Abstract

A grating-lens combination unit is developed to form a scaling self-transform function that can self-image on scale. Then an array of many such grating-lens units is used for the optical interconnection of a two-dimensional neural network, and experiments are carried out. We find that our idea is feasible, the optical interconnection system is simple, and optical adjustment is easy.

© 1998 Optical Society of America

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References

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    [CrossRef]

1997

J. Hua, L. Liu, G. Li, “Dual self-transform function,” J. Phys. A 30, 1–4 (1997).
[CrossRef]

1996

1995

A. Lakhlakia, “Physical fractals: self-similarity and square-integrability,” Speculat. Sci. Technol. 18, 153–156 (1995).

1994

L. Liu, “Periodic self-Fourier–Fresnel function,” J. Phys. A 27, L285–L289 (1994).
[CrossRef]

V. Arrizon, J. Ojeda-Castaneda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
[CrossRef] [PubMed]

E. Bonet, P. Andrés, J. C. Barreiro, A. Pons, “Self-imaging properties of periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[CrossRef]

1992

1991

M. J. Caola, “Self-Fourier function,” J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

1990

1989

F. Ito, K. Kitayama, “Optical implementation of the Hopfield neural network using multiple fiber nets,” Appl. Opt. 28, 4176–4181 (1989).
[CrossRef] [PubMed]

S. Lin, L. Liu, Z. Wang, “Optical implementation of the 2-D Hopfield model for a 1-D associative memory,” Opt. Commun. 76, 87–91 (1989).
[CrossRef]

1988

1987

1986

1985

1983

1982

J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982).
[CrossRef] [PubMed]

Andrés, P.

E. Bonet, P. Andrés, J. C. Barreiro, A. Pons, “Self-imaging properties of periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[CrossRef]

Arrizon, V.

Athale, R. A.

Barreiro, J. C.

E. Bonet, P. Andrés, J. C. Barreiro, A. Pons, “Self-imaging properties of periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[CrossRef]

Bonet, E.

E. Bonet, P. Andrés, J. C. Barreiro, A. Pons, “Self-imaging properties of periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[CrossRef]

Brady, D.

D. Psaltis, D. Brady, X-G. Gu, S. Lin, “Holograph in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Caola, M. J.

M. J. Caola, “Self-Fourier function,” J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

Choen-Sabban, Y.

Dunning, G. J.

Farhat, N. H.

Friedlander, C. B.

Gregory, D. A.

Gu, X-G.

D. Psaltis, D. Brady, X-G. Gu, S. Lin, “Holograph in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Hopfield, J. J.

J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982).
[CrossRef] [PubMed]

Hua, J.

J. Hua, L. Liu, G. Li, “Dual self-transform function,” J. Phys. A 30, 1–4 (1997).
[CrossRef]

J. Hua, L. Liu, “Exact periodic self-Fourier–Fresnel function,” Optik 103, 75–76 (1996).

Ito, F.

Joyeux, D.

Kitayama, K.

Lakhlakia, A.

A. Lakhlakia, “Physical fractals: self-similarity and square-integrability,” Speculat. Sci. Technol. 18, 153–156 (1995).

Leger, J. R.

Li, G.

J. Hua, L. Liu, G. Li, “Dual self-transform function,” J. Phys. A 30, 1–4 (1997).
[CrossRef]

Lin, S.

D. Psaltis, D. Brady, X-G. Gu, S. Lin, “Holograph in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

S. Lin, L. Liu, Z. Wang, “Optical implementation of the 2-D Hopfield model for a 1-D associative memory,” Opt. Commun. 76, 87–91 (1989).
[CrossRef]

Liu, L.

J. Hua, L. Liu, G. Li, “Dual self-transform function,” J. Phys. A 30, 1–4 (1997).
[CrossRef]

J. Hua, L. Liu, “Exact periodic self-Fourier–Fresnel function,” Optik 103, 75–76 (1996).

L. Liu, “Periodic self-Fourier–Fresnel function,” J. Phys. A 27, L285–L289 (1994).
[CrossRef]

L. Liu, X. Liu, L. Ye, “Joint Talbot effect and logic-operated moiré patterns,” J. Opt. Soc. Am. A 7, 970–976 (1990).
[CrossRef]

C. Pan, L. Liu, “Study of fill factor in self-imaging aperture filling of phase-locked arrays,” Opt. Commun. 77, 210–214 (1990).
[CrossRef]

S. Lin, L. Liu, Z. Wang, “Optical implementation of the 2-D Hopfield model for a 1-D associative memory,” Opt. Commun. 76, 87–91 (1989).
[CrossRef]

Liu, T.

Liu, X.

Lohmann, A. W.

Lopez-Olazagasti, E.

Lu, T.

Marom, E.

Mendlovic, D.

Ojeda-Castaneda, J.

Owechko, Y.

Pan, C.

C. Pan, L. Liu, “Study of fill factor in self-imaging aperture filling of phase-locked arrays,” Opt. Commun. 77, 210–214 (1990).
[CrossRef]

Peak, E.

Pons, A.

E. Bonet, P. Andrés, J. C. Barreiro, A. Pons, “Self-imaging properties of periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[CrossRef]

Prata, A.

Psaltis, D.

D. Psaltis, D. Brady, X-G. Gu, S. Lin, “Holograph in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

N. H. Farhat, D. Psaltis, A. Prata, E. Peak, “Optical implementation of the Hopfield model,” Appl. Opt. 24, 1469–1475 (1985).
[CrossRef] [PubMed]

Serrano-Heredia, A.

Soffer, B. H.

Swanson, G. J.

Szu, H. H.

Wang, Z.

S. Lin, L. Liu, Z. Wang, “Optical implementation of the 2-D Hopfield model for a 1-D associative memory,” Opt. Commun. 76, 87–91 (1989).
[CrossRef]

White, H. J.

Wright, W. A.

Yang, X.

Ye, L.

Yu, F. T. S.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

M. J. Caola, “Self-Fourier function,” J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

L. Liu, “Periodic self-Fourier–Fresnel function,” J. Phys. A 27, L285–L289 (1994).
[CrossRef]

J. Hua, L. Liu, G. Li, “Dual self-transform function,” J. Phys. A 30, 1–4 (1997).
[CrossRef]

Nature

D. Psaltis, D. Brady, X-G. Gu, S. Lin, “Holograph in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Opt. Commun.

S. Lin, L. Liu, Z. Wang, “Optical implementation of the 2-D Hopfield model for a 1-D associative memory,” Opt. Commun. 76, 87–91 (1989).
[CrossRef]

E. Bonet, P. Andrés, J. C. Barreiro, A. Pons, “Self-imaging properties of periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[CrossRef]

A. W. Lohmann, “Array illuminators and complexity theory,” Opt. Commun. 89, 167–172 (1992).
[CrossRef]

C. Pan, L. Liu, “Study of fill factor in self-imaging aperture filling of phase-locked arrays,” Opt. Commun. 77, 210–214 (1990).
[CrossRef]

Opt. Lett.

Optik

J. Hua, L. Liu, “Exact periodic self-Fourier–Fresnel function,” Optik 103, 75–76 (1996).

Proc. Natl. Acad. Sci. USA

J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982).
[CrossRef] [PubMed]

Speculat. Sci. Technol.

A. Lakhlakia, “Physical fractals: self-similarity and square-integrability,” Speculat. Sci. Technol. 18, 153–156 (1995).

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Figures (9)

Fig. 1
Fig. 1

Sketch of self-imaging on scale. The term g(x) is a periodic function whose period is d; f is the focal length.

Fig. 2
Fig. 2

Binary-amplitude grating and its magnified self-image: (a) The binary-amplitude grating. (b) The magnified self-image. The photographic scale is 1.84:1, and the units are millimeters.

Fig. 3
Fig. 3

Interconnection between the state vector v of the neuron and the IWM T.

Fig. 4
Fig. 4

Geometry for the optical neural network by use of SSIF: v is the state vector, T is the IWM, and AGLU is the array of grating-lens units.

Fig. 5
Fig. 5

System for the complete operation of the neural network with self-imaging on scale: v is the state vector, T is the IWM, and AGLU is the array of grating-lens units.

Fig. 6
Fig. 6

State vector v and the interconnection result formed by the array of grating-lens units: (a) The state vector v with 10 × 10 cells. The filled cells represent 1, and the open cells represent 0. Units are in millimeters.

Fig. 7
Fig. 7

Self-image intensity distribution of a grating. The period of the grating is 0.1 mm, and the width of each transparent grid is 6 μm. The grating contains 10 cycles.

Fig. 8
Fig. 8

First scheme for optical interconnection: The geometrical size of the IWM mask is K × K p . The filled squares represent the self-image of the outermost grating-lens unit. I 3 represents half the geometrical size of the self-image: I 1 = (K - 1)S g /2, I 2 = p + (K - 2)K p /2, and I 3 = MS g /2. GLU: grating-lens unit.

Fig. 9
Fig. 9

Second scheme for the optical interconnection: The geometrical size of the IWM mask is K × Kp. I 5 represents half the geometrical size of the self-image: I 4 = p + (K - 2)Kp/2, and I 5 = MS g /2. GLU: grating-lens unit.

Equations (27)

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- +   g s exp i π λ Z i s - x 2 d s 2 = | g x | 2 ,
Z i = α   2 d 2 λ ,     α = 1 ,   2 , .
I x = - +   g s exp i π λ f   s 2 exp i π λ Z x - s 2 d s 2 = - +   g s exp - i π λ Zf / Z + f × s - x 1 + Z / f 2 d s 2 = - +   g s exp i π λ Zf / Z + f s - x 1 + Z / f 2 d s 2 .
Zf Z + f = Z i ,
Z = Z i f f - Z i ,
M = 1 + Z / f = f f - Z i .
Md = fd f - Z i .
v j n + 1 = o k = 1 K   T jk v k n ,     j = 1 ,   2 , ,   K ,
v lm n + 1 = o j = 1 K k = 1 K   T lmjk v jk n ,
Md = pK ,
f = pK pK - d   Z i .
D v = JK ± 1 p ,
rd β λ ,
Mrd = η p ,     η 1 ,
K η d β λ ,
K lim = η d β λ .
K = df p f - Z i .
l 1 + l 2 l 3 = ξ ,     ξ 1 ,
l 1 = K - 1 2   S g ,
l 2 = K - 2 2   Kp + p ,
l 3 = MS g 2 ,
S g = K - 2 K + 2 1 - K + ξ Kp / d   p .
ψ = 2 S g f .
l 4 l 5 = ξ ,     ξ 1 ,
l 4 = K - 2 2   Kp + p ,
l 5 = S g M / 2 .
S g = K - 2 + 2 / K ξ   d .

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