Abstract

We find instabilities and self-pulsation in an optical ring cavity with a photorefractive two-wave mixer. The mean-field limit is used to obtain a single criterion for instabilities that originate from a Hopf bifurcation.

© 1989 Optical Society of America

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References

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  1. R. W. Boyd, M. G. Raymer, L. M. Narducci, eds., Optical Instabilities (Cambridge U. Press, Cambridge, UK, 1986).
  2. B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Opt. Lett. 6, 519 (1981).
    [CrossRef] [PubMed]
  3. See, e.g., S. Kwong, M. Cronin-Golomb, A. Yariv, Appl. Phys. Lett. 45, 1016 (1984); G. C. Valley, G. J. Dunning, Opt. Lett. 9, 513 (1984).
    [CrossRef] [PubMed]
  4. D. M. Lininger, P. J. Martin, D. Z. Anderson, Opt. Lett. 14, 697 (1989).
    [CrossRef] [PubMed]
  5. S. Weiss, B. Fischer, Opt. Commun. 70, 515 (1989).
    [CrossRef]
  6. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  7. S. Weiss, B. Fischer, submitted to Phys. Rev. A.
  8. L. A. Lugiato, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. 21, p. 69.
    [CrossRef]
  9. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, Ferroelectrics 22, 949 (1979).
    [CrossRef]
  10. G. C. Valley, J. F. Lam, in Photorefractive Materials and Their Applications I, P. Günter, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1988), p. 75.
    [CrossRef]
  11. The combination of a moving grating and an applied field was first studied by J. P. Huignard, A. Marrakchi, Opt. Commun. 38, 249 (1981). A comprehensive study is given by Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
    [CrossRef]
  12. S. Ducharme, J. Feinberg, J. Appl. Phys. 56, 839 (1984).
    [CrossRef]
  13. J. O. White, M. Cronin-Golomb, B. Fischer, A. Yariv, Appl. Phys. Lett. 40, 450 (1982).
    [CrossRef]
  14. H. Gang, Y. Guo-jian, Phys. Rev. A 38, 1979 (1988).
    [CrossRef]
  15. See, e.g., H. Haken, Synergetics—An Introduction (Springer-Verlag, Berlin, 1983), p. 124.
  16. I. Bar-Joseph, Y. Silberberg, Opt. Commun. 48, 53 (1983); J. Opt. Soc. Am. B 1, 662 (1984); Phys. Rev. A 36, 1731 (1987).
    [CrossRef] [PubMed]

1989 (2)

1988 (1)

H. Gang, Y. Guo-jian, Phys. Rev. A 38, 1979 (1988).
[CrossRef]

1984 (2)

See, e.g., S. Kwong, M. Cronin-Golomb, A. Yariv, Appl. Phys. Lett. 45, 1016 (1984); G. C. Valley, G. J. Dunning, Opt. Lett. 9, 513 (1984).
[CrossRef] [PubMed]

S. Ducharme, J. Feinberg, J. Appl. Phys. 56, 839 (1984).
[CrossRef]

1983 (1)

I. Bar-Joseph, Y. Silberberg, Opt. Commun. 48, 53 (1983); J. Opt. Soc. Am. B 1, 662 (1984); Phys. Rev. A 36, 1731 (1987).
[CrossRef] [PubMed]

1982 (1)

J. O. White, M. Cronin-Golomb, B. Fischer, A. Yariv, Appl. Phys. Lett. 40, 450 (1982).
[CrossRef]

1981 (2)

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Opt. Lett. 6, 519 (1981).
[CrossRef] [PubMed]

The combination of a moving grating and an applied field was first studied by J. P. Huignard, A. Marrakchi, Opt. Commun. 38, 249 (1981). A comprehensive study is given by Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, Ferroelectrics 22, 949 (1979).
[CrossRef]

Anderson, D. Z.

Bar-Joseph, I.

I. Bar-Joseph, Y. Silberberg, Opt. Commun. 48, 53 (1983); J. Opt. Soc. Am. B 1, 662 (1984); Phys. Rev. A 36, 1731 (1987).
[CrossRef] [PubMed]

Cronin-Golomb, M.

See, e.g., S. Kwong, M. Cronin-Golomb, A. Yariv, Appl. Phys. Lett. 45, 1016 (1984); G. C. Valley, G. J. Dunning, Opt. Lett. 9, 513 (1984).
[CrossRef] [PubMed]

J. O. White, M. Cronin-Golomb, B. Fischer, A. Yariv, Appl. Phys. Lett. 40, 450 (1982).
[CrossRef]

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Opt. Lett. 6, 519 (1981).
[CrossRef] [PubMed]

Ducharme, S.

S. Ducharme, J. Feinberg, J. Appl. Phys. 56, 839 (1984).
[CrossRef]

Feinberg, J.

S. Ducharme, J. Feinberg, J. Appl. Phys. 56, 839 (1984).
[CrossRef]

Fischer, B.

S. Weiss, B. Fischer, Opt. Commun. 70, 515 (1989).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, A. Yariv, Appl. Phys. Lett. 40, 450 (1982).
[CrossRef]

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Opt. Lett. 6, 519 (1981).
[CrossRef] [PubMed]

S. Weiss, B. Fischer, submitted to Phys. Rev. A.

Gang, H.

H. Gang, Y. Guo-jian, Phys. Rev. A 38, 1979 (1988).
[CrossRef]

Guo-jian, Y.

H. Gang, Y. Guo-jian, Phys. Rev. A 38, 1979 (1988).
[CrossRef]

Haken, H.

See, e.g., H. Haken, Synergetics—An Introduction (Springer-Verlag, Berlin, 1983), p. 124.

Huignard, J. P.

The combination of a moving grating and an applied field was first studied by J. P. Huignard, A. Marrakchi, Opt. Commun. 38, 249 (1981). A comprehensive study is given by Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, Ferroelectrics 22, 949 (1979).
[CrossRef]

Kwong, S.

See, e.g., S. Kwong, M. Cronin-Golomb, A. Yariv, Appl. Phys. Lett. 45, 1016 (1984); G. C. Valley, G. J. Dunning, Opt. Lett. 9, 513 (1984).
[CrossRef] [PubMed]

Lam, J. F.

G. C. Valley, J. F. Lam, in Photorefractive Materials and Their Applications I, P. Günter, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1988), p. 75.
[CrossRef]

Lininger, D. M.

Lugiato, L. A.

L. A. Lugiato, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. 21, p. 69.
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, Ferroelectrics 22, 949 (1979).
[CrossRef]

Marrakchi, A.

The combination of a moving grating and an applied field was first studied by J. P. Huignard, A. Marrakchi, Opt. Commun. 38, 249 (1981). A comprehensive study is given by Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Martin, P. J.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, Ferroelectrics 22, 949 (1979).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Silberberg, Y.

I. Bar-Joseph, Y. Silberberg, Opt. Commun. 48, 53 (1983); J. Opt. Soc. Am. B 1, 662 (1984); Phys. Rev. A 36, 1731 (1987).
[CrossRef] [PubMed]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, Ferroelectrics 22, 949 (1979).
[CrossRef]

Valley, G. C.

G. C. Valley, J. F. Lam, in Photorefractive Materials and Their Applications I, P. Günter, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1988), p. 75.
[CrossRef]

Vinetski, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, Ferroelectrics 22, 949 (1979).
[CrossRef]

Weiss, S.

S. Weiss, B. Fischer, Opt. Commun. 70, 515 (1989).
[CrossRef]

S. Weiss, B. Fischer, submitted to Phys. Rev. A.

White, J. O.

J. O. White, M. Cronin-Golomb, B. Fischer, A. Yariv, Appl. Phys. Lett. 40, 450 (1982).
[CrossRef]

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Opt. Lett. 6, 519 (1981).
[CrossRef] [PubMed]

Yariv, A.

See, e.g., S. Kwong, M. Cronin-Golomb, A. Yariv, Appl. Phys. Lett. 45, 1016 (1984); G. C. Valley, G. J. Dunning, Opt. Lett. 9, 513 (1984).
[CrossRef] [PubMed]

J. O. White, M. Cronin-Golomb, B. Fischer, A. Yariv, Appl. Phys. Lett. 40, 450 (1982).
[CrossRef]

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Opt. Lett. 6, 519 (1981).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

J. O. White, M. Cronin-Golomb, B. Fischer, A. Yariv, Appl. Phys. Lett. 40, 450 (1982).
[CrossRef]

See, e.g., S. Kwong, M. Cronin-Golomb, A. Yariv, Appl. Phys. Lett. 45, 1016 (1984); G. C. Valley, G. J. Dunning, Opt. Lett. 9, 513 (1984).
[CrossRef] [PubMed]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, Ferroelectrics 22, 949 (1979).
[CrossRef]

J. Appl. Phys. (1)

S. Ducharme, J. Feinberg, J. Appl. Phys. 56, 839 (1984).
[CrossRef]

Opt. Commun. (3)

The combination of a moving grating and an applied field was first studied by J. P. Huignard, A. Marrakchi, Opt. Commun. 38, 249 (1981). A comprehensive study is given by Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

S. Weiss, B. Fischer, Opt. Commun. 70, 515 (1989).
[CrossRef]

I. Bar-Joseph, Y. Silberberg, Opt. Commun. 48, 53 (1983); J. Opt. Soc. Am. B 1, 662 (1984); Phys. Rev. A 36, 1731 (1987).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. A (1)

H. Gang, Y. Guo-jian, Phys. Rev. A 38, 1979 (1988).
[CrossRef]

Other (6)

See, e.g., H. Haken, Synergetics—An Introduction (Springer-Verlag, Berlin, 1983), p. 124.

R. W. Boyd, M. G. Raymer, L. M. Narducci, eds., Optical Instabilities (Cambridge U. Press, Cambridge, UK, 1986).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

S. Weiss, B. Fischer, submitted to Phys. Rev. A.

L. A. Lugiato, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. 21, p. 69.
[CrossRef]

G. C. Valley, J. F. Lam, in Photorefractive Materials and Their Applications I, P. Günter, J. P. Huignard, eds. (Springer-Verlag, Berlin, 1988), p. 75.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Photorefractive unidirectional ring cavity with an injected signal. C, crystal; V, voltage source; BS’s, beam splitters; M’s, mirrors.

Fig. 2
Fig. 2

Instabilities in photorefractive optical bistability. (a) The steady-state-normalized output intensity X versus the normalized input intensity Y for C = 5, σ = −1, Δ0 = 4, θ = 0.5, k = ν0 = 1, and E0 = 2.5 kV/cm. (b) The Hopf bifurcation condition f versus the normalized output intensity X. (c) The phase plane portrait of the same system when driven with normalized input intensity y = 5.5 (Y = 30.25); the variables are x1 = Re{x} and x2 = Im{x}. (d) x1 and x2 versus (t/τcav); one division is 1(t/τcav).

Equations (14)

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( t + c z ) A 1 = σ i ω n 0 G A 4 ,
( t + c z ) A 4 * = + σ i ω n 0 G A 1 * ,
G t = ν 0 [ i n 1 2 f ( E 0 ) A 1 A 4 * G ( I 0 + i Δ 0 ) ] ,
A 4 * ( 0 , t ) = T A 4 in + R e i δ 0 A 4 * ( l , t Δ t ) , A 1 ( 0 , t ) = const . ,
γ 0 l 0 , T 0 , δ 0 0 ,
( t + c l L z ) A ˜ 1 = τ p 1 { [ A 1 ss ( l ) A 1 ss ( 0 ) ] σ i ω n 0 l c G ˜ A ˜ 4 } ,
( t + c l L z ) A ˜ 4 * = τ cav 1 [ ( 1 + i θ ) A ˜ 4 * + A 4 in T + σ i ω n 0 l c T G ˜ A ˜ 1 * ] ,
G ˜ t = t 0 1 [ i n 1 2 f ( E 0 ) A ˜ 1 A ˜ 4 * G ˜ ( I ˜ 0 + i Δ 0 ) ] ,
x ˙ = k [ ( 1 + i θ ) x y + 2 σ C g ] ,
g ˙ = ν 0 [ ( F + i F ) x ( 1 + | x | 2 + i Δ 0 ) g ] ,
Y = X { [ 1 2 σ C ( 1 + X ) F ( 1 + X ) 2 + Δ 0 2 ] 2 + [ θ + 2 σ C Δ 0 F ( 1 + X ) 2 + Δ 0 2 ] 2 + 4 σ C F ( 1 + X ) 2 + Δ 0 2 [ σ C F Δ 0 θ ( 1 + X ) ] } .
[ δ x ˙ δ x ˙ * δ g ˙ δ g ˙ * ] = [ k ( 1 + i θ ) 0 2 σ C k 0 0 k ( 1 i θ ) 0 2 σ C k ν 0 [ ( F + i F ) x s * g s ] ν 0 x s g s ν 0 ( 1 + X s + i Δ 0 ) 0 ν 0 x s * g s * ν 0 [ ( F i F ) x s g s * ] 0 ν 0 ( 1 + X s i Δ 0 ) ] [ δ x δ x * δ g δ g * ] ,
λ 4 + a 1 λ 3 + a 2 λ 2 + a 3 λ + a 4 = 0 .
f = a 1 a 2 a 3 a 1 2 a 4 a 3 2 = 0 .

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