Abstract

The dynamics of an externally pumped phase-conjugate resonator that uses four-wave mixing in BaTiO3 is investigated experimentally. The emergence of spatiotemporal instabilities as the degree of transverse confinement is varied by changing the Fresnel number is described in detail. Local intensity time series show that relaxing the transverse confinement leads the system from a stationary state to periodic, then to quasi-periodic, motions and finally to chaotic behavior. In some regions of parameter space two frequencies are identified in the power spectra of the time series, indicating a route to chaos following the Ruelle–Takens–Newhouse scheme [ Commun. Math. Phys. 64, 35– 40 ( 1978)]. Wave-front topological defects are identified by interferometry. As the system’s confinement is varied, the phase-defect density and the spatial correlation index are found to follow similar trends, indicating that the observed spatiotemporal dynamics may indeed be an example of defect-mediated turbulence.

© 1992 Optical Society of America

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1992 (2)

K. P. Lo and G. Indebetouw, “Iterative image processing using a cavity with a phase-conjugate mirror,” Appl. Opt. 10, 1745–1753 (1992).
[CrossRef]

S. R. Liu and G. Indebetouw, “Dynamics of a phase conjugate resonator: transient buildup and decay rates,” Appl. Phys. B 54, 247–258 (1992).
[CrossRef]

1991 (4)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” and “II. Variational principle for pattern selection, spatial multistability and laser hydrodynamics,” Phys. Rev. A 43, 5090–5120 (1991).
[CrossRef] [PubMed]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, C. O. McDuff, R. Weiss, and C. Tamm, “Interferometeric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

M. R. Belic, D. Timotijevic, and W. Krolikowski, “Multigrating phase conjugation: chaotic results,” J. Opt. Soc. Am. B 8, 1723–1731 (1991).
[CrossRef]

1990 (4)

W. Krolikowski, M. R. Belic, 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]

S. S. Lafleur and R. C. Montgomery, “Real-time dynamic holographic image storage device,” Appl. Opt. 29, 3976, (1990);U.S. patent4,913,534 (April3, 1990).

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]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

1989 (3)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–407 (1989).
[CrossRef]

W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2laser,” Phys. Rev. A 39, 919–922 (1989).
[CrossRef] [PubMed]

P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
[CrossRef] [PubMed]

1988 (3)

1987 (4)

G. Reiner, P. Meystre, and E. M. Wright, “Transverse dynamics of a phase-conjugate resonator. I: sluggish nonlinear medium,” J. Opt. Soc. Am. B 4, 675–686 (1987).
[CrossRef]

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

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

A. Brandstater and H. L. Swinney, “Strange attractors in weakly turbulent Couette–Taylor flow,” Phys. Rev. A 35, 2207–2220 (1987).
[CrossRef] [PubMed]

1986 (3)

1985 (1)

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

1984 (1)

1983 (3)

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

P. Grassberger and I. Procaccia, “Estimation of the Kolomogrov entropy from a chaotic signal,” Phys. Rev. A 28, 2591–2593 (1983).
[CrossRef]

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V V. Shkunov, and B. Y. Zeldovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. B 5, 525–528 (1983).

1978 (1)

S. Newhouse, D. Ruelle, and F. Takens, “Occurrence of strange axiom A attractors near quasiperiodic flows on Tm, m ≥ 3,” Commun. Math. Phys. 64, 35–40 (1978).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wavetrains,” Proc. R. Soc. London 336, 156–190 (1974).

1971 (1)

D. Ruelle and F. Takens, “On the nature of turbulence,” Commun. Math. Phys. 20, 167–192 (1971).
[CrossRef]

Abraham, N. B.

N. B. Abraham, A. M. Albano, and N. B. Tufillaro, “Complexity and chaos,” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, eds. (Plenum, New York, 1990), pp. 1–27.

Albano, A. M.

N. B. Abraham, A. M. Albano, and N. B. Tufillaro, “Complexity and chaos,” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, eds. (Plenum, New York, 1990), pp. 1–27.

Albers, J.

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

Anderson, D. Z.

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

D. Z. Anderson, “Coherent optical eigenstate memory,” Opt. Lett. 11, 56–58 (1986).
[CrossRef] [PubMed]

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Baranova, N. B.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V V. Shkunov, and B. Y. Zeldovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. B 5, 525–528 (1983).

Battipede, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” and “II. Variational principle for pattern selection, spatial multistability and laser hydrodynamics,” Phys. Rev. A 43, 5090–5120 (1991).
[CrossRef] [PubMed]

Belic, M. R.

Berge, P.

P. Berge, Y. Pomeau, and C. Vidal, Order within Chaos (Wiley, New York, 1984).

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wavetrains,” Proc. R. Soc. London 336, 156–190 (1974).

Bledowski, A.

Bougrenet, J. L.

Boyd, R. W.

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

Brambilla, M.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” and “II. Variational principle for pattern selection, spatial multistability and laser hydrodynamics,” Phys. Rev. A 43, 5090–5120 (1991).
[CrossRef] [PubMed]

Brandstater, A.

A. Brandstater and H. L. Swinney, “Strange attractors in weakly turbulent Couette–Taylor flow,” Phys. Rev. A 35, 2207–2220 (1987).
[CrossRef] [PubMed]

Coullet, P.

P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
[CrossRef] [PubMed]

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–407 (1989).
[CrossRef]

P. Coullet and J. Lega, “Defect-mediated turbulence in wave patterns,” Europhys. Lett. 7, 511–516 (1988).
[CrossRef]

P. Coullet, “Defect-induced spatio-temporal chaos,” in Ref. 22, pp. 367–373.

Cronin-Golomb, M.

Dunning, G.

Dunning, G. J.

Erie, M. C.

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

Fisher, R. A.

Gauthier, D. J.

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

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Gil, L.

P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
[CrossRef] [PubMed]

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–407 (1989).
[CrossRef]

Grassberger, P.

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

P. Grassberger and I. Procaccia, “Estimation of the Kolomogrov entropy from a chaotic signal,” Phys. Rev. A 28, 2591–2593 (1983).
[CrossRef]

Günter, P.

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

Heckenberg, N. R.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, C. O. McDuff, R. Weiss, and C. Tamm, “Interferometeric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Huignard, J. P.

Indebetouw, G.

K. P. Lo and G. Indebetouw, “Iterative image processing using a cavity with a phase-conjugate mirror,” Appl. Opt. 10, 1745–1753 (1992).
[CrossRef]

S. R. Liu and G. Indebetouw, “Dynamics of a phase conjugate resonator: transient buildup and decay rates,” Appl. Phys. B 54, 247–258 (1992).
[CrossRef]

G. Indebetouw and S. R. Liu, “Defect-mediated spatial complexity and chaos in a phase-conjugate resonator,” submitted to Opt. Commun.

Klische, W.

W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2laser,” Phys. Rev. A 39, 919–922 (1989).
[CrossRef] [PubMed]

Krolikowski, W.

Lafleur, S. S.

S. S. Lafleur and R. C. Montgomery, “Real-time dynamic holographic image storage device,” Appl. Opt. 29, 3976, (1990);U.S. patent4,913,534 (April3, 1990).

Lega, J.

P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
[CrossRef] [PubMed]

P. Coullet and J. Lega, “Defect-mediated turbulence in wave patterns,” Europhys. Lett. 7, 511–516 (1988).
[CrossRef]

Liu, S. R.

S. R. Liu and G. Indebetouw, “Dynamics of a phase conjugate resonator: transient buildup and decay rates,” Appl. Phys. B 54, 247–258 (1992).
[CrossRef]

G. Indebetouw and S. R. Liu, “Defect-mediated spatial complexity and chaos in a phase-conjugate resonator,” submitted to Opt. Commun.

Lo, K. P.

K. P. Lo and G. Indebetouw, “Iterative image processing using a cavity with a phase-conjugate mirror,” Appl. Opt. 10, 1745–1753 (1992).
[CrossRef]

Lugiato, L. A.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” and “II. Variational principle for pattern selection, spatial multistability and laser hydrodynamics,” Phys. Rev. A 43, 5090–5120 (1991).
[CrossRef] [PubMed]

Mamaev, A. V.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V V. Shkunov, and B. Y. Zeldovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. B 5, 525–528 (1983).

Marom, E.

McDuff, C. O.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, C. O. McDuff, R. Weiss, and C. Tamm, “Interferometeric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Meystre, P.

Montgomery, R. C.

S. S. Lafleur and R. C. Montgomery, “Real-time dynamic holographic image storage device,” Appl. Opt. 29, 3976, (1990);U.S. patent4,913,534 (April3, 1990).

Moore, T. R.

Narducci, L. M.

G. Oppo, M. A. Pernigo, and L. M. Narducci, “Characterization of spatiotemporal structures in lasers: a progress report,” in Ref. 22, pp. 395–404.

Narum, D.

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

Newhouse, S.

S. Newhouse, D. Ruelle, and F. Takens, “Occurrence of strange axiom A attractors near quasiperiodic flows on Tm, m ≥ 3,” Commun. Math. Phys. 64, 35–40 (1978).
[CrossRef]

Nowak, A. V.

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wavetrains,” Proc. R. Soc. London 336, 156–190 (1974).

Oppo, G.

G. Oppo, M. A. Pernigo, and L. M. Narducci, “Characterization of spatiotemporal structures in lasers: a progress report,” in Ref. 22, pp. 395–404.

Owechko, Y.

Pellat-Finet, P.

Penna, V.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” and “II. Variational principle for pattern selection, spatial multistability and laser hydrodynamics,” Phys. Rev. A 43, 5090–5120 (1991).
[CrossRef] [PubMed]

Pernigo, M. A.

G. Oppo, M. A. Pernigo, and L. M. Narducci, “Characterization of spatiotemporal structures in lasers: a progress report,” in Ref. 22, pp. 395–404.

Pilipetsky, N. F.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V V. Shkunov, and B. Y. Zeldovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. B 5, 525–528 (1983).

Pomeau, Y.

P. Berge, Y. Pomeau, and C. Vidal, Order within Chaos (Wiley, New York, 1984).

Prati, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” and “II. Variational principle for pattern selection, spatial multistability and laser hydrodynamics,” Phys. Rev. A 43, 5090–5120 (1991).
[CrossRef] [PubMed]

Procaccia, I.

P. Grassberger and I. Procaccia, “Estimation of the Kolomogrov entropy from a chaotic signal,” Phys. Rev. A 28, 2591–2593 (1983).
[CrossRef]

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

Ramazza, P. L.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Reiner, G.

Residori, S.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–407 (1989).
[CrossRef]

Rubinsztein-Dunlop, H.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, C. O. McDuff, R. Weiss, and C. Tamm, “Interferometeric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Ruelle, D.

S. Newhouse, D. Ruelle, and F. Takens, “Occurrence of strange axiom A attractors near quasiperiodic flows on Tm, m ≥ 3,” Commun. Math. Phys. 64, 35–40 (1978).
[CrossRef]

D. Ruelle and F. Takens, “On the nature of turbulence,” Commun. Math. Phys. 20, 167–192 (1971).
[CrossRef]

Schuster, H. G.

H. G. Schuster, Deterministic Chaos (VCH Verlagsgessel-schaft, Weinheim, 1988).

Shkunov, V V.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V V. Shkunov, and B. Y. Zeldovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. B 5, 525–528 (1983).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Smith, C. P.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, C. O. McDuff, R. Weiss, and C. Tamm, “Interferometeric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Soffer, B. H.

Swinney, H. L.

A. Brandstater and H. L. Swinney, “Strange attractors in weakly turbulent Couette–Taylor flow,” Phys. Rev. A 35, 2207–2220 (1987).
[CrossRef] [PubMed]

Takens, F.

S. Newhouse, D. Ruelle, and F. Takens, “Occurrence of strange axiom A attractors near quasiperiodic flows on Tm, m ≥ 3,” Commun. Math. Phys. 64, 35–40 (1978).
[CrossRef]

D. Ruelle and F. Takens, “On the nature of turbulence,” Commun. Math. Phys. 20, 167–192 (1971).
[CrossRef]

F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, vol. 898 of Lecture Notes in Mathematics, D. A. Rand and L.-S. Young, eds. (Springer-Verlag, Berlin, 1981), pp. 366–381.

Tamm, C.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, C. O. McDuff, R. Weiss, and C. Tamm, “Interferometeric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” and “II. Variational principle for pattern selection, spatial multistability and laser hydrodynamics,” Phys. Rev. A 43, 5090–5120 (1991).
[CrossRef] [PubMed]

Timotijevic, D.

Tufillaro, N. B.

N. B. Abraham, A. M. Albano, and N. B. Tufillaro, “Complexity and chaos,” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, eds. (Plenum, New York, 1990), pp. 1–27.

Valley, G.

Vidal, C.

P. Berge, Y. Pomeau, and C. Vidal, Order within Chaos (Wiley, New York, 1984).

Volt, E.

P. Günter, E. Volt, 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, C. O.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” and “II. Variational principle for pattern selection, spatial multistability and laser hydrodynamics,” Phys. Rev. A 43, 5090–5120 (1991).
[CrossRef] [PubMed]

W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2laser,” Phys. Rev. A 39, 919–922 (1989).
[CrossRef] [PubMed]

Weiss, R.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, C. O. McDuff, R. Weiss, and C. Tamm, “Interferometeric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Wellegehausen, B.

W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2laser,” Phys. Rev. A 39, 919–922 (1989).
[CrossRef] [PubMed]

White, A. G.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, C. O. McDuff, R. Weiss, and C. Tamm, “Interferometeric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Wright, E. M.

Wright, F. J.

F. J. Wright, “Wavefront dislocations and their analysis using catastrophe theory,” in Structural Stability in Physics, W. Güttinger and H. Eikemeier, eds. (Springer-Verlag, Heidelberg, 1979), pp. 141–156.
[CrossRef]

Wu, X.

Ye, P.

Zeldovich, B. Y.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V V. Shkunov, and B. Y. Zeldovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. B 5, 525–528 (1983).

Zha, M. Z.

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

Zhang, Z.

Appl. Opt. (2)

K. P. Lo and G. Indebetouw, “Iterative image processing using a cavity with a phase-conjugate mirror,” Appl. Opt. 10, 1745–1753 (1992).
[CrossRef]

S. S. Lafleur and R. C. Montgomery, “Real-time dynamic holographic image storage device,” Appl. Opt. 29, 3976, (1990);U.S. patent4,913,534 (April3, 1990).

Appl. Phys. B (1)

S. R. Liu and G. Indebetouw, “Dynamics of a phase conjugate resonator: transient buildup and decay rates,” Appl. Phys. B 54, 247–258 (1992).
[CrossRef]

Commun. Math. Phys. (2)

S. Newhouse, D. Ruelle, and F. Takens, “Occurrence of strange axiom A attractors near quasiperiodic flows on Tm, m ≥ 3,” Commun. Math. Phys. 64, 35–40 (1978).
[CrossRef]

D. Ruelle and F. Takens, “On the nature of turbulence,” Commun. Math. Phys. 20, 167–192 (1971).
[CrossRef]

Europhys. Lett. (1)

P. Coullet and J. Lega, “Defect-mediated turbulence in wave patterns,” Europhys. Lett. 7, 511–516 (1988).
[CrossRef]

J. Mod. Opt. (1)

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, C. O. McDuff, R. Weiss, and C. Tamm, “Interferometeric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

J. Opt. Soc. Am. B (8)

Opt. Commun. (2)

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

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–407 (1989).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (3)

Phys. Rev. A (4)

W. Klische, C. O. Weiss, and B. Wellegehausen, “Spatiotemporal chaos from a continuous Na2laser,” Phys. Rev. A 39, 919–922 (1989).
[CrossRef] [PubMed]

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” and “II. Variational principle for pattern selection, spatial multistability and laser hydrodynamics,” Phys. Rev. A 43, 5090–5120 (1991).
[CrossRef] [PubMed]

A. Brandstater and H. L. Swinney, “Strange attractors in weakly turbulent Couette–Taylor flow,” Phys. Rev. A 35, 2207–2220 (1987).
[CrossRef] [PubMed]

P. Grassberger and I. Procaccia, “Estimation of the Kolomogrov entropy from a chaotic signal,” Phys. Rev. A 28, 2591–2593 (1983).
[CrossRef]

Phys. Rev. Lett. (5)

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

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

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

P. Coullet, L. Gil, and J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1989).
[CrossRef] [PubMed]

Proc. R. Soc. London (1)

J. F. Nye and M. V. Berry, “Dislocations in wavetrains,” Proc. R. Soc. London 336, 156–190 (1974).

Other (12)

F. J. Wright, “Wavefront dislocations and their analysis using catastrophe theory,” in Structural Stability in Physics, W. Güttinger and H. Eikemeier, eds. (Springer-Verlag, Heidelberg, 1979), pp. 141–156.
[CrossRef]

P. Berge, Y. Pomeau, and C. Vidal, Order within Chaos (Wiley, New York, 1984).

N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, eds., Measures of Complexity and Chaos (Plenum, New York, 1989).

F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, vol. 898 of Lecture Notes in Mathematics, D. A. Rand and L.-S. Young, eds. (Springer-Verlag, Berlin, 1981), pp. 366–381.

G. Indebetouw and S. R. Liu, “Defect-mediated spatial complexity and chaos in a phase-conjugate resonator,” submitted to Opt. Commun.

D. K. Bandy, A. N. Oraevsky, and J. R. Tredice, eds., feature on photorefractive materials, effects, and devices, J. Opt. Soc. Am. B5, 875–1215 (1988);D. M. Pepper, ed.,special issue on nonlinear optical phase conjugation, IEEE J. Quantum Electron.QE-25, 312–647 (1988).

H. G. Schuster, Deterministic Chaos (VCH Verlagsgessel-schaft, Weinheim, 1988).

N. B. Abraham, A. M. Albano, and N. B. Tufillaro, “Complexity and chaos,” in Measures of Complexity and Chaos, N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, eds. (Plenum, New York, 1990), pp. 1–27.

N. B. Abraham and W. J. Firth, eds., feature on transverse effects in nonlinear-optical systems, J. Opt. Soc. Am. B7, 945–1157, 1261–1373 (1990).

P. Coullet, “Defect-induced spatio-temporal chaos,” in Ref. 22, pp. 367–373.

G. Oppo, M. A. Pernigo, and L. M. Narducci, “Characterization of spatiotemporal structures in lasers: a progress report,” in Ref. 22, pp. 395–404.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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Figures (24)

Fig. 1
Fig. 1

Schematic diagram of the phase-conjugate resonator setup: λ/2’s and P’s are half-wave plates and polarizer assemblies used to vary the intensity of the beams individually. All beam polarizations are extraordinary. The two apertures A1 and A2 are 1 focal length from the lens (f = 16 cm). The Ar+ laser operates in a single mode at 514 nm. The phase-conjugate mirror (PCM) is a single crystal of BaTiO3. D1 and D2 are photo-multiplier tubes measuring the local intensity at two different locations; D3 monitors the laser output, and the charge-coupled-device camera (CCD) is used to capture the dynamics of the intensity distribution.

Fig. 2
Fig. 2

Cavity decay time versus pump ratio, showing the range of pump ratios for which the cavity is above threshold.

Fig. 3
Fig. 3

Output intensity versus time at a point within aperture A1 imaged through the crystal for different values of the Fresnel number F. The total power of the two pumps is 60 mW, and the pump ratio is 1. The time series show (a) a stationary output, (b) stable oscillations, (c) the emergence of subharmonics, (d) the development of chaotic oscillations, (e) unstable oscillatory motions with intermittent bursts, and (f) another chaotic motion.

Fig. 4
Fig. 4

Normalized power spectra calculated from the data of Fig. 3, showing (a) a fundamental frequency and its higher harmonics, (b) a fundamental frequency and its subharmonics multiples of 1/3, (c) a broadband spectrum with the emergence of two frequency peaks, (d) two unrelated frequencies and their linear combinations.

Fig. 5
Fig. 5

Phase-space portraits of time series containing N = 8000 data points. I(i) is plotted versus I(i + n), with a time delay given by Δt = nτ. τ is the sampling interval. One can identify (a) a limit cycle, (b) a stable period-three motion, (c) an irregular filling of phase space, and (d) an unstable limit cycle smeared by a cloud of irregularly distributed points.

Fig. 6
Fig. 6

Time series of the local intensity for F = 5.8, showing a quasi-periodic motion interrupted by intermittent bursts.

Fig. 7
Fig. 7

Log–log plots of the correlation integral C2() versus distance , with increasing values of the embedding dimension d, for the chaotic output of Fig. 3(d).

Fig. 8
Fig. 8

Local slope ν() of the correlation integral of Fig. 7 versus log . The slopes are averaged over seven local points. The plateau indicates the range of linear scaling for each d. The slope in the region of the plateau converges toward the correlation dimension D2.

Fig. 9
Fig. 9

Plot of the correlation dimension for each embedding dimension d of Figs. 7 and 8. The correlation dimension appears to converge to a value D2 ≈ 5.2.

Fig. 10
Fig. 10

Value of the entropy K2,d calculated from the data of Fig. 7 versus the embedding dimension d averaged over the linear scaling region in . The asymptotic value K2 ≈ 0.16 s−1 is a lower bound of the Kolmogorov entropy.

Fig. 11
Fig. 11

a, Sequence of snapshots of the irradiance distribution in aperture A1 imaged through the crystal with F = 2.2; the time interval between snapshots is 1 s. b, Corresponding sequence of interferograms, revealing a pair of defects moving along a horizontal diameter.

Fig. 12
Fig. 12

Instantaneous interferogram with F = 2.2, showing a pair of defects of opposite signs.

Fig. 13
Fig. 13

Sequence of snapshots with F = 3.6 (time interval 1 s) revealing a more complicated but still periodic motion.

Fig. 14
Fig. 14

Instantaneous interferogram with F = 3.6, showing three pairs of defects.

Fig. 15
Fig. 15

Sketches of some typical motions of the defects observed with 2 ≤ F ≤ 5. See text for details.

Fig. 16
Fig. 16

Sequence of snapshots with F = 4.1 (time interval 1 s) revealing a nonperiodic motion.

Fig. 17
Fig. 17

Instantaneous interferogram with F = 4.1, showing four pairs of defects.

Fig. 18
Fig. 18

Sequence of snapshots with F = 5.8 (time interval 1 s), revealing a more complicated pattern but executing a simpler motion.

Fig. 19
Fig. 19

Instantaneous interferogram with F = 5.8, showing only one pair of defects.

Fig. 20
Fig. 20

Instantaneous interferogram with F = 8.6, showing at least eight pairs of defects executing a chaotic dance.

Fig. 21
Fig. 21

Maximum number N of defects observed in the transverse pattern at any given time versus Fresnel number F. The expected scaling law NF2 is interrupted in the region 4 < F < 6.

Fig. 22
Fig. 22

Spatial correlation index K [Eq. (6)] versus Fresnel number F. A comparison with Fig. 21 reveals the relationship between the spatial correlation and the number of defects.

Fig. 23
Fig. 23

Sequence of patterns at equal intervals of time obtained by superimposing three TEM modes 01, 10, and 20, with respective weight coefficients 1, 1, and 1/3 and respective detunings 0.5, 1, and ( 5 + 1 ) / 2. a and b indicate the locations of the two conjugate defects, and the arrow shows the defects’ directions of motion.

Fig. 24
Fig. 24

Motion of two conjugate defects projected on a plane. The defects’ locations are solutions of Re E = Im E = 0, where E is the field amplitude resulting from the superposition of the three modes of Fig. 22. (a) Short-time motion (see text for details); (b) long-time defect trajectories.

Equations (8)

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F = ( D 1 D 2 ) / ( λ f ) ,
C 2 ( ) = 1 N 2 m , n = 1 N θ ( Δ m , n ) ,
Δ m , n = i = 0 d 1 | X n + i X m + i | 2
C 2 ( ; d ) ν exp ( d τ K 2 ) .
C ( r 1 , r 2 , Γ ) = n = 1 N [ I ( r 1 , n ) I ¯ ( r 1 ) ] [ I ( r 2 , n + Γ ) I ¯ ( r 2 ) ] S ( r 1 ) S ( r 2 ) ,
S ( r j ) = { n = 1 N [ I ( r j , n ) I ¯ ( r j ) ] 2 } 1 / 2 ,
K ( | r 1 r 2 | ) = max C ( r 1 , r 2 , Γ ) , Γ .
E m n ( x , y ; t ) = A m n U m ( 2 x w ) U n ( 2 y w ) × exp { i [ ( ω + δ m n ) t + ϕ m n ] x 2 + y 2 w 2 } ,

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