Abstract

I derive and analyze coupled-mode equations for a double phase-conjugating resonator that is loaded with a volume-holographic recording medium in which multiplexed images have been stored. If the only nonlinearity present is the saturating gain of the conjugators, there is no multistability: The most deeply imprinted image always wins the competition for saturation. If several images are imprinted with equal depth, any slowly varying superposition of these images is neutrally stable. However, if a nonlinearity, arising from a space-charge-dependent cubic polarization, is present the resonator does exhibit multiple basins of stability that correspond to the faithful reconstruction of each stored image. The momentary injection of an initial image will tip the system into the basin that corresponds to the stored image that it most closely resembles, even if that image is not the most deeply imprinted one.

© 1991 Optical Society of America

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  1. M. S. Cohen, “Self organization categorization, and abstraction in a phase conjugating resonator,” in Optical Computing, J. A. Neff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.625, 214–219 (1986).
  2. M. S. Cohen, “Multiple correlations in holographic resonators,” in Neural Networks for Optical Computing, R. A. Athale, J. Davis, eds., Proc. Soc. Photo-Opt. Instrum. Eng.882, 122–131 (1988).
    [CrossRef]
  3. B. H. Soffer, G. J. Dunning, Y. Owechko, E. Marom, “Associative holographic memory with feedback using phase conjugate mirrors,” Opt. Lett. 11, 118–120 (1986).
    [CrossRef]
  4. D. Z. Anderson, C. E. Marie, “Resonator memories and optical novelty filters,” Opt. Eng. 26, 434–444 (1987).
    [CrossRef]
  5. M. S. Cohen, “Design of a new medium for volume holographic information processing,” Appl. Opt. 25, 2288–2294 (1986).
    [CrossRef] [PubMed]
  6. M. S. Cohen, W. H. Julian, “Multistability, chains, and cycles in optical multiwave mixing processes,” Appl. Opt. (to be published).
  7. A. E. Siegman, P. A. Belanger, A. Hardy, “Optical resonators using phase-conjugating mirrors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 465–535.
    [CrossRef]
  8. A. C. Newell, “Bifurcation and nonlinear focusing,” in Pattern Formation and Pattern Recognition, H. Haken, ed. (Springer-Verlag, Berlin, 1980), pp. 244–265.
  9. M. Schubert, B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New York, 1986).
  10. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskuv, V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 949 (1979).
    [CrossRef]
  11. P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  12. A. Borshch, M. Brodin, V. Volkov, N. Kukhtarev, V. Starkov, “Optical phase conjugation by degenerate six-photon mixing,” J. Opt. Soc. Am. A 1, 40 (1984).
    [CrossRef]
  13. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1984).
    [CrossRef]
  14. D. L. Bobroff, H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390–403 (1967).
    [CrossRef]
  15. W. W. Rigrod, R. A. Fisher, B. J. Feldman, “Transient analysis of nearly degenerate four-wave mixing,” Opt. Lett. 5, 105–107 (1980).
    [CrossRef] [PubMed]
  16. R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A, 23, 3071–3083 (1981).
    [CrossRef]
  17. F. John, Partial Differential Equations (Springer-Verlag, New York, 1982), p. 48.
  18. M. Sargent, M. O. Scully, W. Lamb, Laser Physics (Addison-Wesley, New York, 1987).

1987

D. Z. Anderson, C. E. Marie, “Resonator memories and optical novelty filters,” Opt. Eng. 26, 434–444 (1987).
[CrossRef]

1986

1985

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

1984

A. Borshch, M. Brodin, V. Volkov, N. Kukhtarev, V. Starkov, “Optical phase conjugation by degenerate six-photon mixing,” J. Opt. Soc. Am. A 1, 40 (1984).
[CrossRef]

J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1984).
[CrossRef]

1981

R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A, 23, 3071–3083 (1981).
[CrossRef]

1980

1979

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskuv, V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 949 (1979).
[CrossRef]

1967

D. L. Bobroff, H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390–403 (1967).
[CrossRef]

Anderson, D. Z.

D. Z. Anderson, C. E. Marie, “Resonator memories and optical novelty filters,” Opt. Eng. 26, 434–444 (1987).
[CrossRef]

Belanger, P. A.

A. E. Siegman, P. A. Belanger, A. Hardy, “Optical resonators using phase-conjugating mirrors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 465–535.
[CrossRef]

Bobroff, D. L.

D. L. Bobroff, H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390–403 (1967).
[CrossRef]

Borshch, A.

Brodin, M.

Cohen, M. S.

M. S. Cohen, “Design of a new medium for volume holographic information processing,” Appl. Opt. 25, 2288–2294 (1986).
[CrossRef] [PubMed]

M. S. Cohen, W. H. Julian, “Multistability, chains, and cycles in optical multiwave mixing processes,” Appl. Opt. (to be published).

M. S. Cohen, “Self organization categorization, and abstraction in a phase conjugating resonator,” in Optical Computing, J. A. Neff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.625, 214–219 (1986).

M. S. Cohen, “Multiple correlations in holographic resonators,” in Neural Networks for Optical Computing, R. A. Athale, J. Davis, eds., Proc. Soc. Photo-Opt. Instrum. Eng.882, 122–131 (1988).
[CrossRef]

Dunning, G. J.

Feldman, B. J.

R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A, 23, 3071–3083 (1981).
[CrossRef]

W. W. Rigrod, R. A. Fisher, B. J. Feldman, “Transient analysis of nearly degenerate four-wave mixing,” Opt. Lett. 5, 105–107 (1980).
[CrossRef] [PubMed]

Fisher, R. A.

R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A, 23, 3071–3083 (1981).
[CrossRef]

W. W. Rigrod, R. A. Fisher, B. J. Feldman, “Transient analysis of nearly degenerate four-wave mixing,” Opt. Lett. 5, 105–107 (1980).
[CrossRef] [PubMed]

Hardy, A.

A. E. Siegman, P. A. Belanger, A. Hardy, “Optical resonators using phase-conjugating mirrors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 465–535.
[CrossRef]

Haus, H. A.

D. L. Bobroff, H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390–403 (1967).
[CrossRef]

Hopfield, J.

J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1984).
[CrossRef]

Huignard, H. P.

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

John, F.

F. John, Partial Differential Equations (Springer-Verlag, New York, 1982), p. 48.

Julian, W. H.

M. S. Cohen, W. H. Julian, “Multistability, chains, and cycles in optical multiwave mixing processes,” Appl. Opt. (to be published).

Kukhtarev, N.

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskuv, V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Lamb, W.

M. Sargent, M. O. Scully, W. Lamb, Laser Physics (Addison-Wesley, New York, 1987).

Marie, C. E.

D. Z. Anderson, C. E. Marie, “Resonator memories and optical novelty filters,” Opt. Eng. 26, 434–444 (1987).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskuv, V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Marom, E.

Newell, A. C.

A. C. Newell, “Bifurcation and nonlinear focusing,” in Pattern Formation and Pattern Recognition, H. Haken, ed. (Springer-Verlag, Berlin, 1980), pp. 244–265.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskuv, V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Owechko, Y.

Rajbenbach, H.

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refregier, P.

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Rigrod, W. W.

Sargent, M.

M. Sargent, M. O. Scully, W. Lamb, Laser Physics (Addison-Wesley, New York, 1987).

Schubert, M.

M. Schubert, B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New York, 1986).

Scully, M. O.

M. Sargent, M. O. Scully, W. Lamb, Laser Physics (Addison-Wesley, New York, 1987).

Siegman, A. E.

A. E. Siegman, P. A. Belanger, A. Hardy, “Optical resonators using phase-conjugating mirrors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 465–535.
[CrossRef]

Soffer, B. H.

Solymar, C.

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Soskuv, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskuv, V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Starkov, V.

Suydam, B. R.

R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A, 23, 3071–3083 (1981).
[CrossRef]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskuv, V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 949 (1979).
[CrossRef]

Volkov, V.

Wilhelmi, B.

M. Schubert, B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New York, 1986).

Appl. Opt.

Ferroelectrics

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskuv, V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 949 (1979).
[CrossRef]

J. Appl. Phys.

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

D. L. Bobroff, H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390–403 (1967).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

D. Z. Anderson, C. E. Marie, “Resonator memories and optical novelty filters,” Opt. Eng. 26, 434–444 (1987).
[CrossRef]

Opt. Lett.

Phys. Rev. A

R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A, 23, 3071–3083 (1981).
[CrossRef]

Proc. Natl. Acad. Sci. USA

J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1984).
[CrossRef]

Other

F. John, Partial Differential Equations (Springer-Verlag, New York, 1982), p. 48.

M. Sargent, M. O. Scully, W. Lamb, Laser Physics (Addison-Wesley, New York, 1987).

M. S. Cohen, “Self organization categorization, and abstraction in a phase conjugating resonator,” in Optical Computing, J. A. Neff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.625, 214–219 (1986).

M. S. Cohen, “Multiple correlations in holographic resonators,” in Neural Networks for Optical Computing, R. A. Athale, J. Davis, eds., Proc. Soc. Photo-Opt. Instrum. Eng.882, 122–131 (1988).
[CrossRef]

M. S. Cohen, W. H. Julian, “Multistability, chains, and cycles in optical multiwave mixing processes,” Appl. Opt. (to be published).

A. E. Siegman, P. A. Belanger, A. Hardy, “Optical resonators using phase-conjugating mirrors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 465–535.
[CrossRef]

A. C. Newell, “Bifurcation and nonlinear focusing,” in Pattern Formation and Pattern Recognition, H. Haken, ed. (Springer-Verlag, Berlin, 1980), pp. 244–265.

M. Schubert, B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New York, 1986).

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Figures (3)

Fig. 1
Fig. 1

Schematic. During the recording phase, each wave vector Ajs exp(ikj · x) from the object is split by beam splitter BS and recording mirrors RM1 and RM2 into two parts, Rjs exp(irj · x) and Ljs exp(ilj · x), where Rjs = Ljs = Ajs. These beams interfere inside the volume-holographic medium VH to produce stored gratings ≃AjsĀls exp[i(rjll) · x] and higher-order intermodulation gratings ≃AjsĀksAlsĀms exp[i(rjlk + rllm) · x]. During the reconstruction phase the resonator is seeded at T = 0 with Rj(0)exp(irj · x), j = 1, …, n. Phase-conjugating mirrors PCM1 and PCM2 close a self-consistent cycle of waves rj → −rj → −llrj, with the stored gratings reconstructing all associated waves Ll(T) and Rl(T).

Fig. 2
Fig. 2

Flow of Eq. (13) maximizes plotted here as distance from the origin. (a) If the dominant eigenvalue C1 has only one eigenvector V1, the system, Eq. (13), relaxes to a stable equilibrium along V1. The equilibria along all the other eigenvectors are unstable. (b) If, however, C1 has D eigenvectors, there is an entire D-dimensional hypersphere of neutrally stable equilibria.

Fig. 3
Fig. 3

Energy surface (X1, …, XM) for EI ≠ 0, a quartic surface with pockets along axes XI, I = 1, …, M, which represent stored vectors Vs after diagonalization ( is plotted here as distance from the origin). Each pocket represents a basin of stability, i.e., a local maximum of . An initially input vector V ^ J(0) will relax into the nearest basin of attraction VJ, i.e., that for which 〈Vs, Vj(0)〉 was maximal. The energy surface for all EI = 0, however, is a hypersphere; any vector from the origin that lands on the hypersphere represents a neutrally stable equilibrium.

Equations (59)

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A j s exp ( i k j · x ) exp ( i ω t ) + c . c . ,
j = 1 n A j s exp ( i r j · x ) exp ( i ω t ) + c . c . ,
l = 1 n A l s exp ( i l l · x ) exp ( i ω t ) + c . c . ,
χ ^ 2 , 1 ( ω ) = exp ( i σ ) { s = 1 M c s j = 1 n l = 1 n A ¯ j s A l s × exp [ i ( r j - l l ) · x ] + c . c . } .
2 E - μ 0 E t t = μ 0 P t t ,
E ( x , X , t , T ) = j = 1 n [ R j ( X , T ) exp ( i r j · x ) + R - j ( X , T ) exp ( - i r j · x ) + L j ( X , T ) exp ( i l j · x ) + L - j ( X , T ) exp ( - i l j · x ) exp ( i ω t ) + c . c .
X = a x ,             T = α t ,
P ^ L ( ω ) = [ χ ^ 1 ( ω ) + α χ ^ 2 , 1 ( ω ) ] E ^ ( ω ) .
( T - r ^ j · x ) R j d d τ R j = l = 1 n κ j l L l - ρ R j .
κ j l = 1 2 v V H μ 0 ω 2 exp ( i σ ) ( s = 1 M c s A j s A ¯ l s ) ,
( T + l ^ j · x ) L ¯ - j d d τ L ¯ - j = l = 1 n κ ¯ l j R ¯ - l - ρ L ¯ - j .
R - j = G ( I 0 ) R ¯ j
L j = G ( I 0 ) L ¯ - j ,
I 0 = l = 1 n R l 2
I 0 = l = 1 n L - l 2
L ¯ - l = exp [ - i ( γ + θ ) ] R l
d d τ A j = G ( I 0 ) l = 1 n κ j l A l - ρ A j ,
ξ j = r j - r j - l j l j r j .
Δ A ^ j Δ T G ( I 0 ) l = 1 n K j l A ^ l - β A ^ j ,
K j l = μ 0 ω 2 L VH ( s = 1 M c s A j s A ¯ l s ) s = 1 M C s A j s A ¯ l s             ( Hermitian )
β = μ 0 ω L VH ρ .
G ( I 0 ) = Γ - g I 0 + order ( I 0 2 ) .
Δ A j Δ T = ( Γ - g l = 1 n A l 2 ) l = 1 n K j l A l - β A j ,
Δ V Δ T = ( Γ - α V 2 ) K V - β V ,
V = [ A 1 A n ]
K = s = 1 M C s V s V s H ,
Δ X j Δ T = ( Γ - g X L 2 ) C J X J - β X J .
Δ X j Δ T = C J H X ¯ J ,
H = Γ X L 2 - g 2 ( X L 2 ) 2 - β C L - 1 X L 2 .
X 1 2 = C 1 Γ - β g C 1 ;             X J 2 = 0 ,             J 1.
Δ X J Δ T = ( Γ C J - β ) X J - g C J ( X L 2 ) X J + E J X J 2 X J ,
H = ( Γ - β C L - 1 ) X L 2 - g 2 ( X L 2 ) 2 + E J C J X L 4 .
X I 2 = C I Γ - β g C I - E I ;             X J 2 = 0 ,             J I .
Δ A j Δ T = ( Γ - g A l 2 ) l = 1 n K j l A l - β A j + k l m T j k l m A k A ¯ l A m
Δ V Δ T = ( Γ - g V 2 ) K V - β V + T ( V , V H , V ) .
T j k l m = s = 1 M E s A j s A ¯ k s A l s A ¯ m s
T = E s V s V s H V s V s H ,
I s 2 = { j = 1 n A j s [ exp ( i r j · x ) + exp ( i l j · x ) ] exp ( i ω t ) + c . c . } 4 ,
n s j k l m A j s A ¯ k s A l s A ¯ m s × exp [ i ( r j - l k + r l - l m ) · x ] + c . c .
d R j d τ = k l m τ j k l m ( L k R ¯ l L m + L ¯ - k R - l L m + L k R - l L ¯ - m )
d L ¯ - j d τ = k l m τ ¯ k j m l ( R ¯ - k L - l R ¯ - m + R ¯ k L l R ¯ - m + R ¯ - k L ¯ l R m ) ,
τ j k l m = s = 1 M e s A j s A ¯ k s A l s A ¯ m s
T j k l m = 6 L T v T Γ 2 s = 1 M Re ( e s ) A j s A ¯ k s A l s A ¯ m s s = 1 M E s A j s A ¯ k s A l s A ¯ m s ,
E I C I ( C I Γ - β g C I - E I ) 2
d A j d ξ j d A j d τ ( T - v k j · x ) A j = f j ( A 1 , , A n ) ,
( x 0 , T 0 ) ( x 0 , T 1 ) d A j d τ d τ = A j ( X 0 , T 1 ) - A j ( X 0 , T 0 ) Δ A j ( X 0 ) | T 0 T 1 = ( X 0 , T 0 ) ( X 0 , T 1 ) f j { A 1 [ X ( T ) , T ] , , A n [ X ( T ) , T ] } d τ f ^ j ( X 0 ) Δ T j ,
Δ A ^ j ( X 0 ) Δ T = f ^ j ( X 0 ) f j ( A ^ 1 , , A ^ n ) .
f j ( X , T ) p = 1 Q U [ X ( T p ) , T p ] f j p { A 1 [ X ( T ) , T ] , , A n [ X ( T ) , T ] } ,
δ ( T p ) = lim T p 0 1 T p U ( T p )
δ [ X ( T p ) , T p ] = lim L p 0 ( 1 L p / V p ) U [ X ( T p ) , T p ] .
U [ X ( T p ) , T p ] L p v p · δ [ X ( T p ) , T p ]
f ^ j ( X 0 ) Δ T ( X 0 , T 0 ) ( X 0 , T 1 ) p L p v p δ [ X ( T p ) , T p ] f j p × { A 1 [ X ( T ) , T ] , , A n [ X ( T ) , T ] } d T ,
Δ A j ( X 0 ) Δ T = p = 1 Q L p v p f ^ j p ,
f ^ j p ( 1 / 2 ) ( f j p { A l [ X ( T p - ) , T p - ] } + f j p { A l [ X ( T p + ) , T p + ] } )
f ^ j p = f j p [ A ^ l ( p ) ] .
A ^ = [ A ^ j 1 ( 1 ) A ^ j ( Q ) ]             for j = 1 , , n ;
Δ A ^ Δ T = F ( A ^ ) ,
A ^ = [ R + R ¯ - L + L ¯ - ] ,
V = [ A ^ 1 A ^ n ] ,

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