Abstract

The behavior of the phase-conjugate output from a self-pumped, BaTiO3 phase-conjugate mirror is known to exhibit temporal instabilities under certain conditions. Approaches for investigating these experimentally observed fluctuations indicate the presence of chaotic behavior. However, a dynamic system that possesses one or more positive Lyapunov exponents is by definition chaotic. For the first time, to our knowledge, we calculate directly from experimental data the largest nonnegative Lyapunov exponent of these irregular fluctuations and report examples of definite transitions from stable to chaotic behavior.

© 1994 Optical Society of America

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  1. S. Sternklar, S. Weiss, M. Segev, and B. Fischer, “Beam coupling and locking of lasers using photorefractive four-wave mixing,” Opt. Lett. 11, 528–530 (1986).
    [Crossref] [PubMed]
  2. R. W. Eason and A. M. C. Smout, “Bistability and non-commutative behavior of multiple-beam self-pulsing and self-pumping in BaTiO3,” Opt. Lett. 12, 51–53 (1987).
    [Crossref] [PubMed]
  3. J. Feinberg, “Self-pumped, continuous-wave phase conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
    [Crossref] [PubMed]
  4. G. W. Ross and R. W. Eason, “Highly efficient self-pumped phase conjugation at near-infrared wavelengths by using nominally undoped BaTiO3,” Opt. Lett. 17, 1104–1106 (1992).
    [Crossref] [PubMed]
  5. G. W. Ross, P. Hribek, R. W. Eason, M. H. Garrett, and D. Rytz, “Impurity enhanced self-pumped phase conjugation in the near infrared in blue BaTiO3,” Opt. Commun. 101, 60–64 (1993).
    [Crossref]
  6. S. W. James, K. E. Youden, P. M. Jeffrey, R. W. Eason, P. J. Chandler, L. Zhang, and P. D. Townsend, “BaTiO3waveguide self-pumped phase conjugator,” Opt. Lett. 18, 1138–1140 (1993).
    [Crossref]
  7. P. Günter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
    [Crossref]
  8. A. M. C. Smout, R. W. Eason, and M. C. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986).
    [Crossref]
  9. M. C. Gower and P. Hribek, “Mechanisms for internally self-pumped phase-conjugate emission from BaTiO3crystals,” J. Opt. Soc. Am. B 5, 1750–1757 (1988).
    [Crossref]
  10. A. Nowak, T. R. Moore, and R. A. Fisher, “Observation of internal beam production in barium titanate phase conjugators,” J. Opt. Soc. Am. B 5, 1864–1878 (1988).
    [Crossref]
  11. A. K. Majumdar and J. L. Kobesky, “Oscillations and possible transition to optical chaos in phase-conjugate waves in self-pumped BaTiO3at 514.5 nm,” Opt. Commun. 75, 339–346 (1990).
    [Crossref]
  12. T. Rauch, C. Denz, and T. Tschudi, “Analysis of irregular fluctuations in a self-pumped BaTiO3phase-conjugate mirror,” Opt. Commun. 88, 160–166 (1992).
    [Crossref]
  13. J. L. Kaplan and J. A. Yorke, “Chaotic behaviour of multidimensional difference equations,” in Functional Differential Equations and Approximations of Fixed Points, H. O. Peitgen and H. O. Walther, eds., Vol. 730 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1979).
    [Crossref]
  14. P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9, 189–208 (1983).
    [Crossref]
  15. J. D. Farmer, “Chaotic attractors of an infinite-dimensional dynamical system,” Physica D 4, 366–393 (1982).
    [Crossref]
  16. P. Grassberger, “Generalized dimensions of strange attractors,” Phys. Lett A 97, 227–230 (1983).
    [Crossref]
  17. H. G. E. Hentschel and I. Procaccia, “The finite number of generalized dimensions of fractals and strange attractors,” Physica D 8, 435–444 (1983).
    [Crossref]
  18. J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617–656 (1985).
    [Crossref]
  19. P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
    [Crossref]
  20. G. Reiner, P. Meystre, and E. M. Wright, “Transverse dynamics of a phase-conjugate resonator. II: Fast nonlinear medium,” J. Opt. Soc. Am. B 4, 865–874 (1987).
    [Crossref]
  21. G. Reiner, M. R. Belik, and P. Meystre, “Optical turbulence and phase-conjugate resonators,” J. Opt. Soc. Am. B 5, 1193–1209 (1988).
    [Crossref]
  22. W. Krölikowski, M. R. Belić, M. Cronin-Golomb, and A. Bledowski, “Chaos in photorefractive four-wave mixing with a single grating and a single interaction region,” J. Opt. Soc. Am. B 7, 1204–1209 (1990).
    [Crossref]
  23. K. D. Shaw, “Observation of chaos in off-Bragg photorefractive four-wave mixing,” Opt. Commun. 97, 148–156 (1993).
    [Crossref]
  24. D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
    [Crossref] [PubMed]
  25. G. Sugihara and R. M. May, “Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series,” Nature 344, 734–741 (1990).
    [Crossref] [PubMed]
  26. M. Dämmig, C. Boden, and F. Mitschke, “On the detection of deterministic structures in irregular signals,” Appl. Phys. B 55, 121–125 (1992).
    [Crossref]
  27. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
    [Crossref]
  28. A. Wolf and T. Bessoir, “Diagnosing chaos in the space circle,” Physica D 50, 239–258 (1991).
    [Crossref]
  29. D. A. Rand and H. B. Wilson, “Detecting chaos: a critique of the Sugihara–May approach to time-series analysis,” preprint (Warwick Preprints, University of Warwick, Coventry CV4 7AL, UK, 1991).
  30. D. A. Rand and H. B. Wilson, “Detecting chaos in a noisy time-series,” preprint (Warwick Preprints, University of Warwick, Coventry CV4 7AL, UK, 1993).
  31. N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712–716 (1980).
    [Crossref]
  32. S. W. James, G. Hussain, and R. W. Eason, “Dynamic instabilities in the mutually pumped bird wing phase conjugator,” in Technical Digest of Meeting on Photorefractive Materials, Effects, and Devices II (Sociéte Frangaise d’Optique, Aussois, France, 1990), pp. 325–328.
  33. D. Wang, Z. Zhang, X. Wu, and P. Ye, “Instabilities in a mutually pumped phase conjugator of BaTiO3,” J. Opt. Soc. Am. B 7, 2289–2293 (1990).
    [Crossref]

1993 (3)

G. W. Ross, P. Hribek, R. W. Eason, M. H. Garrett, and D. Rytz, “Impurity enhanced self-pumped phase conjugation in the near infrared in blue BaTiO3,” Opt. Commun. 101, 60–64 (1993).
[Crossref]

S. W. James, K. E. Youden, P. M. Jeffrey, R. W. Eason, P. J. Chandler, L. Zhang, and P. D. Townsend, “BaTiO3waveguide self-pumped phase conjugator,” Opt. Lett. 18, 1138–1140 (1993).
[Crossref]

K. D. Shaw, “Observation of chaos in off-Bragg photorefractive four-wave mixing,” Opt. Commun. 97, 148–156 (1993).
[Crossref]

1992 (3)

M. Dämmig, C. Boden, and F. Mitschke, “On the detection of deterministic structures in irregular signals,” Appl. Phys. B 55, 121–125 (1992).
[Crossref]

G. W. Ross and R. W. Eason, “Highly efficient self-pumped phase conjugation at near-infrared wavelengths by using nominally undoped BaTiO3,” Opt. Lett. 17, 1104–1106 (1992).
[Crossref] [PubMed]

T. Rauch, C. Denz, and T. Tschudi, “Analysis of irregular fluctuations in a self-pumped BaTiO3phase-conjugate mirror,” Opt. Commun. 88, 160–166 (1992).
[Crossref]

1991 (1)

A. Wolf and T. Bessoir, “Diagnosing chaos in the space circle,” Physica D 50, 239–258 (1991).
[Crossref]

1990 (4)

D. Wang, Z. Zhang, X. Wu, and P. Ye, “Instabilities in a mutually pumped phase conjugator of BaTiO3,” J. Opt. Soc. Am. B 7, 2289–2293 (1990).
[Crossref]

W. Krölikowski, M. R. Belić, M. Cronin-Golomb, and A. Bledowski, “Chaos in photorefractive four-wave mixing with a single grating and a single interaction region,” J. Opt. Soc. Am. B 7, 1204–1209 (1990).
[Crossref]

G. Sugihara and R. M. May, “Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series,” Nature 344, 734–741 (1990).
[Crossref] [PubMed]

A. K. Majumdar and J. L. Kobesky, “Oscillations and possible transition to optical chaos in phase-conjugate waves in self-pumped BaTiO3at 514.5 nm,” Opt. Commun. 75, 339–346 (1990).
[Crossref]

1988 (3)

1987 (3)

1986 (2)

S. Sternklar, S. Weiss, M. Segev, and B. Fischer, “Beam coupling and locking of lasers using photorefractive four-wave mixing,” Opt. Lett. 11, 528–530 (1986).
[Crossref] [PubMed]

A. M. C. Smout, R. W. Eason, and M. C. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986).
[Crossref]

1985 (3)

J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617–656 (1985).
[Crossref]

P. Günter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[Crossref]

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

1983 (4)

P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
[Crossref]

P. Grassberger, “Generalized dimensions of strange attractors,” Phys. Lett A 97, 227–230 (1983).
[Crossref]

H. G. E. Hentschel and I. Procaccia, “The finite number of generalized dimensions of fractals and strange attractors,” Physica D 8, 435–444 (1983).
[Crossref]

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9, 189–208 (1983).
[Crossref]

1982 (2)

J. D. Farmer, “Chaotic attractors of an infinite-dimensional dynamical system,” Physica D 4, 366–393 (1982).
[Crossref]

J. Feinberg, “Self-pumped, continuous-wave phase conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
[Crossref] [PubMed]

1980 (1)

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712–716 (1980).
[Crossref]

Albers, J.

P. Günter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[Crossref]

Belic, M. R.

Belik, M. R.

Bessoir, T.

A. Wolf and T. Bessoir, “Diagnosing chaos in the space circle,” Physica D 50, 239–258 (1991).
[Crossref]

Bledowski, A.

Boden, C.

M. Dämmig, C. Boden, and F. Mitschke, “On the detection of deterministic structures in irregular signals,” Appl. Phys. B 55, 121–125 (1992).
[Crossref]

Boyd, R. W.

D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
[Crossref] [PubMed]

Chandler, P. J.

Cronin-Golomb, M.

Crutchfield, J. P.

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712–716 (1980).
[Crossref]

Dämmig, M.

M. Dämmig, C. Boden, and F. Mitschke, “On the detection of deterministic structures in irregular signals,” Appl. Phys. B 55, 121–125 (1992).
[Crossref]

Denz, C.

T. Rauch, C. Denz, and T. Tschudi, “Analysis of irregular fluctuations in a self-pumped BaTiO3phase-conjugate mirror,” Opt. Commun. 88, 160–166 (1992).
[Crossref]

Eason, R. W.

G. W. Ross, P. Hribek, R. W. Eason, M. H. Garrett, and D. Rytz, “Impurity enhanced self-pumped phase conjugation in the near infrared in blue BaTiO3,” Opt. Commun. 101, 60–64 (1993).
[Crossref]

S. W. James, K. E. Youden, P. M. Jeffrey, R. W. Eason, P. J. Chandler, L. Zhang, and P. D. Townsend, “BaTiO3waveguide self-pumped phase conjugator,” Opt. Lett. 18, 1138–1140 (1993).
[Crossref]

G. W. Ross and R. W. Eason, “Highly efficient self-pumped phase conjugation at near-infrared wavelengths by using nominally undoped BaTiO3,” Opt. Lett. 17, 1104–1106 (1992).
[Crossref] [PubMed]

R. W. Eason and A. M. C. Smout, “Bistability and non-commutative behavior of multiple-beam self-pulsing and self-pumping in BaTiO3,” Opt. Lett. 12, 51–53 (1987).
[Crossref] [PubMed]

A. M. C. Smout, R. W. Eason, and M. C. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986).
[Crossref]

S. W. James, G. Hussain, and R. W. Eason, “Dynamic instabilities in the mutually pumped bird wing phase conjugator,” in Technical Digest of Meeting on Photorefractive Materials, Effects, and Devices II (Sociéte Frangaise d’Optique, Aussois, France, 1990), pp. 325–328.

Eckmann, J.-P.

J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617–656 (1985).
[Crossref]

Farmer, J. D.

J. D. Farmer, “Chaotic attractors of an infinite-dimensional dynamical system,” Physica D 4, 366–393 (1982).
[Crossref]

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712–716 (1980).
[Crossref]

Feinberg, J.

Fischer, B.

Fisher, R. A.

Garrett, M. H.

G. W. Ross, P. Hribek, R. W. Eason, M. H. Garrett, and D. Rytz, “Impurity enhanced self-pumped phase conjugation in the near infrared in blue BaTiO3,” Opt. Commun. 101, 60–64 (1993).
[Crossref]

Gauthier, D. J.

D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
[Crossref] [PubMed]

Gower, M. C.

M. C. Gower and P. Hribek, “Mechanisms for internally self-pumped phase-conjugate emission from BaTiO3crystals,” J. Opt. Soc. Am. B 5, 1750–1757 (1988).
[Crossref]

A. M. C. Smout, R. W. Eason, and M. C. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986).
[Crossref]

Grassberger, P.

P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
[Crossref]

P. Grassberger, “Generalized dimensions of strange attractors,” Phys. Lett A 97, 227–230 (1983).
[Crossref]

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9, 189–208 (1983).
[Crossref]

Günter, P.

P. Günter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[Crossref]

Hentschel, H. G. E.

H. G. E. Hentschel and I. Procaccia, “The finite number of generalized dimensions of fractals and strange attractors,” Physica D 8, 435–444 (1983).
[Crossref]

Hribek, P.

G. W. Ross, P. Hribek, R. W. Eason, M. H. Garrett, and D. Rytz, “Impurity enhanced self-pumped phase conjugation in the near infrared in blue BaTiO3,” Opt. Commun. 101, 60–64 (1993).
[Crossref]

M. C. Gower and P. Hribek, “Mechanisms for internally self-pumped phase-conjugate emission from BaTiO3crystals,” J. Opt. Soc. Am. B 5, 1750–1757 (1988).
[Crossref]

Hussain, G.

S. W. James, G. Hussain, and R. W. Eason, “Dynamic instabilities in the mutually pumped bird wing phase conjugator,” in Technical Digest of Meeting on Photorefractive Materials, Effects, and Devices II (Sociéte Frangaise d’Optique, Aussois, France, 1990), pp. 325–328.

James, S. W.

S. W. James, K. E. Youden, P. M. Jeffrey, R. W. Eason, P. J. Chandler, L. Zhang, and P. D. Townsend, “BaTiO3waveguide self-pumped phase conjugator,” Opt. Lett. 18, 1138–1140 (1993).
[Crossref]

S. W. James, G. Hussain, and R. W. Eason, “Dynamic instabilities in the mutually pumped bird wing phase conjugator,” in Technical Digest of Meeting on Photorefractive Materials, Effects, and Devices II (Sociéte Frangaise d’Optique, Aussois, France, 1990), pp. 325–328.

Jeffrey, P. M.

Kaplan, J. L.

J. L. Kaplan and J. A. Yorke, “Chaotic behaviour of multidimensional difference equations,” in Functional Differential Equations and Approximations of Fixed Points, H. O. Peitgen and H. O. Walther, eds., Vol. 730 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1979).
[Crossref]

Kobesky, J. L.

A. K. Majumdar and J. L. Kobesky, “Oscillations and possible transition to optical chaos in phase-conjugate waves in self-pumped BaTiO3at 514.5 nm,” Opt. Commun. 75, 339–346 (1990).
[Crossref]

Krölikowski, W.

Majumdar, A. K.

A. K. Majumdar and J. L. Kobesky, “Oscillations and possible transition to optical chaos in phase-conjugate waves in self-pumped BaTiO3at 514.5 nm,” Opt. Commun. 75, 339–346 (1990).
[Crossref]

May, R. M.

G. Sugihara and R. M. May, “Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series,” Nature 344, 734–741 (1990).
[Crossref] [PubMed]

Meystre, P.

Mitschke, F.

M. Dämmig, C. Boden, and F. Mitschke, “On the detection of deterministic structures in irregular signals,” Appl. Phys. B 55, 121–125 (1992).
[Crossref]

Moore, T. R.

Narum, P.

D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
[Crossref] [PubMed]

Nowak, A.

Packard, N. H.

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712–716 (1980).
[Crossref]

Procaccia, I.

P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
[Crossref]

H. G. E. Hentschel and I. Procaccia, “The finite number of generalized dimensions of fractals and strange attractors,” Physica D 8, 435–444 (1983).
[Crossref]

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9, 189–208 (1983).
[Crossref]

Rand, D. A.

D. A. Rand and H. B. Wilson, “Detecting chaos in a noisy time-series,” preprint (Warwick Preprints, University of Warwick, Coventry CV4 7AL, UK, 1993).

D. A. Rand and H. B. Wilson, “Detecting chaos: a critique of the Sugihara–May approach to time-series analysis,” preprint (Warwick Preprints, University of Warwick, Coventry CV4 7AL, UK, 1991).

Rauch, T.

T. Rauch, C. Denz, and T. Tschudi, “Analysis of irregular fluctuations in a self-pumped BaTiO3phase-conjugate mirror,” Opt. Commun. 88, 160–166 (1992).
[Crossref]

Reiner, G.

Ross, G. W.

G. W. Ross, P. Hribek, R. W. Eason, M. H. Garrett, and D. Rytz, “Impurity enhanced self-pumped phase conjugation in the near infrared in blue BaTiO3,” Opt. Commun. 101, 60–64 (1993).
[Crossref]

G. W. Ross and R. W. Eason, “Highly efficient self-pumped phase conjugation at near-infrared wavelengths by using nominally undoped BaTiO3,” Opt. Lett. 17, 1104–1106 (1992).
[Crossref] [PubMed]

Ruelle, D.

J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617–656 (1985).
[Crossref]

Rytz, D.

G. W. Ross, P. Hribek, R. W. Eason, M. H. Garrett, and D. Rytz, “Impurity enhanced self-pumped phase conjugation in the near infrared in blue BaTiO3,” Opt. Commun. 101, 60–64 (1993).
[Crossref]

Segev, M.

Shaw, K. D.

K. D. Shaw, “Observation of chaos in off-Bragg photorefractive four-wave mixing,” Opt. Commun. 97, 148–156 (1993).
[Crossref]

Shaw, R. S.

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, “Geometry from a time series,” Phys. Rev. Lett. 45, 712–716 (1980).
[Crossref]

Smout, A. M. C.

R. W. Eason and A. M. C. Smout, “Bistability and non-commutative behavior of multiple-beam self-pulsing and self-pumping in BaTiO3,” Opt. Lett. 12, 51–53 (1987).
[Crossref] [PubMed]

A. M. C. Smout, R. W. Eason, and M. C. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986).
[Crossref]

Sternklar, S.

Sugihara, G.

G. Sugihara and R. M. May, “Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series,” Nature 344, 734–741 (1990).
[Crossref] [PubMed]

Swift, J. B.

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

Swinney, H. L.

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

Townsend, P. D.

Tschudi, T.

T. Rauch, C. Denz, and T. Tschudi, “Analysis of irregular fluctuations in a self-pumped BaTiO3phase-conjugate mirror,” Opt. Commun. 88, 160–166 (1992).
[Crossref]

Vastano, J. A.

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

Voit, E.

P. Günter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[Crossref]

Wang, D.

Weiss, S.

Wilson, H. B.

D. A. Rand and H. B. Wilson, “Detecting chaos: a critique of the Sugihara–May approach to time-series analysis,” preprint (Warwick Preprints, University of Warwick, Coventry CV4 7AL, UK, 1991).

D. A. Rand and H. B. Wilson, “Detecting chaos in a noisy time-series,” preprint (Warwick Preprints, University of Warwick, Coventry CV4 7AL, UK, 1993).

Wolf, A.

A. Wolf and T. Bessoir, “Diagnosing chaos in the space circle,” Physica D 50, 239–258 (1991).
[Crossref]

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D 16, 285–317 (1985).
[Crossref]

Wright, E. M.

Wu, X.

Ye, P.

Yorke, J. A.

J. L. Kaplan and J. A. Yorke, “Chaotic behaviour of multidimensional difference equations,” in Functional Differential Equations and Approximations of Fixed Points, H. O. Peitgen and H. O. Walther, eds., Vol. 730 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1979).
[Crossref]

Youden, K. E.

Zha, M. Z.

P. Günter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[Crossref]

Zhang, L.

Zhang, Z.

Appl. Phys. B (1)

M. Dämmig, C. Boden, and F. Mitschke, “On the detection of deterministic structures in irregular signals,” Appl. Phys. B 55, 121–125 (1992).
[Crossref]

J. Opt. Soc. Am. B (6)

Nature (1)

G. Sugihara and R. M. May, “Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series,” Nature 344, 734–741 (1990).
[Crossref] [PubMed]

Opt. Commun. (6)

K. D. Shaw, “Observation of chaos in off-Bragg photorefractive four-wave mixing,” Opt. Commun. 97, 148–156 (1993).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Experimental setup used to investigate SPPC in BaTiO3. BS, beam splitter; SH, shutter; VND, variable neutral-density filter; SF, spatial filter; PBS, polarizing beam splitter; IR, mechanical iris; POL, polarizer; PD1, PD2, photodiodes; FR, Faraday rotator; HWP1, HWP2, 514.5-nm half-wave plates.

Fig. 2
Fig. 2

Experimental results showing the variation of the phase-conjugate intensity with transverse position (x) of the input beam along the crystal face (θ = 2°). (a) x = 3.50 mm, (b) x = 3.90 mm, and (c) x = 4.10 mm.

Fig. 3
Fig. 3

Dependence of the largest nonnegative Lyapunov exponent (λ1) as a function of the evolution time Δ for θ = 2°, Iin = 2.1 W cm−2, and x = 3.50 mm.

Fig. 4
Fig. 4

Experimental results showing the calculated largest nonnegative Lyapunov exponent (λ1) as a function of input-beam (Iin = 2.1 W cm−2) position along the x axis of the crystal (5 mm): (a) θ = 2°, (b) θ = 15°, and (c) θ = 50°.

Equations (3)

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D 2 D 1 D 0 ,
λ 1 = 1 t M - t 0 k = 1 M log 2 L ( t k ) L ( t k - 1 ) ,
Δ = t k + 1 - t k ,

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