Abstract

We show that the standard model of four-wave mixing in a photorefractive crystal predicts the appearance of deterministic chaos. In this model there is a single (transmission) grating and no external or internal (intracavity) feedback. The intensity of the phase-conjugate wave is found to exhibit a period-doubling route to chaos on variation of the intensity of the probe beam and the linear absorption coefficient. The crucial elements in obtaining chaotic behavior are operation above the threshold for self-oscillation and the presence of an external electric field, which causes a shift in the optical frequency of the phase-conjugate wave.

© 1990 Optical Society of America

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References

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  1. See, for example, R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983); P. Günter, Phys. Rep. 93, 99 (1982).
    [CrossRef]
  2. See G. C. Valley, Pochi Yeh, eds., J. Opt. Soc. Am. 5, 1681–1821 (1988), and numerous references cited therein.
  3. P. Günter, E. Voit, M. Z. Zha, J. Albers, Opt. Commun. 55, 210 (1985); A. M. C. Smout, R. W. Eason, M. C. Gower, Opt. Commun. 59, 77 (1986).
    [CrossRef]
  4. D. J. Gauthier, P. Narum, R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987).
    [CrossRef] [PubMed]
  5. G. Valley, G. Dunning, Opt. Lett. 9, 420 (1984); G. Reiner, P. Meystre, E. M. Wright, J. Opt. Soc. Am. B 4, 865, (1987); G. Reiner, M. R. Belić, P. Meystre, J. Opt. Soc. Am. B 5, 1193 (1988).
    [CrossRef]
  6. J. Feinberg, Opt. Lett. 7, 486 (1982).
    [CrossRef] [PubMed]
  7. C. Pare, M. Piche, P. A. Belanger, J. Opt. Soc. Am. B 5, 679 (1988); Y. Silberberg, I. Bar-Joseph, Phys. Rev. Lett. 48, 1541 (1982).
    [CrossRef]
  8. S. I. Stepanov, M. P. Petrov, Opt. Commun. 53, 64 (1985).
    [CrossRef]
  9. N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, V. Vinetskii, Ferroelectrics 22, 949 (1979); N. V. Kukhtarev, V. Markov, S. Odulov, Opt. Commun. 23, 338 (1977).
    [CrossRef]
  10. L. Solymar, J. M. Heaton, Opt. Commun. 51, 76 (1984); J. M. Heaton, L. Solymar, Opt. Acta 32, 397 (1985).
    [CrossRef]
  11. Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
    [CrossRef]
  12. B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Opt. Lett. 6, 519 (1981).
    [CrossRef] [PubMed]
  13. W. Królikowski, M. R. Belić, A. Błȩdowski, Phys. Rev. A 31, 2224 (1988).
    [CrossRef]
  14. B. Fischer, S. Sternklar, S. Weiss, IEEE J. Quantum Electron. 25, 550 (1989).
    [CrossRef]
  15. R. W. Boyd, R. M. Raymer, L. M. Narducci, eds., Optical Instabilities (Cambridge U. Press, Cambridge, 1986); N. B. Abraham, L. A. Lugiato, L. M. Narducci, eds., J. Opt. Soc. Am. B 2, 1–264 (1985).
  16. M. Bier, T. Bountis, Phys. Lett. A 104, 239 (1984).
    [CrossRef]
  17. P. Grassberger, I. Procaccia, Phys. Rev. Lett. 50, 346 (1983).
    [CrossRef]
  18. A. Błȩdowski, W. Królikowski, IEEE J. Quantum Electron. 24, 652 (1988).
    [CrossRef]
  19. Although in the case discussed in this paper the external electric field is crucial for the dynamics of the system, sometimes a sufficiently strong nonlinear coupling in the absence of the electric field can cause unstable (but not chaotic) behavior of the photorefractive 4WM. See A. Błȩdowski, W. Królikowski, A. Kujawski, J. Opt. Soc. Am. B 6, 1544 (1989).
    [CrossRef]
  20. It has been shown recently that the absorption coefficient in photorefractive BaTiO3depends on total light intensity. See G. A. Brost, R. A. Motes, J. R. Rotge, J. Opt. Soc. Am. B 5, 1879 (1988).
    [CrossRef]
  21. K. Shaw, M. Cronin-Golomb, Opt. Commun. 65, 301 (1988).
    [CrossRef]

1989 (2)

1988 (6)

See G. C. Valley, Pochi Yeh, eds., J. Opt. Soc. Am. 5, 1681–1821 (1988), and numerous references cited therein.

A. Błȩdowski, W. Królikowski, IEEE J. Quantum Electron. 24, 652 (1988).
[CrossRef]

K. Shaw, M. Cronin-Golomb, Opt. Commun. 65, 301 (1988).
[CrossRef]

It has been shown recently that the absorption coefficient in photorefractive BaTiO3depends on total light intensity. See G. A. Brost, R. A. Motes, J. R. Rotge, J. Opt. Soc. Am. B 5, 1879 (1988).
[CrossRef]

W. Królikowski, M. R. Belić, A. Błȩdowski, Phys. Rev. A 31, 2224 (1988).
[CrossRef]

C. Pare, M. Piche, P. A. Belanger, J. Opt. Soc. Am. B 5, 679 (1988); Y. Silberberg, I. Bar-Joseph, Phys. Rev. Lett. 48, 1541 (1982).
[CrossRef]

1987 (1)

D. J. Gauthier, P. Narum, R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987).
[CrossRef] [PubMed]

1985 (3)

S. I. Stepanov, M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

P. Günter, E. Voit, M. Z. Zha, J. Albers, Opt. Commun. 55, 210 (1985); A. M. C. Smout, R. W. Eason, M. C. Gower, Opt. Commun. 59, 77 (1986).
[CrossRef]

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

1984 (3)

L. Solymar, J. M. Heaton, Opt. Commun. 51, 76 (1984); J. M. Heaton, L. Solymar, Opt. Acta 32, 397 (1985).
[CrossRef]

M. Bier, T. Bountis, Phys. Lett. A 104, 239 (1984).
[CrossRef]

G. Valley, G. Dunning, Opt. Lett. 9, 420 (1984); G. Reiner, P. Meystre, E. M. Wright, J. Opt. Soc. Am. B 4, 865, (1987); G. Reiner, M. R. Belić, P. Meystre, J. Opt. Soc. Am. B 5, 1193 (1988).
[CrossRef]

1983 (1)

P. Grassberger, I. Procaccia, Phys. Rev. Lett. 50, 346 (1983).
[CrossRef]

1982 (1)

1981 (1)

1979 (1)

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, V. Vinetskii, Ferroelectrics 22, 949 (1979); N. V. Kukhtarev, V. Markov, S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Albers, J.

P. Günter, E. Voit, M. Z. Zha, J. Albers, Opt. Commun. 55, 210 (1985); A. M. C. Smout, R. W. Eason, M. C. Gower, Opt. Commun. 59, 77 (1986).
[CrossRef]

Belanger, P. A.

Belic, M. R.

W. Królikowski, M. R. Belić, A. Błȩdowski, Phys. Rev. A 31, 2224 (1988).
[CrossRef]

Bier, M.

M. Bier, T. Bountis, Phys. Lett. A 104, 239 (1984).
[CrossRef]

Bl?dowski, A.

Bountis, T.

M. Bier, T. Bountis, Phys. Lett. A 104, 239 (1984).
[CrossRef]

Boyd, R. W.

D. J. Gauthier, P. Narum, R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987).
[CrossRef] [PubMed]

Brost, G. A.

Cronin-Golomb, M.

Dunning, G.

Feinberg, J.

Fischer, B.

B. Fischer, S. Sternklar, S. Weiss, IEEE J. Quantum Electron. 25, 550 (1989).
[CrossRef]

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

Gauthier, D. J.

D. J. Gauthier, P. Narum, R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987).
[CrossRef] [PubMed]

Grassberger, P.

P. Grassberger, I. Procaccia, Phys. Rev. Lett. 50, 346 (1983).
[CrossRef]

Günter, P.

P. Günter, E. Voit, M. Z. Zha, J. Albers, Opt. Commun. 55, 210 (1985); A. M. C. Smout, R. W. Eason, M. C. Gower, Opt. Commun. 59, 77 (1986).
[CrossRef]

Heaton, J. M.

L. Solymar, J. M. Heaton, Opt. Commun. 51, 76 (1984); J. M. Heaton, L. Solymar, Opt. Acta 32, 397 (1985).
[CrossRef]

Huignard, J. P.

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Królikowski, W.

Kujawski, A.

Kukhtarev, N. V.

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, V. Vinetskii, Ferroelectrics 22, 949 (1979); N. V. Kukhtarev, V. Markov, S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Markov, V.

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, V. Vinetskii, Ferroelectrics 22, 949 (1979); N. V. Kukhtarev, V. Markov, S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Motes, R. A.

Narum, P.

D. J. Gauthier, P. Narum, R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987).
[CrossRef] [PubMed]

Odulov, S.

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, V. Vinetskii, Ferroelectrics 22, 949 (1979); N. V. Kukhtarev, V. Markov, S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Pare, C.

Petrov, M. P.

S. I. Stepanov, M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

Piche, M.

Procaccia, I.

P. Grassberger, I. Procaccia, Phys. Rev. Lett. 50, 346 (1983).
[CrossRef]

Rajbenbach, H.

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Refregier, Ph.

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Rotge, J. R.

Shaw, K.

K. Shaw, M. Cronin-Golomb, Opt. Commun. 65, 301 (1988).
[CrossRef]

Solymar, L.

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

L. Solymar, J. M. Heaton, Opt. Commun. 51, 76 (1984); J. M. Heaton, L. Solymar, Opt. Acta 32, 397 (1985).
[CrossRef]

Soskin, M.

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, V. Vinetskii, Ferroelectrics 22, 949 (1979); N. V. Kukhtarev, V. Markov, S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Stepanov, S. I.

S. I. Stepanov, M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

Sternklar, S.

B. Fischer, S. Sternklar, S. Weiss, IEEE J. Quantum Electron. 25, 550 (1989).
[CrossRef]

Valley, G.

Vinetskii, V.

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, V. Vinetskii, Ferroelectrics 22, 949 (1979); N. V. Kukhtarev, V. Markov, S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Voit, E.

P. Günter, E. Voit, M. Z. Zha, J. Albers, Opt. Commun. 55, 210 (1985); A. M. C. Smout, R. W. Eason, M. C. Gower, Opt. Commun. 59, 77 (1986).
[CrossRef]

Weiss, S.

B. Fischer, S. Sternklar, S. Weiss, IEEE J. Quantum Electron. 25, 550 (1989).
[CrossRef]

White, J. O.

Yariv, A.

Zha, M. Z.

P. Günter, E. Voit, M. Z. Zha, J. Albers, Opt. Commun. 55, 210 (1985); A. M. C. Smout, R. W. Eason, M. C. Gower, Opt. Commun. 59, 77 (1986).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, V. Vinetskii, Ferroelectrics 22, 949 (1979); N. V. Kukhtarev, V. Markov, S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

IEEE J. Quantum Electron. (2)

B. Fischer, S. Sternklar, S. Weiss, IEEE J. Quantum Electron. 25, 550 (1989).
[CrossRef]

A. Błȩdowski, W. Królikowski, IEEE J. Quantum Electron. 24, 652 (1988).
[CrossRef]

J. Appl. Phys. (1)

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

See G. C. Valley, Pochi Yeh, eds., J. Opt. Soc. Am. 5, 1681–1821 (1988), and numerous references cited therein.

J. Opt. Soc. Am. B (3)

Opt. Commun. (4)

K. Shaw, M. Cronin-Golomb, Opt. Commun. 65, 301 (1988).
[CrossRef]

S. I. Stepanov, M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

L. Solymar, J. M. Heaton, Opt. Commun. 51, 76 (1984); J. M. Heaton, L. Solymar, Opt. Acta 32, 397 (1985).
[CrossRef]

P. Günter, E. Voit, M. Z. Zha, J. Albers, Opt. Commun. 55, 210 (1985); A. M. C. Smout, R. W. Eason, M. C. Gower, Opt. Commun. 59, 77 (1986).
[CrossRef]

Opt. Lett. (3)

Phys. Lett. A (1)

M. Bier, T. Bountis, Phys. Lett. A 104, 239 (1984).
[CrossRef]

Phys. Rev. A (1)

W. Królikowski, M. R. Belić, A. Błȩdowski, Phys. Rev. A 31, 2224 (1988).
[CrossRef]

Phys. Rev. Lett. (2)

D. J. Gauthier, P. Narum, R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987).
[CrossRef] [PubMed]

P. Grassberger, I. Procaccia, Phys. Rev. Lett. 50, 346 (1983).
[CrossRef]

Other (2)

See, for example, R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983); P. Günter, Phys. Rep. 93, 99 (1982).
[CrossRef]

R. W. Boyd, R. M. Raymer, L. M. Narducci, eds., Optical Instabilities (Cambridge U. Press, Cambridge, 1986); N. B. Abraham, L. A. Lugiato, L. M. Narducci, eds., J. Opt. Soc. Am. B 2, 1–264 (1985).

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Figures (6)

Fig. 1
Fig. 1

Four-wave mixing geometry considered in the text. The pump waves A1 and A2 are orthogonally polarized, A4 is the probe, and A3 is the phase-conjugate output. V is the high-voltage source of the electric field.

Fig. 2
Fig. 2

(A), (B) Output intensity and (C), (D) amplitude of a conjugate wave in the unstable regime (regular pulsations): E0 = 3 kV/cm, I10 = 0.3, I2l = 0.7, α = 0, γ0l = −4. Note the Period doubling when the probe intensity is increased from (A), (C) I40 = 0.015 to (B), (D) I40 = 0.03.

Fig. 3
Fig. 3

Bifurcation diagram representing PCW intensity I30 as a function of probe intensity I40. The parameters used for this diagram are γ0l = −4, α = 0, I10 = 0.3, I2l = 0.7 (intensities are normalized to total pump intensity).

Fig. 4
Fig. 4

Same as Fig. 3 except that the external electric field has been changed from E0 = 3 kV/cm to E0 = 2 kV/cm.

Fig. 5
Fig. 5

Bifurcation diagram with the linear absorption coefficient as the control parameter (γ0l = −4, I40 = 0.13, E0 = 3 kV/cm). (B) is an enlargement of the region of 0.06 ≤ α ≤ 0.12 from (A).

Fig. 6
Fig. 6

Log–log plot of the correlation integral C(r) versus distance r for a few embedding dimensions. The slopes of these curves in the range in which they are parallel should give the fractal dimension of the attractor.

Equations (5)

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

A 1 / z = Q A 4 α A 1 ,
A 4 * / z = Q A 1 * α A 4 * ,
A 2 * / z = Q A 3 * + α A 2 * ,
A 3 / z = Q A 2 + α A 3 ,
τ Q / t + E D + E q + i E 0 E M + E D + i E 0 Q = γ 0 I 0 E q + E D E D E D + i E 0 E M + E D + i E 0 × ( A 1 A 4 * + A 2 * A 3 ) ,

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