Abstract

Spatial coherence can be created by appropriate spatial arrangements of incoherent point sources. Lau used a source of extended light and two amplitude gratings of identical periods, separated by the quarter Talbot distance, to provide coherent light. Because of the two successive amplitude gratings, most of the power is lost. By modifying the geometry of the second grating, I designed an array illuminator, providing several compression ratios and various topologies of the output plane, with significantly reduced losses. To further improve the power efficiency of the system, I used a longitudinal mirror system to collect the light rays that are lost in the initial Lau setup. Both one- and two-dimensional geometries are considered.

© 2004 Optical Society of America

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References

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  1. M. Barge, H. Hamam, Y. Defosse, R. Chevallier, J. L. de Bougrenet, “Array illuminators using diffractive optical elements,” J. Opt. 27, 151–170 (1996).
    [CrossRef]
  2. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  3. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  4. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–110.
    [CrossRef]
  5. E. Lau, “Beugungserscheinung an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–427 (1948).
  6. M. Robblin, “Modulation spatiale d’intensité produite par l’association de deux réseaux parallèles,” Ph.D. dissertation (Université de Paris VI, Paris, 1973).
  7. R. Sudol, B. Thompson, “Lau effects theory and experiments,” Appl. Opt. 20, 1107–1116 (1981).
    [CrossRef] [PubMed]
  8. K. Patorski, “Incoherent superimposition of multiple self-imaging under plane wave-front illumination,” Appl. Opt. 25, 2396–2404 (1986).
    [CrossRef] [PubMed]
  9. J. Ojeda-Castaneda, P. Andrés, J. Ibarra, “Lensless theta decoder with high light throughput,” Opt. Commun. 67, 256–260 (1988).
    [CrossRef]
  10. J. Barreiro, P. Andrés, J. Ojeda-Castaneda, “Lau effect with only phase gratings,” Opt. Commun. 73, 106–110 (1989).
    [CrossRef]
  11. E. Menzel, Ch. Menzel, “Zur Abbildung optischer Gitter,” Optik (Stuttgart) 1, 22–36 (1948).
  12. C. H. Deckers, “Etude de l’influence de la dimension finie d’un réseau sur la formation des images de Fresnel,” J. Opt. (Paris) 7, 113–119 (1976).
  13. D. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613–2624 (1972).
    [CrossRef] [PubMed]
  14. A. Smirnov, “Depth of focus of Fresnel images,” Opt. Spectrosc. 46, 319–322 (1979).
  15. H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
    [CrossRef]
  16. H. Hamam, “Applet for Lau array illumination,” http://www.umoncton.ca/genie/electrique/Cours/Hamam/Optics/Lau/Lail.htm (July2003).
  17. R. Loseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. 55, 544–547 (1983).
  18. H. Hamam, “Talbot imaging and unification,” Appl. Opt. 42, 7052–7059 (2003).
    [CrossRef] [PubMed]

2003 (1)

1996 (2)

H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
[CrossRef]

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, J. L. de Bougrenet, “Array illuminators using diffractive optical elements,” J. Opt. 27, 151–170 (1996).
[CrossRef]

1990 (1)

1989 (1)

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

1988 (1)

J. Ojeda-Castaneda, P. Andrés, J. Ibarra, “Lensless theta decoder with high light throughput,” Opt. Commun. 67, 256–260 (1988).
[CrossRef]

1986 (1)

1983 (1)

R. Loseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. 55, 544–547 (1983).

1981 (1)

1979 (1)

A. Smirnov, “Depth of focus of Fresnel images,” Opt. Spectrosc. 46, 319–322 (1979).

1976 (1)

C. H. Deckers, “Etude de l’influence de la dimension finie d’un réseau sur la formation des images de Fresnel,” J. Opt. (Paris) 7, 113–119 (1976).

1972 (1)

1967 (1)

1948 (2)

E. Lau, “Beugungserscheinung an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–427 (1948).

E. Menzel, Ch. Menzel, “Zur Abbildung optischer Gitter,” Optik (Stuttgart) 1, 22–36 (1948).

Andrés, P.

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

J. Ojeda-Castaneda, P. Andrés, J. Ibarra, “Lensless theta decoder with high light throughput,” Opt. Commun. 67, 256–260 (1988).
[CrossRef]

Barge, M.

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, J. L. de Bougrenet, “Array illuminators using diffractive optical elements,” J. Opt. 27, 151–170 (1996).
[CrossRef]

Barreiro, J.

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

Chevallier, R.

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, J. L. de Bougrenet, “Array illuminators using diffractive optical elements,” J. Opt. 27, 151–170 (1996).
[CrossRef]

de Bougrenet, J. L.

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, J. L. de Bougrenet, “Array illuminators using diffractive optical elements,” J. Opt. 27, 151–170 (1996).
[CrossRef]

Deckers, C. H.

C. H. Deckers, “Etude de l’influence de la dimension finie d’un réseau sur la formation des images de Fresnel,” J. Opt. (Paris) 7, 113–119 (1976).

Defosse, Y.

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, J. L. de Bougrenet, “Array illuminators using diffractive optical elements,” J. Opt. 27, 151–170 (1996).
[CrossRef]

Hamam, H.

H. Hamam, “Talbot imaging and unification,” Appl. Opt. 42, 7052–7059 (2003).
[CrossRef] [PubMed]

H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
[CrossRef]

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, J. L. de Bougrenet, “Array illuminators using diffractive optical elements,” J. Opt. 27, 151–170 (1996).
[CrossRef]

Ibarra, J.

J. Ojeda-Castaneda, P. Andrés, J. Ibarra, “Lensless theta decoder with high light throughput,” Opt. Commun. 67, 256–260 (1988).
[CrossRef]

Lau, E.

E. Lau, “Beugungserscheinung an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–427 (1948).

Lohmann, A. W.

Loseliani, R.

R. Loseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. 55, 544–547 (1983).

Menzel, Ch.

E. Menzel, Ch. Menzel, “Zur Abbildung optischer Gitter,” Optik (Stuttgart) 1, 22–36 (1948).

Menzel, E.

E. Menzel, Ch. Menzel, “Zur Abbildung optischer Gitter,” Optik (Stuttgart) 1, 22–36 (1948).

Montgomery, W. D.

Ojeda-Castaneda, J.

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

J. Ojeda-Castaneda, P. Andrés, J. Ibarra, “Lensless theta decoder with high light throughput,” Opt. Commun. 67, 256–260 (1988).
[CrossRef]

Patorski, K.

K. Patorski, “Incoherent superimposition of multiple self-imaging under plane wave-front illumination,” Appl. Opt. 25, 2396–2404 (1986).
[CrossRef] [PubMed]

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–110.
[CrossRef]

Robblin, M.

M. Robblin, “Modulation spatiale d’intensité produite par l’association de deux réseaux parallèles,” Ph.D. dissertation (Université de Paris VI, Paris, 1973).

Silva, D.

Smirnov, A.

A. Smirnov, “Depth of focus of Fresnel images,” Opt. Spectrosc. 46, 319–322 (1979).

Sudol, R.

Thomas, J. A.

Thompson, B.

Ann. Phys. (Leipzig) (1)

E. Lau, “Beugungserscheinung an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–427 (1948).

Appl. Opt. (5)

J. Opt. (1)

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, J. L. de Bougrenet, “Array illuminators using diffractive optical elements,” J. Opt. 27, 151–170 (1996).
[CrossRef]

J. Opt. (Paris) (1)

C. H. Deckers, “Etude de l’influence de la dimension finie d’un réseau sur la formation des images de Fresnel,” J. Opt. (Paris) 7, 113–119 (1976).

J. Opt. Soc. Am. (1)

Opt. Commun. (3)

J. Ojeda-Castaneda, P. Andrés, J. Ibarra, “Lensless theta decoder with high light throughput,” Opt. Commun. 67, 256–260 (1988).
[CrossRef]

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

H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
[CrossRef]

Opt. Spectrosc. (2)

R. Loseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. 55, 544–547 (1983).

A. Smirnov, “Depth of focus of Fresnel images,” Opt. Spectrosc. 46, 319–322 (1979).

Optik (Stuttgart) (1)

E. Menzel, Ch. Menzel, “Zur Abbildung optischer Gitter,” Optik (Stuttgart) 1, 22–36 (1948).

Other (3)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–110.
[CrossRef]

M. Robblin, “Modulation spatiale d’intensité produite par l’association de deux réseaux parallèles,” Ph.D. dissertation (Université de Paris VI, Paris, 1973).

H. Hamam, “Applet for Lau array illumination,” http://www.umoncton.ca/genie/electrique/Cours/Hamam/Optics/Lau/Lail.htm (July2003).

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Figures (7)

Fig. 1
Fig. 1

Lau effect: two identical amplitude gratings separated by a distance z = l 0 = 0.5d 2/λ. The distance between any transparent point of the first grating and the point from the second grating is a multiple of the wavelength.

Fig. 2
Fig. 2

Creation of the spatial coherence: (a) Two amplitude gratings are illuminated by an incoherent extended source, the first grating with spatial period d and the second grating with period 2d/ q. (b) This configuration is equivalent to the setup of (a). The diffraction field h(x, z + z′) can be calculated by means of the fractional Talbot effect. For example, we obtain the intensity profile on the right for z′ = (1 - 1/q)2d 2/λ.

Fig. 3
Fig. 3

Two-dimensional gratings with unequal periods: l k 1 k 2 (m 1, m 2) is the distance covered by a ray coming from a point (k 1, k 2) of the first grating and exiting the second grating at the point (m 1, m 2). k 1 = 0. The difference between any two optical paths going through the point (m 1, m 2) is a multiple of the wavelength for q as a multiple of 4.

Fig. 4
Fig. 4

Use of two mirrors M 1 and M 2 and a diffractive phase element to produce a nonconventional topology of diffraction spots. (a) A diffractive phase element can be placed just behind or at a distance z′ from the second grating. (b) Output plane; spots are arranged in a chessboard topology.

Fig. 5
Fig. 5

Use of longitudinal mirrors M 1 and M 2. A reflection of a ray through the mirror is equivalent to this ray coming from a virtual grating, which presents the continuation with the real one.

Fig. 6
Fig. 6

Intensity profile of the replay field of the LAIL [Fig. 2(a)] at a distance z′ = (1 - 1/q)2d 2/λ for when approximation (4) is (a) taken into account and (b) not taken into account.

Fig. 7
Fig. 7

Longitudinal mirrors can be made by a metallic layer covering a transparent parallelepiped with a depth z = (1/q)2d 2/λ. The two arrows indicate where to place the first and second grating.

Equations (17)

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l0=z=1q ZT=2d2qλ.
lk=l02+k2d21/2=2d2qλ1+k2λ24d2 q21/2,
A=k2λ24d2 q2  1,
1+A1+A/2,
lk=l0+k2q4 λ.
b0=k=0N-1 ak.
q2  4d2N2λ2,
l0s=l02+s21/2=2d2qλ1+s2λ24d4 q21/2,
l0s=l00+q4sd2λ.
lks=l02+kd+s21/2=l0s+q2ksd λ+q4 k2λ.
bns=expj q4sd22k=0N-1 ak expjq n-ksd π=expjq nsd πb0s.
bnm 2dq=bnm=expj2 m2q πb00.
hx, z=n=0N-1m=0q/2-1 bnmδx-nd-m 2dq.
lk1k2m1, m2=l002+k1d+m12dq2+k2d+m22dq21/2,
lk1k2m1, m2=2d2λq1+k12+k22λ2q24d2+m12+m22λ2d2+k1m1+k2m2qλ2d21/2.
l00m1, m2=2d2λq+m12+m22λq.
lk1k2=l00m1, m2+k12+k22×λq4+k1m1+k2m2λ.

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