Abstract

A new theoretical model of the Lau effect is presented. The transmittance of a diffraction grating can be expressed in an equivalent form as the sum of transmittances of thin cylindrical lenses. Therefore it is possible to explain the Lau effect on the basis of the well-known imaging properties of lenses. According to the given approach, the Lau fringes are created by overlapped images of the first grating that are formed by a set of lenses corresponding to the second grating in the setup. The theory leads to an exhaustive description of the Lau-effect parameters. In particular, one can indicate the shape of the Lau fringes and localize planes of the fringes dependent on the axial distance between gratings and their periods.

© 2000 Optical Society of America

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  1. E. Lau, “Beugungserscheinungen und Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
    [CrossRef]
  2. J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  3. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [CrossRef]
  4. R. Sudol, B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [CrossRef] [PubMed]
  5. K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
    [CrossRef]
  6. J. Sethuraman, “Bloch function and Lau effect,” Opt. Commun. 52, 377–379 (1985).
    [CrossRef]
  7. K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringe explanation,” Opt. Acta 30, 745–758 (1983).
    [CrossRef]
  8. G. J. Swanson, E. N. Leith, “Lau effect and grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
    [CrossRef]
  9. S. Jutamulia, T. Asakura, H. Fujii, “Lau effect and noncoherent processing,” Opt. Commun. 53, 77–80 (1985).
    [CrossRef]
  10. H. O. Bartelt, J. Jahns, “Interferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
    [CrossRef]
  11. J. Ojeda-Castaneda, J. C. Barreiro, J. Ibarra, “Shardin–Lau interferometer,” Opt. Commun. 67, 325–330 (1988).
    [CrossRef]
  12. J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
    [CrossRef]
  13. P. Andres, E. Tepichin, J. Ojeda-Castaneda, “Lau rings: in-register incoherent superposition of radial self-images,” Opt. Commun. 72, 47–53 (1989).
    [CrossRef]
  14. K. V. Avudainayagam, S. Chitra Nayagam, “Two-grating diffraction and Lau effect under laser illumination,” Appl. Opt. 36, 2029–2033 (1997).
    [CrossRef] [PubMed]
  15. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 3–108.
  16. A. Kolodziejczyk, “Lensless multiple image formation by using a sampling filter,” Opt. Commun. 59, 97–102 (1986).
    [CrossRef]
  17. A. Kolodziejczyk, “Self-imaging effect—a new approach,” Opt. Commun. 65, 84–86 (1988).
    [CrossRef]
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 6, pp. 101–140.
  19. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 159–160, 360.
  20. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  21. W. Klaus, Y. Arimoto, K. Kodate, “Talbot array illuminators providing spatial intensity and phase modulation,” J. Opt. Soc. Am. A 14, 1092–1102 (1997).
    [CrossRef]
  22. T. J. Suleski, “Generation of Lohmann images from binary-phase Talbot array illuminators,” Appl. Opt. 36, 4686–4691 (1997).
    [CrossRef] [PubMed]
  23. W. Klaus, Y. Arimoto, K. Kodate, “High-performance Talbot array illuminators,” Appl. Opt. 37, 4357–4365 (1998).
    [CrossRef]
  24. J. C. Barreiro, P. Andres, J. Ojeda-Castaneda, “Lau effect only with phase gratings,” Opt. Commun. 73, 106–110 (1989).
    [CrossRef]

1998 (1)

1997 (3)

1990 (1)

1989 (3)

J. C. Barreiro, P. Andres, J. Ojeda-Castaneda, “Lau effect only with phase gratings,” Opt. Commun. 73, 106–110 (1989).
[CrossRef]

J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
[CrossRef]

P. Andres, E. Tepichin, J. Ojeda-Castaneda, “Lau rings: in-register incoherent superposition of radial self-images,” Opt. Commun. 72, 47–53 (1989).
[CrossRef]

1988 (2)

J. Ojeda-Castaneda, J. C. Barreiro, J. Ibarra, “Shardin–Lau interferometer,” Opt. Commun. 67, 325–330 (1988).
[CrossRef]

A. Kolodziejczyk, “Self-imaging effect—a new approach,” Opt. Commun. 65, 84–86 (1988).
[CrossRef]

1986 (1)

A. Kolodziejczyk, “Lensless multiple image formation by using a sampling filter,” Opt. Commun. 59, 97–102 (1986).
[CrossRef]

1985 (2)

J. Sethuraman, “Bloch function and Lau effect,” Opt. Commun. 52, 377–379 (1985).
[CrossRef]

S. Jutamulia, T. Asakura, H. Fujii, “Lau effect and noncoherent processing,” Opt. Commun. 53, 77–80 (1985).
[CrossRef]

1983 (2)

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

1982 (1)

1981 (1)

1979 (3)

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

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

H. O. Bartelt, J. Jahns, “Interferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

1948 (1)

E. Lau, “Beugungserscheinungen und Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Andres, P.

J. C. Barreiro, P. Andres, J. Ojeda-Castaneda, “Lau effect only with phase gratings,” Opt. Commun. 73, 106–110 (1989).
[CrossRef]

P. Andres, E. Tepichin, J. Ojeda-Castaneda, “Lau rings: in-register incoherent superposition of radial self-images,” Opt. Commun. 72, 47–53 (1989).
[CrossRef]

Arimoto, Y.

Asakura, T.

S. Jutamulia, T. Asakura, H. Fujii, “Lau effect and noncoherent processing,” Opt. Commun. 53, 77–80 (1985).
[CrossRef]

Avudainayagam, K. V.

Barreiro, J. C.

J. C. Barreiro, P. Andres, J. Ojeda-Castaneda, “Lau effect only with phase gratings,” Opt. Commun. 73, 106–110 (1989).
[CrossRef]

J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
[CrossRef]

J. Ojeda-Castaneda, J. C. Barreiro, J. Ibarra, “Shardin–Lau interferometer,” Opt. Commun. 67, 325–330 (1988).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, J. Jahns, “Interferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

Fujii, H.

S. Jutamulia, T. Asakura, H. Fujii, “Lau effect and noncoherent processing,” Opt. Commun. 53, 77–80 (1985).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 159–160, 360.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 6, pp. 101–140.

Gori, F.

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

Ibarra, J.

J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
[CrossRef]

J. Ojeda-Castaneda, J. C. Barreiro, J. Ibarra, “Shardin–Lau interferometer,” Opt. Commun. 67, 325–330 (1988).
[CrossRef]

Jahns, J.

H. O. Bartelt, J. Jahns, “Interferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Jutamulia, S.

S. Jutamulia, T. Asakura, H. Fujii, “Lau effect and noncoherent processing,” Opt. Commun. 53, 77–80 (1985).
[CrossRef]

Klaus, W.

Kodate, K.

Kolodziejczyk, A.

A. Kolodziejczyk, “Self-imaging effect—a new approach,” Opt. Commun. 65, 84–86 (1988).
[CrossRef]

A. Kolodziejczyk, “Lensless multiple image formation by using a sampling filter,” Opt. Commun. 59, 97–102 (1986).
[CrossRef]

Lau, E.

E. Lau, “Beugungserscheinungen und Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Leith, E. N.

Lohmann, A. W.

A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[CrossRef] [PubMed]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Nayagam, S. Chitra

Ojeda-Castaneda, J.

P. Andres, E. Tepichin, J. Ojeda-Castaneda, “Lau rings: in-register incoherent superposition of radial self-images,” Opt. Commun. 72, 47–53 (1989).
[CrossRef]

J. C. Barreiro, P. Andres, J. Ojeda-Castaneda, “Lau effect only with phase gratings,” Opt. Commun. 73, 106–110 (1989).
[CrossRef]

J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
[CrossRef]

J. Ojeda-Castaneda, J. C. Barreiro, J. Ibarra, “Shardin–Lau interferometer,” Opt. Commun. 67, 325–330 (1988).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

Patorski, K.

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 3–108.

Sethuraman, J.

J. Sethuraman, “Bloch function and Lau effect,” Opt. Commun. 52, 377–379 (1985).
[CrossRef]

Sudol, R.

Suleski, T. J.

Swanson, G. J.

Tepichin, E.

P. Andres, E. Tepichin, J. Ojeda-Castaneda, “Lau rings: in-register incoherent superposition of radial self-images,” Opt. Commun. 72, 47–53 (1989).
[CrossRef]

Thomas, J. A.

Thompson, B. J.

Ann. Phys. (1)

E. Lau, “Beugungserscheinungen und Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Opt. Commun. (12)

S. Jutamulia, T. Asakura, H. Fujii, “Lau effect and noncoherent processing,” Opt. Commun. 53, 77–80 (1985).
[CrossRef]

H. O. Bartelt, J. Jahns, “Interferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

J. Ojeda-Castaneda, J. C. Barreiro, J. Ibarra, “Shardin–Lau interferometer,” Opt. Commun. 67, 325–330 (1988).
[CrossRef]

J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
[CrossRef]

P. Andres, E. Tepichin, J. Ojeda-Castaneda, “Lau rings: in-register incoherent superposition of radial self-images,” Opt. Commun. 72, 47–53 (1989).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

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

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

J. Sethuraman, “Bloch function and Lau effect,” Opt. Commun. 52, 377–379 (1985).
[CrossRef]

A. Kolodziejczyk, “Lensless multiple image formation by using a sampling filter,” Opt. Commun. 59, 97–102 (1986).
[CrossRef]

A. Kolodziejczyk, “Self-imaging effect—a new approach,” Opt. Commun. 65, 84–86 (1988).
[CrossRef]

J. C. Barreiro, P. Andres, J. Ojeda-Castaneda, “Lau effect only with phase gratings,” Opt. Commun. 73, 106–110 (1989).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 6, pp. 101–140.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 159–160, 360.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 3–108.

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Figures (3)

Fig. 1
Fig. 1

Lau optical setup consisting of two diffraction gratings G1 and G2. Lau fringes can be observed in the focal plane (1) of the convergent lens L placed behind grating G2(f1 is the focal length of the lens). Generally, a real set of Lau fringes can also appear behind grating G2 (2), and, moreover, a virtual set can be observed in front of grating G2 (3).

Fig. 2
Fig. 2

Schematic drawing of imaging by the set of cylindrical lenses that correspond to grating G2. The parameters p, q, a, and d2 are described in the text. (Xo), (X), and (X1) are coordinates in the grating G1 plane, the grating G2 plane, and the output plane, respectively. The delta function δ(xo) represents the point object.

Fig. 3
Fig. 3

Schematic drawings of singular-period transmittances of the Ronchi gratings with periods d1, d2 and opening widths Δ1, Δ2.

Equations (24)

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T1(x)=t1(x)uδ(x-ud1),
T2(x)=t2(x)uδ(x-ud2),
uδ(x-ud2)=HC(H)exp-ik2 f(x-Hd2-D)2,
f=d222λmM,d2=d2/Mmevend2/2Mmodd.
C(H)=(1/md2)exp(iπH12/2Mm)×h=0m-1 exp[i2π(h2M-hH1)/m],
T2(x)=HC(H)exp-ik2 f(x-Hd2-D)2t2(x).
1p+1q=1f,a=d2 p+qp,
h(x1)=HC(H)t2x1 fq-Ha-D,
|h(x1)|2=|C|2t2x1 fq2Hδ(x1-Ha-D),
|h(x1)|2=t2x1 fq2Hδ(x1-Ha).
I(x1)=t1-x1 pq2t2x1 fq2uδ(x1-ud1q/p)Hδ(x1-Ha).
I1(x1)=t1-x1 pq2t2x1 fq2uδ(x1-ud1q/p).
a=d1qpnN,
I(x1)=t1-x1 pq2t2x1 fq2Hδx1-Hd1qpN.
p=d1d22λmnNmevend1d2λmnNmodd,
q=d1d222λmnd1Mn-d2Nmevend1d22λmn2d1Mn-d2Nmodd.
t1(x)=rect(x/Δ1),t2(x)=rect(x/Δ2),
Δ2 qfa=d1 qnpN.
I(x1)=rectx1pΔ1qrectx1fΔ2qHδx1-Hd1qpN,
Δ1 qp+Δ2 qfd1 qpN.
Δ1N+d1d2Δ2nMd1mevenΔ1N+2 d1d2Δ2nMd1modd.
Δ1+d1d2Δ2d1,f=d22λβ,p=d1d2λβ,q=d1d22(d1-d2)λβ,
exp-ik2 fx2expik2zx2=izff-z1/2 exp-ik2(f-z)x2,
T2(x)=T2(x)expik2zx2=HC(H)exp-ik2 f(x-Hd2-D)2t2(x),

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