Abstract

I present a general approach for designing Talbot array illuminators (TAIL’s). Beyond the scope of conventional TAIL’s, in which each period of the replay field contains only one bright spot, I examine complex configurations ensuring various topologies of the output spots. The synthesis problem is stated in terms of constraints and degrees of freedom. Both the one-dimensional and two-dimensional cases are treated, and an example of constraints, namely the binarization of the TAIL, is discussed. For illustration, an application is proposed in which I consider a dynamic multilayer interconnection architecture involving a programmable TAIL.

© 1997 Optical Society of America

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References

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    [CrossRef] [PubMed]
  3. A. Kalestynski, B. Smolinska, “Spatial frequency sampling by phase modulation as a method of generating multiple images,” Appl. Opt. 16, 2261–2263 (1977).
    [CrossRef] [PubMed]
  4. H. Hamam, “Hartley holograms,” Appl. Opt. 35, 5286–5292 (1996).
    [CrossRef] [PubMed]
  5. E. Noponen, J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
    [CrossRef]
  6. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), pp. 1–110.
  7. J. Westerholm, J. Turunen, J. Huttunen, “Fresnel diffraction in fractional Talbot planes: a new formulation,” J. Opt. Soc. Am. A 11, 1283–1290 (1994).
    [CrossRef]
  8. H. Hamam, J. L. de Bougrenet de la Tocnaye, “Efficient Fresnel transform algorithm based on fractional Fresnel diffraction,” J. Opt. Soc. Am. A 12, 1920–1931 (1995).
    [CrossRef]
  9. R. E. Loseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. 55, 544–547 (1983).
  10. H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
    [CrossRef]
  11. O. Guyot, H. Hamam, “Logic operations based on the fractional Talbot effect,” Opt. Commun. 127, 96–106 (1996).
    [CrossRef]
  12. B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuk, T. C. Strand, “Architectural implications of a digital optical processor,” Appl. Opt. 23, 3465–3474 (1984).
    [CrossRef] [PubMed]

1996 (3)

H. Hamam, “Hartley holograms,” Appl. Opt. 35, 5286–5292 (1996).
[CrossRef] [PubMed]

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

O. Guyot, H. Hamam, “Logic operations based on the fractional Talbot effect,” Opt. Commun. 127, 96–106 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (2)

1988 (1)

1984 (1)

1983 (1)

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

1977 (1)

Chavel, P.

de Bougrenet de la Tocnaye, J. L.

Forchheimer, R.

Guyot, O.

O. Guyot, H. Hamam, “Logic operations based on the fractional Talbot effect,” Opt. Commun. 127, 96–106 (1996).
[CrossRef]

Hamam, H.

O. Guyot, H. Hamam, “Logic operations based on the fractional Talbot effect,” Opt. Commun. 127, 96–106 (1996).
[CrossRef]

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

H. Hamam, “Hartley holograms,” Appl. Opt. 35, 5286–5292 (1996).
[CrossRef] [PubMed]

H. Hamam, J. L. de Bougrenet de la Tocnaye, “Efficient Fresnel transform algorithm based on fractional Fresnel diffraction,” J. Opt. Soc. Am. A 12, 1920–1931 (1995).
[CrossRef]

Huang, A.

Huttunen, J.

Jahns, J.

Jenkins, B. K.

Kalestynski, A.

Loseliani, R. E.

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

Murdocca, M. J.

Noponen, E.

E. Noponen, J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

Patorski, K.

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

Sawchuk, A. A.

Smolinska, B.

Strand, T. C.

Streibl, N.

Tooley, F. A. P.

Turunen, J.

Wakelin, S.

Westerholm, J.

Appl. Opt. (5)

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

Opt. Commun. (3)

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

O. Guyot, H. Hamam, “Logic operations based on the fractional Talbot effect,” Opt. Commun. 127, 96–106 (1996).
[CrossRef]

E. Noponen, J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

Opt. Spectrosc. (1)

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

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Only two possible binary 1-D TAIL’s ensuring a compression ration of Tc = q/8: (a) and (b) represent the phase distributions, and (c) and (d) represent the corresponding replay fields.

Fig. 2
Fig. 2

Binary 2-D TAIL with a compression ration of Tc = 2 × 2: (a) the phase distribution and (b) the amplitude distribution of the replay field. The possible phases of the bright spots are 0 and π.

Fig. 3
Fig. 3

Two binary 2-D TAIL’s operating at the quarter-Talbot distance: (a) and (b) represent the phase distributions, and (c) and (d) represent the corresponding amplitude distributions of the replay fields.

Fig. 4
Fig. 4

Binary 2-D TAIL operating at the 1/8th Talbot distance: (a) the phase distribution and (b) the amplitude distribution of its replay field.

Fig. 5
Fig. 5

Multilayer interconnection network that uses a multifaceted DOE and a programmable TAIL, which contains four repeated holograms, for switching.

Fig. 6
Fig. 6

Response of a binary TAIL operating at the 1/16th Talbot distance.

Fig. 7
Fig. 7

Optical reconstruction of four repeated holograms of the TAIL from Fig. 5. One of the four binary holograms will be freely activated by the programmable element. The mirror images of the holograms are not presented.

Equations (39)

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hx, z=a=0q2-1Ta, p, qhx-d2+2adq,  Ta, p, q=2qb=0q2-1 exp-iπ2b2qp+bexpiπ4baq,
hx, z=a=0q-1Ta, p, qhx-adq,  Ta, p, q=1qb=0q-1 exp-i2πb2qpexpi2πbaq,
hx, y, z=ay=0qy/2-1ax=0qx/2-1 Tay, py, qyTax, px, qx×hx-dx2+2axdxqx, y-dy2+2aydyqy,
z=pxqxZTx=pyqyZTy,
hx-d2+a2dpq=exp-iπa2aqp-1.
hx+2adpq=exp-i2πa2qp.
hx-dx2+2axpxdxqx, y-dy2+2aypydyqy=exp-iπax2axqxpx-1+ay2ayqypy-1
hx, z=1Nj=0N-1 expiϕjgx-2njdqp, z,
hx=1Nj=0N-1 expiϕjgx-2njdqp.
hx2=1Nj=0N-1expiϕjgx-2njdqp2=1.
j=0N-1expiϕjta, nj2=N, ta, nj=exp-iπa-nj2a-njqp-1.
j=0N-1expiϕjexpiπ4anjqp+nj-2njqp-12=N.
j=0N-1expiφjexpi4πanjqp2=N,
φj=ϕj-π2nj2qp+nj.
j=0N-1expiφjexpi4πaxnjqxpx+aymjqypy2=N,
φj=ϕj-π2nj2qxpx+nj+2mj2qypy+mj.
hx=2aqd=12expiϕ0ta, 0+expiϕ1ta, q4.
ta+q4, nj=ta, nj+q4,
hx=2a+q/4qd=12expiϕ0ta, q4+expiϕ1ta, 0.
h2aqd=12expiϕ0ta, 0+expiϕ1ta, 1+expiϕ2ta, q/4+expiϕ3ta, q/4+1h2a+q/4qd=12expiϕ0ta, q4+expiϕ1×ta, q4+1+expiϕ2 ta, 0+expiϕ3ta, 1.
Sa=1Nj=0N-1expiϕjta, nj=D expiα1, expiα2, ta, nj=exp-iπa-nj2a-njqP-1.
Sa=j=0N-1expiϕjta, nj, ta, nj=exp-iπ2a2qp-a×expiπ4anjqp+nj-2njqp-1
ta, q4=Aata, 0,
ta, 1=-BaCta, 0,
ta, q4+1=AaBaCta, 0,
Sa=expiϕ0ta, O+expiϕ1ta, q4, Sa+q4=expiϕ0ta, q4+expiϕ1ta, O.
Sa=expiϕ0+Aaexpiϕ1ta, 0, Sa+q4=Aaexpiϕ0+expiϕ1ta, O.
Sa=expiϕ0+expiπapexpiϕ1ta, O, Sa+q4=expiπapexpiϕ0+expiϕ1ta, O.
Sa=2b=expiϕ0+expiϕ1, Sa=2b+1=expiϕ0-expiϕ1.
Sa=expiϕ0±i expiπapexpiϕ1ta, O, Sa+q4=±i expiπapexpiϕ0+expiϕ1ta, O.
Sa=expiϕ0-BaCexpiϕ1ta, 0, Sa+q4=expiϕ0+BaCexpiϕ1Aata, 0.
BaCexpiϕ1=±i expiϕ0,
ϕ1=ϕ0+π2qp-4πaqp±π2.
Sa=expiϕ0+Aaexpiϕ2-BaCexpiϕ1-Aaexpiϕ3ta, 0, Sa+q4=expiϕ2+Aaexpiϕ0-BaCexpiϕ3-Aaexpiϕ1ta, 0.
Sa=expiϕ0+expiϕ2-BaCexpiϕ1-expiϕ3ta, 0, Sa+q4=expiϕ2+expiϕ0-BaCexpiϕ3-expiϕ1ta, 0,
Sa=expiϕ0-expiϕ2-BaCexpiϕ1+expiϕ3ta, 0, Sa+q4=expiϕ2-expiϕ0-BaCexpiϕ3+expiϕ1ta, 0.
Sax, ay=expiϕ0tax, 0tay, 0+expiϕ1tax, qx4tay, qy4, Sax+qx4, ay=expiϕ0tax, qx4tay, 0+expiϕ1tax, 0 tay, qy4.
Sax, 0=expiϕ0+AaxA0expiϕ1tax, 0, Sax+qx4, 0=Aaxexpiϕ0+A0expiϕ1tax, 0.
Sax, ay=expiϕ0tax, 0tay, 0+expiϕ1tax, 1tay, 0+expiϕ2tax, qx4tay, qy4+expiϕ3tax, q4+1 tay, qy4+expiϕ4tax, 0tay, 1+expiϕ5tax, 1tay, 1+expiϕ6tax, qx4tay, qy4+1+expiϕ7tax, q4+1 tay, qy4+1.

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