Abstract

It is shown that a new type of instability of a light field in a dissipative medium (spatiotemporal instability, which causes the generation of new types of nonlinear light wave) can be observed by controlling the spatial scale and the topology of the transverse interactions of light fields in a medium with cubic nonlinearity. The excitation conditions for optical reverberators, rotating helical waves, and various dissipative structures are experimentally determined. Transformations and interactions of the structures lead to optical turbulence in both space and time. Physical interpretation of these phenomena is based on the parabolic equation for the nonlinear phase shift. It is found that this theoretical model allows one not only to obtain the excitation conditions but to investigate thoroughly such phenomena as hysteresis and nonlinear interactions of structures.

© 1992 Optical Society of America

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  1. S. A. Akhmanov, V. A. Visloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, Woodbury, NY., 1991).
  2. G. S. Agarval, Nonlinear Fiber Optics (Academic, Orlando, Fla., 1989).
  3. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).
  4. S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Nauka, Moscow, 1964).
  5. N. B. Abraham and W. J. Firth, eds., feature on transverse effects in nonlinear-optical systems, J. Opt. Soc. Am. B 7, 947–1157 (1990).
  6. M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhauzen, “Rotatory instability of the spatial structure of light fields in nonlinear media with 2-D feedback,” in Proceedings of the Third Binational USA–USSR Symposium of Laser Optics of Condensed Matter (Leningrad, 1987) (Plenum, New York, 1988), pp. 507–517.
    [CrossRef]
  7. 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]
  8. S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, “Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of optical turbulence,” JETP Lett. 47, 707–711 (1988).
  9. D. McLaughlin, J. Moloney, and A. Newell, “New class of instabilities in passive optical cavities,” Phys. Rev. Lett. 54, 681–684 (1985).
    [CrossRef] [PubMed]
  10. M. A. Vorontsov, Yu. D. Dumarevski, V. I. Shmalhauzen, and D. V. Pruidze, “Autowave processes in systems with optical feedback,” Izv. Akad. Nauk SSSR Ser. Fiz. 52, 374–376 (1988).
  11. K. Ikeda, A. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
    [CrossRef]
  12. W. J. Firth and I. Galbraith, “Diffusion transverse coupling of bistable elements—switching waves and cross-talk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
    [CrossRef]
  13. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
    [CrossRef] [PubMed]
  14. A. A. Vasiliev, D. Kasasent, I. N. Kompanets, and A. V. Parfenov, Spatial Light Modulators (Radio and Svyaz, Moscow, 1987).
  15. N. N. Rosanov and G. V. Khodova, “Formation of switching waves in bistable systems,” Kvantovaya Electron. (Moscow) 13, 368–377 (1986).
  16. W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1012 (1985).
    [CrossRef]
  17. M. A. Vorontsov, N. I. Zheleznykh, and V. Yu. Ivanov, “Transverse interactions in 2-D feedback nonlinear systems,” Opt. Quantum Electron. 22, 505–515 (1990).
    [CrossRef]
  18. N. I. Zheleznykh, M. A. Vorontsov, and A. V. Larichev, “2-D dynamics neural network optical system with simplest types of large-scale interactions,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1402, 154–164 (1990).
  19. A. A. Vasiliev, I. N. Kompanets, and A. V. Parfenov, “Advantages in development and application of optically controlled SLM,” Kvantovaya Elektron. (Moscow) 10, 1074–1088 (1983).
  20. J. P. Crutchfield, “Spatio-temporal complexity in nonlinear image processing,” IEEE Trans. Circuits Syst. 35, 770–780 (1988).
    [CrossRef]
  21. M. A. Vorontsov and N. I. Zheleznykh, “Transverse bistability and multistability in nonlinear optical systems with 2-D feedback,” Mat. Modelirov. 2, 31–39 (1990).
  22. V. Yu. Ivanov, A. V. Larichev, and M. A. Vorontsov, “1-D rotatory waves in the optical systems with nonlinear large-scale field interactions,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1402, 145–153 (1990).
    [CrossRef]
  23. K. Otsuka and K. Ikeda, “Cooperative dynamics and functions in a collective nonlinear optical element system,” Phys. Rev. A 39, 5209–5228 (1989).
    [CrossRef] [PubMed]
  24. S. A. Akhmanov, M. A. Vorontsov, and V. Yu Ivanov, “Generation of the structures in optical systems with 2-D feedback: to creation of nonlinear optical analogs of neural networks,” in New Physical Principles of Optical Information Processes, S. A. Akhmanov and M. A. Vorontsov, eds. (Nauka, Moscow, 1989), pp. 263–325.
  25. M. A. Vorontsov and K. V. Shishakov, “Phase effects in passive nonlinear cavities,” J. Opt. Soc. Am. B (to be published).
  26. E. F. Kobzev and M. A. Vorontsov, “Optical implementation of Winner-take-all models of neural networks,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1402, 165–174 (1990).
    [CrossRef]
  27. D. Z. Anderson, “Nonlinear optical neural networks: dynamic ring oscillators,” in Neural Computers, R. Echmiller and Ch. V. D. Malsburg, eds. (Springer-Verlag, Berlin, 1988), p. 417.
  28. D. Z. Anderson and R. Saxena, “Theory of multimode operation of a unidirectional ring oscillator having photorefractive gain: weak-field limit,” J. Opt. Soc. Am. B 4, 164–176 (1987).
    [CrossRef]
  29. H. Haken, Synergetics. An Introduction, 3rd ed. (Springer-Verlag, New York, 1983).
    [CrossRef]
  30. V. A. Vasiliev, Yu. M. Romanovsky, D. S. Chernavsky, and V. G. Yahno, Autowave Processes in Kinetic Systems (VEB Deutscher Verlag, Berlin, 1987).
    [CrossRef]
  31. M. Kreuzer, W. Balzer, and T. Tschudi, “Formation of spatial structures in bistable optical elements containing nematic liquid crystals,” Appl. Opt. 29, 579–582 (1989).
    [CrossRef]

1990 (4)

N. B. Abraham and W. J. Firth, eds., feature on transverse effects in nonlinear-optical systems, J. Opt. Soc. Am. B 7, 947–1157 (1990).

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]

M. A. Vorontsov, N. I. Zheleznykh, and V. Yu. Ivanov, “Transverse interactions in 2-D feedback nonlinear systems,” Opt. Quantum Electron. 22, 505–515 (1990).
[CrossRef]

M. A. Vorontsov and N. I. Zheleznykh, “Transverse bistability and multistability in nonlinear optical systems with 2-D feedback,” Mat. Modelirov. 2, 31–39 (1990).

1989 (2)

K. Otsuka and K. Ikeda, “Cooperative dynamics and functions in a collective nonlinear optical element system,” Phys. Rev. A 39, 5209–5228 (1989).
[CrossRef] [PubMed]

M. Kreuzer, W. Balzer, and T. Tschudi, “Formation of spatial structures in bistable optical elements containing nematic liquid crystals,” Appl. Opt. 29, 579–582 (1989).
[CrossRef]

1988 (3)

J. P. Crutchfield, “Spatio-temporal complexity in nonlinear image processing,” IEEE Trans. Circuits Syst. 35, 770–780 (1988).
[CrossRef]

S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, “Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of optical turbulence,” JETP Lett. 47, 707–711 (1988).

M. A. Vorontsov, Yu. D. Dumarevski, V. I. Shmalhauzen, and D. V. Pruidze, “Autowave processes in systems with optical feedback,” Izv. Akad. Nauk SSSR Ser. Fiz. 52, 374–376 (1988).

1987 (2)

1986 (1)

N. N. Rosanov and G. V. Khodova, “Formation of switching waves in bistable systems,” Kvantovaya Electron. (Moscow) 13, 368–377 (1986).

1985 (3)

W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1012 (1985).
[CrossRef]

D. McLaughlin, J. Moloney, and A. Newell, “New class of instabilities in passive optical cavities,” Phys. Rev. Lett. 54, 681–684 (1985).
[CrossRef] [PubMed]

W. J. Firth and I. Galbraith, “Diffusion transverse coupling of bistable elements—switching waves and cross-talk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
[CrossRef]

1983 (1)

A. A. Vasiliev, I. N. Kompanets, and A. V. Parfenov, “Advantages in development and application of optically controlled SLM,” Kvantovaya Elektron. (Moscow) 10, 1074–1088 (1983).

1980 (1)

K. Ikeda, A. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

Agarval, G. S.

G. S. Agarval, Nonlinear Fiber Optics (Academic, Orlando, Fla., 1989).

Akhmanov, S. A.

S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, “Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of optical turbulence,” JETP Lett. 47, 707–711 (1988).

S. A. Akhmanov, V. A. Visloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, Woodbury, NY., 1991).

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Nauka, Moscow, 1964).

S. A. Akhmanov, M. A. Vorontsov, and V. Yu Ivanov, “Generation of the structures in optical systems with 2-D feedback: to creation of nonlinear optical analogs of neural networks,” in New Physical Principles of Optical Information Processes, S. A. Akhmanov and M. A. Vorontsov, eds. (Nauka, Moscow, 1989), pp. 263–325.

Akimoto, O.

K. Ikeda, A. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

Anderson, D. Z.

D. Z. Anderson and R. Saxena, “Theory of multimode operation of a unidirectional ring oscillator having photorefractive gain: weak-field limit,” J. Opt. Soc. Am. B 4, 164–176 (1987).
[CrossRef]

D. Z. Anderson, “Nonlinear optical neural networks: dynamic ring oscillators,” in Neural Computers, R. Echmiller and Ch. V. D. Malsburg, eds. (Springer-Verlag, Berlin, 1988), p. 417.

Arecchi, F. T.

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]

Balzer, W.

Chernavsky, D. S.

V. A. Vasiliev, Yu. M. Romanovsky, D. S. Chernavsky, and V. G. Yahno, Autowave Processes in Kinetic Systems (VEB Deutscher Verlag, Berlin, 1987).
[CrossRef]

Chirkin, A. S.

S. A. Akhmanov, V. A. Visloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, Woodbury, NY., 1991).

Crutchfield, J. P.

J. P. Crutchfield, “Spatio-temporal complexity in nonlinear image processing,” IEEE Trans. Circuits Syst. 35, 770–780 (1988).
[CrossRef]

Daido, A.

K. Ikeda, A. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

Dumarevski, Yu. D.

M. A. Vorontsov, Yu. D. Dumarevski, V. I. Shmalhauzen, and D. V. Pruidze, “Autowave processes in systems with optical feedback,” Izv. Akad. Nauk SSSR Ser. Fiz. 52, 374–376 (1988).

Firth, W. J.

W. J. Firth and I. Galbraith, “Diffusion transverse coupling of bistable elements—switching waves and cross-talk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
[CrossRef]

W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1012 (1985).
[CrossRef]

Galbraith, I.

W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1012 (1985).
[CrossRef]

W. J. Firth and I. Galbraith, “Diffusion transverse coupling of bistable elements—switching waves and cross-talk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
[CrossRef]

Giacomelli, G.

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]

Gibbs, H. M.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

Haken, H.

H. Haken, Synergetics. An Introduction, 3rd ed. (Springer-Verlag, New York, 1983).
[CrossRef]

Ikeda, K.

K. Otsuka and K. Ikeda, “Cooperative dynamics and functions in a collective nonlinear optical element system,” Phys. Rev. A 39, 5209–5228 (1989).
[CrossRef] [PubMed]

K. Ikeda, A. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

Ivanov, V. Yu

S. A. Akhmanov, M. A. Vorontsov, and V. Yu Ivanov, “Generation of the structures in optical systems with 2-D feedback: to creation of nonlinear optical analogs of neural networks,” in New Physical Principles of Optical Information Processes, S. A. Akhmanov and M. A. Vorontsov, eds. (Nauka, Moscow, 1989), pp. 263–325.

Ivanov, V. Yu.

M. A. Vorontsov, N. I. Zheleznykh, and V. Yu. Ivanov, “Transverse interactions in 2-D feedback nonlinear systems,” Opt. Quantum Electron. 22, 505–515 (1990).
[CrossRef]

S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, “Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of optical turbulence,” JETP Lett. 47, 707–711 (1988).

M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhauzen, “Rotatory instability of the spatial structure of light fields in nonlinear media with 2-D feedback,” in Proceedings of the Third Binational USA–USSR Symposium of Laser Optics of Condensed Matter (Leningrad, 1987) (Plenum, New York, 1988), pp. 507–517.
[CrossRef]

V. Yu. Ivanov, A. V. Larichev, and M. A. Vorontsov, “1-D rotatory waves in the optical systems with nonlinear large-scale field interactions,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1402, 145–153 (1990).
[CrossRef]

Kasasent, D.

A. A. Vasiliev, D. Kasasent, I. N. Kompanets, and A. V. Parfenov, Spatial Light Modulators (Radio and Svyaz, Moscow, 1987).

Khodova, G. V.

N. N. Rosanov and G. V. Khodova, “Formation of switching waves in bistable systems,” Kvantovaya Electron. (Moscow) 13, 368–377 (1986).

Khokhlov, R. V.

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Nauka, Moscow, 1964).

Kobzev, E. F.

E. F. Kobzev and M. A. Vorontsov, “Optical implementation of Winner-take-all models of neural networks,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1402, 165–174 (1990).
[CrossRef]

Kompanets, I. N.

A. A. Vasiliev, I. N. Kompanets, and A. V. Parfenov, “Advantages in development and application of optically controlled SLM,” Kvantovaya Elektron. (Moscow) 10, 1074–1088 (1983).

A. A. Vasiliev, D. Kasasent, I. N. Kompanets, and A. V. Parfenov, Spatial Light Modulators (Radio and Svyaz, Moscow, 1987).

Kreuzer, M.

Larichev, A. V.

V. Yu. Ivanov, A. V. Larichev, and M. A. Vorontsov, “1-D rotatory waves in the optical systems with nonlinear large-scale field interactions,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1402, 145–153 (1990).
[CrossRef]

N. I. Zheleznykh, M. A. Vorontsov, and A. V. Larichev, “2-D dynamics neural network optical system with simplest types of large-scale interactions,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1402, 154–164 (1990).

Lefever, R.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

Lugiato, L. A.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

McLaughlin, D.

D. McLaughlin, J. Moloney, and A. Newell, “New class of instabilities in passive optical cavities,” Phys. Rev. Lett. 54, 681–684 (1985).
[CrossRef] [PubMed]

Moloney, J.

D. McLaughlin, J. Moloney, and A. Newell, “New class of instabilities in passive optical cavities,” Phys. Rev. Lett. 54, 681–684 (1985).
[CrossRef] [PubMed]

Newell, A.

D. McLaughlin, J. Moloney, and A. Newell, “New class of instabilities in passive optical cavities,” Phys. Rev. Lett. 54, 681–684 (1985).
[CrossRef] [PubMed]

Otsuka, K.

K. Otsuka and K. Ikeda, “Cooperative dynamics and functions in a collective nonlinear optical element system,” Phys. Rev. A 39, 5209–5228 (1989).
[CrossRef] [PubMed]

Parfenov, A. V.

A. A. Vasiliev, I. N. Kompanets, and A. V. Parfenov, “Advantages in development and application of optically controlled SLM,” Kvantovaya Elektron. (Moscow) 10, 1074–1088 (1983).

A. A. Vasiliev, D. Kasasent, I. N. Kompanets, and A. V. Parfenov, Spatial Light Modulators (Radio and Svyaz, Moscow, 1987).

Pruidze, D. V.

M. A. Vorontsov, Yu. D. Dumarevski, V. I. Shmalhauzen, and D. V. Pruidze, “Autowave processes in systems with optical feedback,” Izv. Akad. Nauk SSSR Ser. Fiz. 52, 374–376 (1988).

Ramazza, P. L.

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]

Residori, S.

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]

Romanovsky, Yu. M.

V. A. Vasiliev, Yu. M. Romanovsky, D. S. Chernavsky, and V. G. Yahno, Autowave Processes in Kinetic Systems (VEB Deutscher Verlag, Berlin, 1987).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov and G. V. Khodova, “Formation of switching waves in bistable systems,” Kvantovaya Electron. (Moscow) 13, 368–377 (1986).

Saxena, R.

Shishakov, K. V.

M. A. Vorontsov and K. V. Shishakov, “Phase effects in passive nonlinear cavities,” J. Opt. Soc. Am. B (to be published).

Shmalhauzen, V. I.

M. A. Vorontsov, Yu. D. Dumarevski, V. I. Shmalhauzen, and D. V. Pruidze, “Autowave processes in systems with optical feedback,” Izv. Akad. Nauk SSSR Ser. Fiz. 52, 374–376 (1988).

M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhauzen, “Rotatory instability of the spatial structure of light fields in nonlinear media with 2-D feedback,” in Proceedings of the Third Binational USA–USSR Symposium of Laser Optics of Condensed Matter (Leningrad, 1987) (Plenum, New York, 1988), pp. 507–517.
[CrossRef]

Tschudi, T.

Vasiliev, A. A.

A. A. Vasiliev, I. N. Kompanets, and A. V. Parfenov, “Advantages in development and application of optically controlled SLM,” Kvantovaya Elektron. (Moscow) 10, 1074–1088 (1983).

A. A. Vasiliev, D. Kasasent, I. N. Kompanets, and A. V. Parfenov, Spatial Light Modulators (Radio and Svyaz, Moscow, 1987).

Vasiliev, V. A.

V. A. Vasiliev, Yu. M. Romanovsky, D. S. Chernavsky, and V. G. Yahno, Autowave Processes in Kinetic Systems (VEB Deutscher Verlag, Berlin, 1987).
[CrossRef]

Visloukh, V. A.

S. A. Akhmanov, V. A. Visloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, Woodbury, NY., 1991).

Vorontsov, M. A.

M. A. Vorontsov, N. I. Zheleznykh, and V. Yu. Ivanov, “Transverse interactions in 2-D feedback nonlinear systems,” Opt. Quantum Electron. 22, 505–515 (1990).
[CrossRef]

M. A. Vorontsov and N. I. Zheleznykh, “Transverse bistability and multistability in nonlinear optical systems with 2-D feedback,” Mat. Modelirov. 2, 31–39 (1990).

M. A. Vorontsov, Yu. D. Dumarevski, V. I. Shmalhauzen, and D. V. Pruidze, “Autowave processes in systems with optical feedback,” Izv. Akad. Nauk SSSR Ser. Fiz. 52, 374–376 (1988).

S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, “Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of optical turbulence,” JETP Lett. 47, 707–711 (1988).

M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhauzen, “Rotatory instability of the spatial structure of light fields in nonlinear media with 2-D feedback,” in Proceedings of the Third Binational USA–USSR Symposium of Laser Optics of Condensed Matter (Leningrad, 1987) (Plenum, New York, 1988), pp. 507–517.
[CrossRef]

N. I. Zheleznykh, M. A. Vorontsov, and A. V. Larichev, “2-D dynamics neural network optical system with simplest types of large-scale interactions,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1402, 154–164 (1990).

V. Yu. Ivanov, A. V. Larichev, and M. A. Vorontsov, “1-D rotatory waves in the optical systems with nonlinear large-scale field interactions,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1402, 145–153 (1990).
[CrossRef]

E. F. Kobzev and M. A. Vorontsov, “Optical implementation of Winner-take-all models of neural networks,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1402, 165–174 (1990).
[CrossRef]

M. A. Vorontsov and K. V. Shishakov, “Phase effects in passive nonlinear cavities,” J. Opt. Soc. Am. B (to be published).

S. A. Akhmanov, M. A. Vorontsov, and V. Yu Ivanov, “Generation of the structures in optical systems with 2-D feedback: to creation of nonlinear optical analogs of neural networks,” in New Physical Principles of Optical Information Processes, S. A. Akhmanov and M. A. Vorontsov, eds. (Nauka, Moscow, 1989), pp. 263–325.

Wright, E. M.

Yahno, V. G.

V. A. Vasiliev, Yu. M. Romanovsky, D. S. Chernavsky, and V. G. Yahno, Autowave Processes in Kinetic Systems (VEB Deutscher Verlag, Berlin, 1987).
[CrossRef]

Zheleznykh, N. I.

M. A. Vorontsov and N. I. Zheleznykh, “Transverse bistability and multistability in nonlinear optical systems with 2-D feedback,” Mat. Modelirov. 2, 31–39 (1990).

M. A. Vorontsov, N. I. Zheleznykh, and V. Yu. Ivanov, “Transverse interactions in 2-D feedback nonlinear systems,” Opt. Quantum Electron. 22, 505–515 (1990).
[CrossRef]

N. I. Zheleznykh, M. A. Vorontsov, and A. V. Larichev, “2-D dynamics neural network optical system with simplest types of large-scale interactions,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1402, 154–164 (1990).

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

W. J. Firth and I. Galbraith, “Diffusion transverse coupling of bistable elements—switching waves and cross-talk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

J. P. Crutchfield, “Spatio-temporal complexity in nonlinear image processing,” IEEE Trans. Circuits Syst. 35, 770–780 (1988).
[CrossRef]

Izv. Akad. Nauk SSSR Ser. Fiz. (1)

M. A. Vorontsov, Yu. D. Dumarevski, V. I. Shmalhauzen, and D. V. Pruidze, “Autowave processes in systems with optical feedback,” Izv. Akad. Nauk SSSR Ser. Fiz. 52, 374–376 (1988).

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

JETP Lett. (1)

S. A. Akhmanov, M. A. Vorontsov, and V. Yu. Ivanov, “Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of optical turbulence,” JETP Lett. 47, 707–711 (1988).

Kvantovaya Electron. (Moscow) (1)

N. N. Rosanov and G. V. Khodova, “Formation of switching waves in bistable systems,” Kvantovaya Electron. (Moscow) 13, 368–377 (1986).

Kvantovaya Elektron. (Moscow) (1)

A. A. Vasiliev, I. N. Kompanets, and A. V. Parfenov, “Advantages in development and application of optically controlled SLM,” Kvantovaya Elektron. (Moscow) 10, 1074–1088 (1983).

Mat. Modelirov. (1)

M. A. Vorontsov and N. I. Zheleznykh, “Transverse bistability and multistability in nonlinear optical systems with 2-D feedback,” Mat. Modelirov. 2, 31–39 (1990).

Opt. Quantum Electron. (1)

M. A. Vorontsov, N. I. Zheleznykh, and V. Yu. Ivanov, “Transverse interactions in 2-D feedback nonlinear systems,” Opt. Quantum Electron. 22, 505–515 (1990).
[CrossRef]

Phys. Rev. A (1)

K. Otsuka and K. Ikeda, “Cooperative dynamics and functions in a collective nonlinear optical element system,” Phys. Rev. A 39, 5209–5228 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

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]

K. Ikeda, A. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

D. McLaughlin, J. Moloney, and A. Newell, “New class of instabilities in passive optical cavities,” Phys. Rev. Lett. 54, 681–684 (1985).
[CrossRef] [PubMed]

Other (14)

M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhauzen, “Rotatory instability of the spatial structure of light fields in nonlinear media with 2-D feedback,” in Proceedings of the Third Binational USA–USSR Symposium of Laser Optics of Condensed Matter (Leningrad, 1987) (Plenum, New York, 1988), pp. 507–517.
[CrossRef]

S. A. Akhmanov, V. A. Visloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, Woodbury, NY., 1991).

G. S. Agarval, Nonlinear Fiber Optics (Academic, Orlando, Fla., 1989).

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Nauka, Moscow, 1964).

A. A. Vasiliev, D. Kasasent, I. N. Kompanets, and A. V. Parfenov, Spatial Light Modulators (Radio and Svyaz, Moscow, 1987).

N. I. Zheleznykh, M. A. Vorontsov, and A. V. Larichev, “2-D dynamics neural network optical system with simplest types of large-scale interactions,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1402, 154–164 (1990).

S. A. Akhmanov, M. A. Vorontsov, and V. Yu Ivanov, “Generation of the structures in optical systems with 2-D feedback: to creation of nonlinear optical analogs of neural networks,” in New Physical Principles of Optical Information Processes, S. A. Akhmanov and M. A. Vorontsov, eds. (Nauka, Moscow, 1989), pp. 263–325.

M. A. Vorontsov and K. V. Shishakov, “Phase effects in passive nonlinear cavities,” J. Opt. Soc. Am. B (to be published).

E. F. Kobzev and M. A. Vorontsov, “Optical implementation of Winner-take-all models of neural networks,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1402, 165–174 (1990).
[CrossRef]

D. Z. Anderson, “Nonlinear optical neural networks: dynamic ring oscillators,” in Neural Computers, R. Echmiller and Ch. V. D. Malsburg, eds. (Springer-Verlag, Berlin, 1988), p. 417.

V. Yu. Ivanov, A. V. Larichev, and M. A. Vorontsov, “1-D rotatory waves in the optical systems with nonlinear large-scale field interactions,” in Nonlinear Optics in Diagnostics, Modeling, and Control of Biological Objects and Processes, S. A. Akhmanov and V. N. Zadkov, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1402, 145–153 (1990).
[CrossRef]

H. Haken, Synergetics. An Introduction, 3rd ed. (Springer-Verlag, New York, 1983).
[CrossRef]

V. A. Vasiliev, Yu. M. Romanovsky, D. S. Chernavsky, and V. G. Yahno, Autowave Processes in Kinetic Systems (VEB Deutscher Verlag, Berlin, 1987).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Block diagram of an optical system with 2D feedback.

Fig. 2
Fig. 2

Base field transformations in the feedback circuit: various types of 2D feedback.

Fig. 3
Fig. 3

Block scheme of experimental setup: 1, interferometer stabilization scheme; 2, TV camera; 3, computer; 4, device for field rotation.

Fig. 4
Fig. 4

Dissipative optical structures in a nonlinear system with a field shift in the 2D feedback.

Fig. 5
Fig. 5

Instant photographs of rotating nonlinear structures (optical reverberators) formed with different values of the nonlinear parameter K and of the angle of field rotation Δ: (a),(b) without mask; (c), (d) optical reverberators with mask, composed of one or three concentric circular regions in the feedback loop.

Fig. 6
Fig. 6

(a), (b) Spiral nonlinear structures in the system with 2D feedback. (c), (d) Transition to optical turbulence.

Fig. 7
Fig. 7

(a) Simplified schematic of a nonlinear Fizeau interferometer with degenerate 2D feedback (the model of two coupled resonators), yielding spatial optical bistability and multistability. M’s, mirrors; NL, nonlinear medium. (b) Dependence of stationary solutions û1 and û2 on the nonlinearity parameter K. (c), (d) Experimental results: (c) optical bistability, (d) spatial fragmentation of the beam.

Fig. 8
Fig. 8

Spatial intensity distribution in optical bistability (numerical simulation): (a) optical bistability, (b) spatial fragmentation of the beam.

Fig. 9
Fig. 9

Boundaries of the rotatory structures excitation areas.

Fig. 10
Fig. 10

(a) Dependence of the rotation frequency ωm on the angle of rotation Δ. (b) Comparison between theoretical (curve) and experimental (asterisks) dependences. (c), (d) Hysteresis of spatial structures. Dependences ωm = ωm(Δ) for opposite directions of rotation (A): (c) experiment, (d) numerical simulation.

Fig. 11
Fig. 11

Shape of the rotating waves. (a), (b) Strongly nonlinear phase modulation: (a) numerical simulation, (b) experiment. (c) Shape of the wave near the boundary of transition, experiment.

Fig. 12
Fig. 12

Hysteresis of spatial structures: Numerical solution of nonlinear equation (15) (solid curves) and dependence Am(Δ) (dashed curves) at D* = 0.026, K = 2.5, γ = 0.9 (3, 4, and 5: numbers of petals).

Fig. 13
Fig. 13

Coexistence of optical reverberators with different numbers of petals: (a) experiment, (b) numerical simulation.

Equations (29)

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A z - i 1 2 2 k ω 2 2 A t 2 + i k n 2 n 0 A 2 A = 0.
A z - i 1 2 k 2 A + i k n 2 n 0 A 2 A = 0.
n ( r , t ) = n 0 + n 2 P ( r , t ) ,
τ P t ( r , t ) + P ( r , t ) = A 2 ,
A ( r , t ) = exp ( - g l / 2 + i l k n ) A ( r , t ) ,
τ n t + n = n 0 + n 2 A 2 .
A ( r , t ) = ( 1 - R ) 1 / 2 A in ( r , t ) + η R A ( r , t ) .
A ( r , t ) = ( 1 - R ) 1 / 2 A in ( r , t ) + B exp [ i l k n ( r , t ) ] A ( r , t ) ,
A ( r , t ) = ( 1 - R ) 1 / 2 A in ( r , t ) + B exp [ i l k n ( r , t ) ] × [ ( 1 - R ) 1 / 2 A in ( r , t ) + B exp [ i l k n ( r , t ) ] A ( r , t ) ] ,
A ( r , t ) = ( 1 - R ) 1 / 2 { A in ( r , t ) + A in ( r , t ) B exp [ i l k n ( r , t ) ] } .
I ( r , t ) = ( 1 - R ) I in { 1 + 2 B cos [ φ ( r , t ) ] } ,
τ u t ( r , t ) + u ( r , t ) = K { 1 + γ cos [ u ( r , t ) + φ 0 ] } ,
τ u t ( r , t ) + u ( r , t ) = D 2 u ( r , t ) + K { 1 + γ cos [ u ( r , t ) + φ 0 ] } .
τ u t ( r , t ) + u ( r , t ) = D 2 u ( r , t ) + K { 1 + γ cos [ u ( r , t - T ) + φ 0 ] } .
τ u t ( x , y , t ) + u ( x , y , t ) = D 2 u ( x , y , t ) + K ( 1 + γ cos [ u ( - x , y , t ) + φ 0 ] } .
τ u 1 t + u 1 = K [ 1 + γ cos ( u 2 + φ 0 ) ] , τ u 2 t + u 2 = K [ 1 + γ cos ( u 1 + φ 0 ) ] ,
τ u t ( θ , t ) + u ( θ , t ) = D * 2 u θ 2 ( θ , t ) + K { 1 + γ cos [ u ( θ + Δ , t ) + φ 0 ] } .
u ( 0 , t ) = u ( 2 π , t ) ,             u θ ( 0 , t ) = u θ ( 2 π , t ) .
u ( θ , t ) = u ^ ( t ) + A ( t ) cos ( ω t - κ θ ) ,
τ d A d t + ( 1 + D * m 2 ) A = - 2 K γ J 1 ( A ) sin ( u ^ + φ 0 ) cos ( m Δ ) ,
τ d u ^ d t + u ^ = K [ 1 + γ J 0 ( A ) cos ( u ^ + φ 0 ) ] ,
τ ω A - 2 K γ J 1 ( A ) sin ( u ^ + φ 0 ) sin ( m Δ ) = 0.
J 1 ( A m ) = 1 + D * m 2 2 K γ sin ( u ^ m + φ 0 ) cos ( m Δ ) A m ,
u ^ m = K [ 1 + γ J 0 ( A m ) cos ( u ^ m + φ 0 ) ] .
ω m = - ( 1 + D * m 2 ) tan ( m Δ ) / τ .
τ d A d t + ( 1 + D * m 2 ) A = Λ cos ( m Δ ) A ,
τ d u ^ d t + u ^ = K [ 1 + γ cos ( u ^ + φ 0 ) ] ,
A ( t ) = A 0 exp [ δ m ( t / τ ) ] ,
- K m γ sin ( u ^ + φ 0 ) cos ( m Δ ) = 1 + D * m 2 ,

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