. F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).
  2. E. Bar-Ziv, “Effect of diffraction on the moire image. I. Theory,” J. Opt. Soc. Am. 2, 371–379 (1985).
    [CrossRef]
  3. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).
  4. L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront, and their use,” Appl. Opt. 28, 4668–4677 (1989).
    [CrossRef] [PubMed]
  5. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [CrossRef]
  6. G. J. Swanson, E. N. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. A 2, 789–793 (1985).
    [CrossRef]
  7. E. N. Leith, R. Hershey, “Transfer functions and spatial filtering in grating interferometers,” Appl. Opt. 24, 237–239 (1985).
    [CrossRef] [PubMed]
  8. D. Crespo, J. Alonso, E. Bernabeu, “Generalized grating imaging using a monochromatic extended light source,” J. Opt. Soc. Am A 17, 1231–1240 (2000).
    [CrossRef]
  9. K. Patorski, “Influence of the type of illumination and separation of gratings on the intensity distribution in moiré fringes,” in Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), Chap. 4, pp. 69–98.
  10. E. Keren, O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2, 111–120 (1985).
    [CrossRef]
  11. D. Crespo, J. Alonso, T. Morlanes, E. Bernabeu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817–824 (2000).
    [CrossRef]

2000 (2)

D. Crespo, J. Alonso, E. Bernabeu, “Generalized grating imaging using a monochromatic extended light source,” J. Opt. Soc. Am A 17, 1231–1240 (2000).
[CrossRef]

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

1989 (1)

1985 (4)

1979 (1)

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

1948 (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).

1836 (1)

F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).

Alonso, J.

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

D. Crespo, J. Alonso, E. Bernabeu, “Generalized grating imaging using a monochromatic extended light source,” J. Opt. Soc. Am A 17, 1231–1240 (2000).
[CrossRef]

Bar-Ziv, E.

E. Bar-Ziv, “Effect of diffraction on the moire image. I. Theory,” J. Opt. Soc. Am. 2, 371–379 (1985).
[CrossRef]

Bernabeu, E.

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

D. Crespo, J. Alonso, E. Bernabeu, “Generalized grating imaging using a monochromatic extended light source,” J. Opt. Soc. Am A 17, 1231–1240 (2000).
[CrossRef]

Crespo, D.

D. Crespo, J. Alonso, E. Bernabeu, “Generalized grating imaging using a monochromatic extended light source,” J. Opt. Soc. Am A 17, 1231–1240 (2000).
[CrossRef]

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

Gori, F.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Hershey, R.

Kafri, O.

Keren, E.

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).

Leith, E. N.

Liu, L.

Morlanes, T.

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

Patorski, K.

K. Patorski, “Influence of the type of illumination and separation of gratings on the intensity distribution in moiré fringes,” in Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), Chap. 4, pp. 69–98.

Swanson, G. J.

Talbot, F.

F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).

Ann. Phys. (Leipzig) (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).

Appl. Opt. (2)

J. Opt. Soc. Am A (1)

D. Crespo, J. Alonso, E. Bernabeu, “Generalized grating imaging using a monochromatic extended light source,” J. Opt. Soc. Am A 17, 1231–1240 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

E. Bar-Ziv, “Effect of diffraction on the moire image. I. Theory,” J. Opt. Soc. Am. 2, 371–379 (1985).
[CrossRef]

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

Opt. Commun. (1)

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Opt. Eng. (1)

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

Philos. Mag. (1)

F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).

Other (1)

K. Patorski, “Influence of the type of illumination and separation of gratings on the intensity distribution in moiré fringes,” in Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), Chap. 4, pp. 69–98.

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Figures (6)

Fig. 1
Fig. 1

Schematic view of a double diffraction arrangement with an extended light source.

Fig. 2
Fig. 2

Profile of the fringes obtained with the CCD camera located on the detection plane. The comparison is shown with a triangular profile.

Fig. 3
Fig. 3

(a) Experimental measurements of the contrast of the fringes for different values of z 2. (b) Theoretical predictions for the modulation of the fringes for the same values. Each letter indicates a curve for a different value of z 2: A, z 2 = 5 mm; B, z 2 = 9 mm; C, z 2 = 13 mm; D, z 2 = 17 mm; E, z 2 = 21 mm; F, z 2 = 25 mm; G, z 2 = 29 mm; H, z 2 = 33 mm; I, z 2 = 37 mm; and J, z 2 = 41 mm.

Fig. 4
Fig. 4

Theoretical and experimental curves showing the value of Z 1 for which a maximum in the modulation is obtained, as a function of Z 2.

Fig. 5
Fig. 5

Comparison between theoretical and experimental curves of the maximum value of the modulation obtained for different values of Z 2. Both curves are normalized to their respective highest values.

Fig. 6
Fig. 6

Curves showing the width at half-maximum of the modulation peaks as a function of Z 2. The experimental and the theoretical curves are shown, as well as the linear approximation obtained from Eq. (14).

Equations (15)

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t1x=n an expinq1x,
t2x=m bm expimq2x,
R=q2q1, Z0=q12z0πk, Z1=q12z1πk, Z2=q12z2πk, ZT=q12zTπk,
ISx=nnmm anan*bmbm*expixq1ZtZ0×n-n+RZ0+Z1m-m×expiπ2Z0ZTn2-n2Z1+Z2×expiπ2Z0+Z1ZTm2-m2R2Z2×expiπZ0ZTnm-nmRZ2×sincSZT q1n-nZ1+Z2+m-mRZ2.
ISx=nnmm anan*bmbm*×expixq1n-n+m-m×expiπ2n2-n2Z1+Z2×expiπ2m2-m2Z2×expiπnm-nmZ2×sincSq1n-nZ1+Z2+m-mZ2.
ISx=lexp-ix q1F Z1lexp-iπ2 Z1l2×nm anan-1*bmbm+l* exp-iπZ1ml,
Z1+Z2nI+RZ2mI0,
|nI|  Sq1πZT RZ2, |mI|  Sq1πZTZ1+Z2.
ISxI0+j0 dj expix Z1Z2 q1jnI×sincj Sq1ZTZ1+Z2nI+RZ2mI,
dj=expiπ2 RZ1nImIj2n an+jnIan*×m bm+jmIbm* exp-iπRZ1nImj.
M1π2cosπ2 RZ1sincSq1ZTZ1+Z2-RZ2.
V=maxIS-minISmaxIS+minIS,
Z1+1-RZ20,
RZ12n
fZ1=sincSq1ZTZ1+Z2-RZ2,

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