Abstract

The capabilities of photorefractive crystals as media for holographic interconnections in neural networks are examined. Limitations on the density of interconnections and the number of holographic associations which can be stored in photorefractive crystals are derived. Optical architectures for implementing various neural schemes are described. Experimental results are presented for one of these architectures.

© 1988 Optical Society of America

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References

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  1. Y. S. Abu-Mostafa, D. Psaltis, “Optical Neural Computers,” Sci. Am. 256, 88 (1987).
    [CrossRef]
  2. D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).
  3. D. Psaltis, X. G. Gu, H. Lee, J. Yu, “Optical Interconnections Implemented with Volume Holograms,” to be published.
  4. P. J. van Heerden, “Theory of Optical Information Storage in Solids,” Appl. Opt. 2, 393 (1963).
    [CrossRef]
  5. M. Cohen, “Design of a New Medium for Volume Holographic Information Processing,” Appl. Opt. 25, 2288 (1986).
    [CrossRef] [PubMed]
  6. K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 752, 16 (1987).
  7. K. Wagner, D. Psaltis, “Nonlinear Etalons in Adaptive Optical Neural Computers,” presented at IEEE First Annual International Conference on Neural Networks, San Diego, 21–24 June 1987.
  8. K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Appl. Opt. 26, 5061 (1987).
    [CrossRef] [PubMed]
  9. D. Z. Anderson, “Adaptable Interconnects for Optical Neuromorphs: Demonstration of a Photorefractive Projection Operator,” in Proceedings, International Conference on Neural Networks, San Diego (June1987).
  10. T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, Berlin, 1984).
  11. J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. U.S.A. 79, 2554 (1982).
    [CrossRef] [PubMed]
  12. S. S. Venkatesh, D. Psaltis, “Information Storage and Retrieval in Two Associative Nets,” presented at Conference on Neural Network Models for Computing, Santa Barbara, CA (April 1985).
  13. L. Personnaz, I. Guyon, G. Dreyfus, “Information Storage and Retrieval in Spin-Glass Like Neural Networks,” J. Phys. Lett. 46, L359 (1985).
    [CrossRef]
  14. D. Psaltis, C. Park, “Nonlinear Discriminant Functions and Associative Memories,” APS Conf. Proc. 151, 370 (1986).
    [CrossRef]
  15. T. Maxwell, C. L. Giles, Y. C. Lee, H. H. Chen, “Nonlinear Dynamics of Artificial Neural Systems,” APS Conf. Proc. 151, 299 (1986).
    [CrossRef]
  16. E. B. Baum, “On the Capabilities of Multilayer Perceptrons,” to be published.
  17. D. Psaltis, N. H. Farhat, “Optical Information Processing Based on an Associative Memory Model of Neural Nets with Thresholding and Feedback,” Opt. Lett. 10, 98 (1985).
    [CrossRef] [PubMed]
  18. Y. Owechko, G. J. Dunning, E. Marom, B. H. Soffer, “Holographic Associative Memory with Nonlinearities in the Correlation Domain,” Appl. Opt. 26, 1900 (1987).
    [CrossRef] [PubMed]
  19. B. Kosko, C. Guest, “Optical Bidirectional Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 758, (1987).
  20. R. A. Athale, H. H. Szu, C. B. Friedlander, “Optical Implementation of Associative Memory with Controlled Nonlinearity in the Correlation Domain,” Opt. Lett. 11, 482 (1986).
    [CrossRef] [PubMed]
  21. F. Rosenblatt, Principles of Neurodynamics: Perceptron and the Theory of Brain Mechanisms (Spartan Books, Washington, DC, 1961).
  22. B. Widrow, M. E. Hoff, “Adaptive Switching Circuits,” IRE WESCON Conv. Rec. 4, 96 (1960).
  23. D. E. Rumelhart, J. L. McClelland, Eds., Parallel Distributed Processing, Vol. 1 (MIT Press, Cambridge, MA, 1986).
  24. D. B. Parker, “Learning Logic,” Invention Report S81-64, File 1, Office of Technology Licensing, Stanford U. (Oct.1982).
  25. J. D. Denker, Ed., “Neural Networks for Computing,” APS Conf. Proc.151 (1986).
    [CrossRef]
  26. A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1986).
  27. K. Fukushima, “A Hierarchical Neural Network Model for Associative Memory,” Biol. Cybern. 50, 105 (1984).
    [CrossRef] [PubMed]
  28. S. Grossberg, Studies of Mind and Brain (Reidel, Boston, 1982).
    [CrossRef]
  29. N. V. Kuktarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic Storage in Electrooptic Crystals. I: Steady State,” Ferroelectrics 22, 949 (1979).
    [CrossRef]
  30. J. W. Goodman, “Fan-In and Fan-Out with Optical Interconnections,” Opt. Acta 32, 1489 (1985).
    [CrossRef]
  31. D. L. Staebler, W. Phillips, “Fe-Doped LiNbO3 for Read-Write Applications,” Appl. Opt. 13, 788 (1974).
    [CrossRef] [PubMed]

1987

Y. S. Abu-Mostafa, D. Psaltis, “Optical Neural Computers,” Sci. Am. 256, 88 (1987).
[CrossRef]

K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 752, 16 (1987).

K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Appl. Opt. 26, 5061 (1987).
[CrossRef] [PubMed]

Y. Owechko, G. J. Dunning, E. Marom, B. H. Soffer, “Holographic Associative Memory with Nonlinearities in the Correlation Domain,” Appl. Opt. 26, 1900 (1987).
[CrossRef] [PubMed]

B. Kosko, C. Guest, “Optical Bidirectional Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 758, (1987).

1986

R. A. Athale, H. H. Szu, C. B. Friedlander, “Optical Implementation of Associative Memory with Controlled Nonlinearity in the Correlation Domain,” Opt. Lett. 11, 482 (1986).
[CrossRef] [PubMed]

A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1986).

M. Cohen, “Design of a New Medium for Volume Holographic Information Processing,” Appl. Opt. 25, 2288 (1986).
[CrossRef] [PubMed]

D. Psaltis, C. Park, “Nonlinear Discriminant Functions and Associative Memories,” APS Conf. Proc. 151, 370 (1986).
[CrossRef]

T. Maxwell, C. L. Giles, Y. C. Lee, H. H. Chen, “Nonlinear Dynamics of Artificial Neural Systems,” APS Conf. Proc. 151, 299 (1986).
[CrossRef]

1985

D. Psaltis, N. H. Farhat, “Optical Information Processing Based on an Associative Memory Model of Neural Nets with Thresholding and Feedback,” Opt. Lett. 10, 98 (1985).
[CrossRef] [PubMed]

L. Personnaz, I. Guyon, G. Dreyfus, “Information Storage and Retrieval in Spin-Glass Like Neural Networks,” J. Phys. Lett. 46, L359 (1985).
[CrossRef]

J. W. Goodman, “Fan-In and Fan-Out with Optical Interconnections,” Opt. Acta 32, 1489 (1985).
[CrossRef]

1984

K. Fukushima, “A Hierarchical Neural Network Model for Associative Memory,” Biol. Cybern. 50, 105 (1984).
[CrossRef] [PubMed]

1982

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. U.S.A. 79, 2554 (1982).
[CrossRef] [PubMed]

1979

N. V. Kuktarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic Storage in Electrooptic Crystals. I: Steady State,” Ferroelectrics 22, 949 (1979).
[CrossRef]

1974

1963

1960

B. Widrow, M. E. Hoff, “Adaptive Switching Circuits,” IRE WESCON Conv. Rec. 4, 96 (1960).

Abu-Mostafa, Y. S.

Y. S. Abu-Mostafa, D. Psaltis, “Optical Neural Computers,” Sci. Am. 256, 88 (1987).
[CrossRef]

Anderson, D. Z.

D. Z. Anderson, “Adaptable Interconnects for Optical Neuromorphs: Demonstration of a Photorefractive Projection Operator,” in Proceedings, International Conference on Neural Networks, San Diego (June1987).

Athale, R. A.

Baum, E. B.

E. B. Baum, “On the Capabilities of Multilayer Perceptrons,” to be published.

Chen, H. H.

T. Maxwell, C. L. Giles, Y. C. Lee, H. H. Chen, “Nonlinear Dynamics of Artificial Neural Systems,” APS Conf. Proc. 151, 299 (1986).
[CrossRef]

Cohen, M.

Dreyfus, G.

L. Personnaz, I. Guyon, G. Dreyfus, “Information Storage and Retrieval in Spin-Glass Like Neural Networks,” J. Phys. Lett. 46, L359 (1985).
[CrossRef]

Dunning, G. J.

Farhat, N. H.

Fisher, A. D.

A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1986).

Friedlander, C. B.

Fukuda, R. C.

A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1986).

Fukushima, K.

K. Fukushima, “A Hierarchical Neural Network Model for Associative Memory,” Biol. Cybern. 50, 105 (1984).
[CrossRef] [PubMed]

Giles, C. L.

T. Maxwell, C. L. Giles, Y. C. Lee, H. H. Chen, “Nonlinear Dynamics of Artificial Neural Systems,” APS Conf. Proc. 151, 299 (1986).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Fan-In and Fan-Out with Optical Interconnections,” Opt. Acta 32, 1489 (1985).
[CrossRef]

Grossberg, S.

S. Grossberg, Studies of Mind and Brain (Reidel, Boston, 1982).
[CrossRef]

Gu, X. G.

D. Psaltis, X. G. Gu, H. Lee, J. Yu, “Optical Interconnections Implemented with Volume Holograms,” to be published.

D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

Guest, C.

B. Kosko, C. Guest, “Optical Bidirectional Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 758, (1987).

Guyon, I.

L. Personnaz, I. Guyon, G. Dreyfus, “Information Storage and Retrieval in Spin-Glass Like Neural Networks,” J. Phys. Lett. 46, L359 (1985).
[CrossRef]

Hoff, M. E.

B. Widrow, M. E. Hoff, “Adaptive Switching Circuits,” IRE WESCON Conv. Rec. 4, 96 (1960).

Hopfield, J. J.

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. U.S.A. 79, 2554 (1982).
[CrossRef] [PubMed]

Kohonen, T.

T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, Berlin, 1984).

Kosko, B.

B. Kosko, C. Guest, “Optical Bidirectional Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 758, (1987).

Kuktarev, N. V.

N. V. Kuktarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic Storage in Electrooptic Crystals. I: Steady State,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Lee, H.

D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

D. Psaltis, X. G. Gu, H. Lee, J. Yu, “Optical Interconnections Implemented with Volume Holograms,” to be published.

Lee, J. N.

A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1986).

Lee, Y. C.

T. Maxwell, C. L. Giles, Y. C. Lee, H. H. Chen, “Nonlinear Dynamics of Artificial Neural Systems,” APS Conf. Proc. 151, 299 (1986).
[CrossRef]

Markov, V. B.

N. V. Kuktarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic Storage in Electrooptic Crystals. I: Steady State,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Marom, E.

Maxwell, T.

T. Maxwell, C. L. Giles, Y. C. Lee, H. H. Chen, “Nonlinear Dynamics of Artificial Neural Systems,” APS Conf. Proc. 151, 299 (1986).
[CrossRef]

Odulov, S. G.

N. V. Kuktarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic Storage in Electrooptic Crystals. I: Steady State,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Owechko, Y.

Park, C.

D. Psaltis, C. Park, “Nonlinear Discriminant Functions and Associative Memories,” APS Conf. Proc. 151, 370 (1986).
[CrossRef]

Parker, D. B.

D. B. Parker, “Learning Logic,” Invention Report S81-64, File 1, Office of Technology Licensing, Stanford U. (Oct.1982).

Personnaz, L.

L. Personnaz, I. Guyon, G. Dreyfus, “Information Storage and Retrieval in Spin-Glass Like Neural Networks,” J. Phys. Lett. 46, L359 (1985).
[CrossRef]

Phillips, W.

Psaltis, D.

K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Appl. Opt. 26, 5061 (1987).
[CrossRef] [PubMed]

Y. S. Abu-Mostafa, D. Psaltis, “Optical Neural Computers,” Sci. Am. 256, 88 (1987).
[CrossRef]

K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 752, 16 (1987).

D. Psaltis, C. Park, “Nonlinear Discriminant Functions and Associative Memories,” APS Conf. Proc. 151, 370 (1986).
[CrossRef]

D. Psaltis, N. H. Farhat, “Optical Information Processing Based on an Associative Memory Model of Neural Nets with Thresholding and Feedback,” Opt. Lett. 10, 98 (1985).
[CrossRef] [PubMed]

K. Wagner, D. Psaltis, “Nonlinear Etalons in Adaptive Optical Neural Computers,” presented at IEEE First Annual International Conference on Neural Networks, San Diego, 21–24 June 1987.

D. Psaltis, X. G. Gu, H. Lee, J. Yu, “Optical Interconnections Implemented with Volume Holograms,” to be published.

S. S. Venkatesh, D. Psaltis, “Information Storage and Retrieval in Two Associative Nets,” presented at Conference on Neural Network Models for Computing, Santa Barbara, CA (April 1985).

D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

Rosenblatt, F.

F. Rosenblatt, Principles of Neurodynamics: Perceptron and the Theory of Brain Mechanisms (Spartan Books, Washington, DC, 1961).

Soffer, B. H.

Soskin, M. S.

N. V. Kuktarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic Storage in Electrooptic Crystals. I: Steady State,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Staebler, D. L.

Szu, H. H.

van Heerden, P. J.

Venkatesh, S. S.

S. S. Venkatesh, D. Psaltis, “Information Storage and Retrieval in Two Associative Nets,” presented at Conference on Neural Network Models for Computing, Santa Barbara, CA (April 1985).

Vinetskii, V. L.

N. V. Kuktarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic Storage in Electrooptic Crystals. I: Steady State,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Wagner, K.

K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Appl. Opt. 26, 5061 (1987).
[CrossRef] [PubMed]

K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 752, 16 (1987).

K. Wagner, D. Psaltis, “Nonlinear Etalons in Adaptive Optical Neural Computers,” presented at IEEE First Annual International Conference on Neural Networks, San Diego, 21–24 June 1987.

Widrow, B.

B. Widrow, M. E. Hoff, “Adaptive Switching Circuits,” IRE WESCON Conv. Rec. 4, 96 (1960).

Yu, J.

D. Psaltis, X. G. Gu, H. Lee, J. Yu, “Optical Interconnections Implemented with Volume Holograms,” to be published.

D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

Appl. Opt.

APS Conf. Proc.

D. Psaltis, C. Park, “Nonlinear Discriminant Functions and Associative Memories,” APS Conf. Proc. 151, 370 (1986).
[CrossRef]

T. Maxwell, C. L. Giles, Y. C. Lee, H. H. Chen, “Nonlinear Dynamics of Artificial Neural Systems,” APS Conf. Proc. 151, 299 (1986).
[CrossRef]

Biol. Cybern.

K. Fukushima, “A Hierarchical Neural Network Model for Associative Memory,” Biol. Cybern. 50, 105 (1984).
[CrossRef] [PubMed]

Ferroelectrics

N. V. Kuktarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic Storage in Electrooptic Crystals. I: Steady State,” Ferroelectrics 22, 949 (1979).
[CrossRef]

IRE WESCON Conv. Rec.

B. Widrow, M. E. Hoff, “Adaptive Switching Circuits,” IRE WESCON Conv. Rec. 4, 96 (1960).

J. Phys. Lett.

L. Personnaz, I. Guyon, G. Dreyfus, “Information Storage and Retrieval in Spin-Glass Like Neural Networks,” J. Phys. Lett. 46, L359 (1985).
[CrossRef]

Opt. Acta

J. W. Goodman, “Fan-In and Fan-Out with Optical Interconnections,” Opt. Acta 32, 1489 (1985).
[CrossRef]

Opt. Lett.

Proc. Natl. Acad. Sci. U.S.A.

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. U.S.A. 79, 2554 (1982).
[CrossRef] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng.

K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 752, 16 (1987).

B. Kosko, C. Guest, “Optical Bidirectional Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 758, (1987).

A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1986).

Sci. Am.

Y. S. Abu-Mostafa, D. Psaltis, “Optical Neural Computers,” Sci. Am. 256, 88 (1987).
[CrossRef]

Other

D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

D. Psaltis, X. G. Gu, H. Lee, J. Yu, “Optical Interconnections Implemented with Volume Holograms,” to be published.

E. B. Baum, “On the Capabilities of Multilayer Perceptrons,” to be published.

S. S. Venkatesh, D. Psaltis, “Information Storage and Retrieval in Two Associative Nets,” presented at Conference on Neural Network Models for Computing, Santa Barbara, CA (April 1985).

D. Z. Anderson, “Adaptable Interconnects for Optical Neuromorphs: Demonstration of a Photorefractive Projection Operator,” in Proceedings, International Conference on Neural Networks, San Diego (June1987).

T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, Berlin, 1984).

F. Rosenblatt, Principles of Neurodynamics: Perceptron and the Theory of Brain Mechanisms (Spartan Books, Washington, DC, 1961).

K. Wagner, D. Psaltis, “Nonlinear Etalons in Adaptive Optical Neural Computers,” presented at IEEE First Annual International Conference on Neural Networks, San Diego, 21–24 June 1987.

D. E. Rumelhart, J. L. McClelland, Eds., Parallel Distributed Processing, Vol. 1 (MIT Press, Cambridge, MA, 1986).

D. B. Parker, “Learning Logic,” Invention Report S81-64, File 1, Office of Technology Licensing, Stanford U. (Oct.1982).

J. D. Denker, Ed., “Neural Networks for Computing,” APS Conf. Proc.151 (1986).
[CrossRef]

S. Grossberg, Studies of Mind and Brain (Reidel, Boston, 1982).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

General neural network architecture.

Fig. 2
Fig. 2

Two-layer network with lateral inhibition. Connections ending with an open circle are inhibitory.

Fig. 3
Fig. 3

Optical neural computer architecture.

Fig. 4
Fig. 4

Optical architecture for backward error propagation learning.

Fig. 5
Fig. 5

Simple photorefractive learning system: PB is a polarizing beam splitter; L1 and L2 are imaging lenses; WP is a quarterwave plate; SH is a shutter; P is a polarizer; D is a detector; M is a mirror.

Fig. 6
Fig. 6

Experimental learning curves.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

w i j = f i j [ x ( m ) , y ( m ) ] m = 1 M .
w i j = m = 1 M x i ( m ) y i ( m ) .
Δ w i j ( k ) = f i j [ x ( m ) , w r s ( k ) , y ( m ) ] .
Δ w i j ( k ) = f i j [ z i { w r s ( k ) , y ( m ) , x ( m ) } , z j { w r s ( k ) , y ( m ) , x ( m ) } ] .
Δ w i j ( k ) = f i j [ x ( m ) , w r s ( k ) ] .
Δ χ i j = A i A j * exp ( j K i j · r ) ,
k α k β = K i j .
n ( r ) = ν S n ν exp ( j k ν · r ) ,
k ν = [ ν x ( 2 π L x ) x ˆ + ν y ( 2 π L y ) y ˆ + ν z ( 2 π L z ) z ˆ ] ν i = 0 , ± 1 , ± 2 ,
E i in exp ( j k i · r ) exp ( j ϕ i ) + c . c . = j η i j exp ( j ψ i j ) × exp ( j K i j · r ) × E j out exp ( j k j · r ) × exp ( j ϕ j ) + c . c .
R { i j η i j exp ( j ψ i j ) exp ( i K i j · r ) } η 0 ,
η 1 η 0 N 2 .
y i = f ( | j w i j x j | 2 ) ,
y i = f ( j w i j x j ) .
I in = κ | j N η i j exp ( j ψ i j ) E j out exp ( j ϕ j ) | 2 = N η 1 2 I out = I out G coherent .
G coherent = 1 2 η 0 .
I in = κ j N η i j 2 | E j out | 2 = N η 1 2 I out = I out G incoherent .
G incoherent = N 2 η 0 .
η 0 Δ 2 π λ L ,
A m = A 0 [ 1 exp ( t m τ r ) ] exp ( m = m + 1 M t m τ e ) ,
[ 1 exp ( t m τ r ) ] exp ( t m + 1 τ e ) = [ 1 exp ( t m + 1 τ r ) ] .
t m = τ e ln ( m m 1 ) m > 1 ,
A m = A M = A 0 M .
t m 0 + 1 t m 0 1 + t m 0 τ e = ( ρ m 0 1 + ρ m 0 ) τ e ,
ρ m 0 + 1 = ρ m 0 1 + ρ m 0 .
ρ m = 1 ( m m 0 ) + 1 ρ m 0 .
ρ m 1 m ,
t m = τ e m .
A m = A M = A 0 [ 1 exp ( τ e M τ r ) ]
M τ e τ r A 0 A m .
Δ w i j ( 2 ) = α ε i f ( x i in ) x j out ,
Δ w l m ( 1 ) = i α ε i f ( x i in ) w i l ( 2 ) f ( x l in ) x m 0 ,
i ε i f ( x i in ) w i l ( 2 ) .

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