Abstract

Strong nonlinear optical effects and optical pattern-forming systems can be designed with the optical system architectures introduced here on the basis of large-scale arrays of optoelectronic feedback circuits. Experiments were performed with a liquid-crystal television panel as a large-scale array of phase modulators and a CCD camera as a photoarray. By synthesizing various nonlinearities and using controllable spatial coupling, we obtained a variety of transversal optical patterns, localized states, waves, and chaotic regimes.

© 2000 Optical Society of America

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  1. H. M. Gibbs, Optical Bistability—Controlling Light with Light (Academic, Orlando, Fla., 1985); I. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  2. J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1991); L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” Adv. At. Mol. Opt. Phys. 40, 229–306 (1998); M. A. Vorontsov and W. B. Miller, eds., Self-Organization in Optical Systems and Applications in Information Technology (Springer, Berlin, 1995); C. O. Weiss, “Spatio-temporal structures. Part II,” Phys. Rep. 219, 311–328 (1992).
    [CrossRef]
  3. G. Hausler and M. Simon, “Generation of space and time picture oscillations by active incoherent feedback,” Opt. Acta 25, 327–336 (1978); J. P. Crutchfield, “Space-time dynamics in video-feedback,” Physica D 10, 229–338 (1984); G. Hausler, G. Seckmeyer, and T. Weiss, “Chaos and cooperation in nonlinear pictorial feedback systems,” Appl. Opt. APOPAI 25, 4656–4663 (1986).
    [CrossRef]
  4. U. Efron, ed., Spatial Light Modulator Technology: Materials, Devices, and Applications (Marcel Dekker, New York, 1995); V. G. Chigrinov, Liquid Crystal Devices: Physics and Applications (Artech House, Norwood, Mass., 1999).
  5. M. A. Vorontsov, Yu. D. Dumarevsky, D. V. Pruidze, and V. I. Shmalhauzen, “Autowave processes in optical feedback systems,” 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, 611–614 (1988).
  6. S. A. Akhmanov, M. A. Vorontsov, V. Yu. Ivanov, A. V. Larichev, and N. I. Zheleznykh, Controlling transverse-wave interactions in nonlinear optics: generation and interaction of spatiotemporal structures,” J. Opt. Soc. Am. B 9, 78–90 (1992); R. Neubecker, G. L. Oppo, B. Thuering, and T. Tschudi, “Pattern formation in a liquid-crystal light valve with feedback, including polarization, saturation, and internal threshold effects,” Phys. Rev. A 52, 791–808 (1995); P. L. Ramazza, E. Pampaloni, S. Residori, and F. T. Arecchi, “Optical pattern formation in a Kerr-like medium with feedback,” Physica D PDNPDT 96, 259–271 (1996).
    [CrossRef] [PubMed]
  7. M. A. Vorontsov, “Information processing with nonlinear optical two-dimensional feedback systems,” J. Opt. B: Quantum Semiclass. Opt. 1, R1–R10 (1999).
    [CrossRef]
  8. C. A. Mead, “Neuromorphic electronic systems,” Proc. IEEE 78, 1629–1640 (1990); R. P. Lippmann and D. S. Touretzky, eds., Neural Information Processing Systems, (Morgan Kaufmann, San Mateo, Calif., 1995), Vol. 3.
    [CrossRef]
  9. M. C. Wu, “Micromachining for optical and opto-electronic systems,” Proc. IEEE 85, 1833–1997 (1997); G. V. Vdovin and P. M. Sarro, “Flexible mirror micromachined in silicon,” Appl. Opt. 34, 2968–2972 (1995).
    [CrossRef] [PubMed]
  10. S. Serati, G. Sharp, R. Serati, D. McKnight, and J. Stookley, “128×128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent and T.-H. Chao, eds., Proc. SPIE 2490, 55–63 (1995); http://www.bnonlinear.com.
  11. Currently developed arrays of turntable micromirrors as well as arrays of LC-on-silicon phase modulators may have 512×512 actuators with actuator (pixel) size less than 100 μm for MEMS and 15 μm for LC devices.
  12. High-spatial resolution of both the micromirror and the photoarray allows the use of continuous-form mathematical models.
  13. L. A. Lugiato and M. S. El Nashie, eds., Special issue on nonlinear optical structures, patterns, and chaos, Chaos, Solitons, and Fractals 4, 1251–1844 (1994).
    [CrossRef]
  14. A. G. Andreou and K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits and Signal Process. 9, 141–153 (1996).
    [CrossRef]
  15. G. Giusfredy, J. F. Valley, R. Pon, G. Khitrova, and H. M. Gibbs, “Optical instabilities in sodium vapor,” J. Opt. Soc. Am. B 5, 1181–1192 (1988).
    [CrossRef]
  16. W. J. Firth, “Spatial instabilities in a Kerr medium with a single feedback mirror,” J. Mod. Opt. 37, 151–155 (1990); G. P. D’Alessandro and W. J. Firth, “Hexagon spatial patterns for a Kerr slice with a feedback mirror,” Phys. Rev. A 46, 537–548 (1992); M. A. Vorontsov and W. Firth, “Pattern formation and competition in nonlinear optical systems with two-dimensional feedback,” Phys. Rev. A PLRAAN 49, 2891–2903 (1994).
    [CrossRef] [PubMed]
  17. W. F. Ames, Numerical Methods for Partial Differential Equations (Academic, San Diego, Calif., 1992).
  18. R. Neubecker, B. Thuering, and T. Tschudi, “Formation and characterization of hexagonal patterns in a single feedback experiment,” in a special issue on nonlinear optical structures, patterns, and chaos, Chaos, Solitons, and Fractals 4, L. A. Lugiato and M. S. El Nashie eds., 1307–1322 (1994); M. Tamburrini and E. Ciaramella, “Hexagonal beam filamentation in a liquid crystal film with single feedback mirror, 1355–1367.
    [CrossRef]
  19. H. Adachihara and H. Faid, “Two-dimensional nonlinear-interferometer pattern analysis and decay of spirals,” J. Opt. Soc. Am. B 10, 1242–1253 (1993); N. I. Zheleznikh, M. Le Berre, F. Ressayre, and A. Tallet, “Rotating spiral waves in a nonlinear optical system with spatial interaction,” in Chaos, Solitons and Fractals, L. A. Lugiato and M. S. El Nashie, eds. (Pergamon, New York, 1994).
    [CrossRef]
  20. M. A. Vorontsov, “‘Akhseals’ as a new class of spatio-temporal light field instabilities,” Quantum Electron. 23, 269–271 (1993); M. A. Vorontsov, N. G. Iroshnikov, and R. Abernathy, “Diffractive patterns in a nonlinear optical 2D-feedback system with field rotation,” in Chaos, Solitons and Fractals, L. A. Lugiato and M. S. El Nashie, eds. (Pergamon, New York, 1994).
    [CrossRef]
  21. E. Pampaloni, P. L. Ramazza, S. Residori, and F. T. Arecchi, “Two-dimensional crystals and quasicrystals in nonlinear optics,” Phys. Rev. Lett. 74, 258–261 (1995); B. Y. Rubinstein and L. M. Pismen, “Resonant two-dimensional patterns in optical cavities with rotated beam,” Phys. Rev. A 56, 4264–4272 (1997).
    [CrossRef] [PubMed]
  22. F. T. Arecchi, S. Boccaletti, G. Giacomelli, P. L. Ramazza, and S. Residori, “Pattern and vortex dynamics in photorefractive oscillators,” in Self-Organization in Optical Systems and Applications in Information Technology, M. A. Vorontsov and W. B. Miller, eds. (Springer, New York, 1995).
  23. F. T. Arecchi, “Optical morphogenesis: pattern formation and control in nonlinear optics,” II Nuovo Cimento A 107, 1111–1121 (1994).
    [CrossRef]
  24. N. N. Rosanov and G. V. Khodova, “Diffractive autosolitons in nonlinear interferometers,” J. Opt. Soc. Am. B 7, 1057–1065 (1990); D. V. McLaughlin, J. V. Moloney, and A. C. Newel, “Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring resonator,” Phys. Rev. Lett. 51, 75–78 (1983); G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B JOBPDE 7, 1328–1335 (1990); Y. S. Kivshar and Xiaoping Yang, “Dynamics of dark solitons,” in Chaos, Solitons and Fractals, L. A. Lugiato and M. S. El Nashie, eds. (Pergamon, New York, 1994), p. 1745.
    [CrossRef]
  25. M. A. Vorontsov and B. A. Samson, “Nonlinear dynamics in an optical system with controlled 2D-feedback: black-eye patterns and related phenomena,” Phys. Rev. A 57, 3040–3049 (1998).
    [CrossRef]
  26. R. Martin, A. J. Scroggie, G. L. Oppo, and W. J. Firth, “Stabilization and tracking of unstable patterns by Fourier space techniques,” Phys. Rev. Lett. 77, 4007–4012 (1996); W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
    [CrossRef] [PubMed]
  27. J. N. Mait, “Diffractive beauty,” Opt. Photon. News 9, 21–25 (1998).
    [CrossRef]
  28. H. Haken, Synergetics, an Introduction (Springer-Verlag, Berlin, 1997); G. Nicolis, Introduction to Nonlinear Science (Cambridge U. Press, Cambridge, UK, 1995).
  29. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer-Verlag, Berlin, 1984); G. H. Gunaratne, Q. Ouyang, and H. L. Swinney, “Pattern formation in the presence of symmetries,” Phys. Rev. E 50, 2802–2820 (1994).
    [CrossRef]
  30. E. V. Degtiarev and V. G. Watagin, “Stability analysis of a two-component nonlinear system,” Opt. Commun. 124, 309–313 (1996).
    [CrossRef]
  31. C.-M. Ho and Y.-C. Tai, “Micro-electro-mechanical-systems and fluid flows,” Annu. Rev. Fluid Mech. 30, 579–612 (1998).
    [CrossRef]

1999 (1)

M. A. Vorontsov, “Information processing with nonlinear optical two-dimensional feedback systems,” J. Opt. B: Quantum Semiclass. Opt. 1, R1–R10 (1999).
[CrossRef]

1998 (3)

M. A. Vorontsov and B. A. Samson, “Nonlinear dynamics in an optical system with controlled 2D-feedback: black-eye patterns and related phenomena,” Phys. Rev. A 57, 3040–3049 (1998).
[CrossRef]

J. N. Mait, “Diffractive beauty,” Opt. Photon. News 9, 21–25 (1998).
[CrossRef]

C.-M. Ho and Y.-C. Tai, “Micro-electro-mechanical-systems and fluid flows,” Annu. Rev. Fluid Mech. 30, 579–612 (1998).
[CrossRef]

1996 (2)

E. V. Degtiarev and V. G. Watagin, “Stability analysis of a two-component nonlinear system,” Opt. Commun. 124, 309–313 (1996).
[CrossRef]

A. G. Andreou and K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits and Signal Process. 9, 141–153 (1996).
[CrossRef]

1994 (2)

F. T. Arecchi, “Optical morphogenesis: pattern formation and control in nonlinear optics,” II Nuovo Cimento A 107, 1111–1121 (1994).
[CrossRef]

L. A. Lugiato and M. S. El Nashie, eds., Special issue on nonlinear optical structures, patterns, and chaos, Chaos, Solitons, and Fractals 4, 1251–1844 (1994).
[CrossRef]

1988 (1)

Andreou, A. G.

A. G. Andreou and K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits and Signal Process. 9, 141–153 (1996).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, “Optical morphogenesis: pattern formation and control in nonlinear optics,” II Nuovo Cimento A 107, 1111–1121 (1994).
[CrossRef]

Boahen, K. A.

A. G. Andreou and K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits and Signal Process. 9, 141–153 (1996).
[CrossRef]

Degtiarev, E. V.

E. V. Degtiarev and V. G. Watagin, “Stability analysis of a two-component nonlinear system,” Opt. Commun. 124, 309–313 (1996).
[CrossRef]

El Nashie, M. S.

L. A. Lugiato and M. S. El Nashie, eds., Special issue on nonlinear optical structures, patterns, and chaos, Chaos, Solitons, and Fractals 4, 1251–1844 (1994).
[CrossRef]

Gibbs, H. M.

Giusfredy, G.

Ho, C.-M.

C.-M. Ho and Y.-C. Tai, “Micro-electro-mechanical-systems and fluid flows,” Annu. Rev. Fluid Mech. 30, 579–612 (1998).
[CrossRef]

Khitrova, G.

Lugiato, L. A.

L. A. Lugiato and M. S. El Nashie, eds., Special issue on nonlinear optical structures, patterns, and chaos, Chaos, Solitons, and Fractals 4, 1251–1844 (1994).
[CrossRef]

Mait, J. N.

J. N. Mait, “Diffractive beauty,” Opt. Photon. News 9, 21–25 (1998).
[CrossRef]

Pon, R.

Samson, B. A.

M. A. Vorontsov and B. A. Samson, “Nonlinear dynamics in an optical system with controlled 2D-feedback: black-eye patterns and related phenomena,” Phys. Rev. A 57, 3040–3049 (1998).
[CrossRef]

Tai, Y.-C.

C.-M. Ho and Y.-C. Tai, “Micro-electro-mechanical-systems and fluid flows,” Annu. Rev. Fluid Mech. 30, 579–612 (1998).
[CrossRef]

Valley, J. F.

Vorontsov, M. A.

M. A. Vorontsov, “Information processing with nonlinear optical two-dimensional feedback systems,” J. Opt. B: Quantum Semiclass. Opt. 1, R1–R10 (1999).
[CrossRef]

M. A. Vorontsov and B. A. Samson, “Nonlinear dynamics in an optical system with controlled 2D-feedback: black-eye patterns and related phenomena,” Phys. Rev. A 57, 3040–3049 (1998).
[CrossRef]

Watagin, V. G.

E. V. Degtiarev and V. G. Watagin, “Stability analysis of a two-component nonlinear system,” Opt. Commun. 124, 309–313 (1996).
[CrossRef]

Analog Integr. Circuits and Signal Process. (1)

A. G. Andreou and K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits and Signal Process. 9, 141–153 (1996).
[CrossRef]

Annu. Rev. Fluid Mech. (1)

C.-M. Ho and Y.-C. Tai, “Micro-electro-mechanical-systems and fluid flows,” Annu. Rev. Fluid Mech. 30, 579–612 (1998).
[CrossRef]

Chaos, Solitons, and Fractals (1)

L. A. Lugiato and M. S. El Nashie, eds., Special issue on nonlinear optical structures, patterns, and chaos, Chaos, Solitons, and Fractals 4, 1251–1844 (1994).
[CrossRef]

II Nuovo Cimento A (1)

F. T. Arecchi, “Optical morphogenesis: pattern formation and control in nonlinear optics,” II Nuovo Cimento A 107, 1111–1121 (1994).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (1)

M. A. Vorontsov, “Information processing with nonlinear optical two-dimensional feedback systems,” J. Opt. B: Quantum Semiclass. Opt. 1, R1–R10 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

E. V. Degtiarev and V. G. Watagin, “Stability analysis of a two-component nonlinear system,” Opt. Commun. 124, 309–313 (1996).
[CrossRef]

Opt. Photon. News (1)

J. N. Mait, “Diffractive beauty,” Opt. Photon. News 9, 21–25 (1998).
[CrossRef]

Phys. Rev. A (1)

M. A. Vorontsov and B. A. Samson, “Nonlinear dynamics in an optical system with controlled 2D-feedback: black-eye patterns and related phenomena,” Phys. Rev. A 57, 3040–3049 (1998).
[CrossRef]

Other (22)

R. Martin, A. J. Scroggie, G. L. Oppo, and W. J. Firth, “Stabilization and tracking of unstable patterns by Fourier space techniques,” Phys. Rev. Lett. 77, 4007–4012 (1996); W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

N. N. Rosanov and G. V. Khodova, “Diffractive autosolitons in nonlinear interferometers,” J. Opt. Soc. Am. B 7, 1057–1065 (1990); D. V. McLaughlin, J. V. Moloney, and A. C. Newel, “Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring resonator,” Phys. Rev. Lett. 51, 75–78 (1983); G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B JOBPDE 7, 1328–1335 (1990); Y. S. Kivshar and Xiaoping Yang, “Dynamics of dark solitons,” in Chaos, Solitons and Fractals, L. A. Lugiato and M. S. El Nashie, eds. (Pergamon, New York, 1994), p. 1745.
[CrossRef]

W. J. Firth, “Spatial instabilities in a Kerr medium with a single feedback mirror,” J. Mod. Opt. 37, 151–155 (1990); G. P. D’Alessandro and W. J. Firth, “Hexagon spatial patterns for a Kerr slice with a feedback mirror,” Phys. Rev. A 46, 537–548 (1992); M. A. Vorontsov and W. Firth, “Pattern formation and competition in nonlinear optical systems with two-dimensional feedback,” Phys. Rev. A PLRAAN 49, 2891–2903 (1994).
[CrossRef] [PubMed]

W. F. Ames, Numerical Methods for Partial Differential Equations (Academic, San Diego, Calif., 1992).

R. Neubecker, B. Thuering, and T. Tschudi, “Formation and characterization of hexagonal patterns in a single feedback experiment,” in a special issue on nonlinear optical structures, patterns, and chaos, Chaos, Solitons, and Fractals 4, L. A. Lugiato and M. S. El Nashie eds., 1307–1322 (1994); M. Tamburrini and E. Ciaramella, “Hexagonal beam filamentation in a liquid crystal film with single feedback mirror, 1355–1367.
[CrossRef]

H. Adachihara and H. Faid, “Two-dimensional nonlinear-interferometer pattern analysis and decay of spirals,” J. Opt. Soc. Am. B 10, 1242–1253 (1993); N. I. Zheleznikh, M. Le Berre, F. Ressayre, and A. Tallet, “Rotating spiral waves in a nonlinear optical system with spatial interaction,” in Chaos, Solitons and Fractals, L. A. Lugiato and M. S. El Nashie, eds. (Pergamon, New York, 1994).
[CrossRef]

M. A. Vorontsov, “‘Akhseals’ as a new class of spatio-temporal light field instabilities,” Quantum Electron. 23, 269–271 (1993); M. A. Vorontsov, N. G. Iroshnikov, and R. Abernathy, “Diffractive patterns in a nonlinear optical 2D-feedback system with field rotation,” in Chaos, Solitons and Fractals, L. A. Lugiato and M. S. El Nashie, eds. (Pergamon, New York, 1994).
[CrossRef]

E. Pampaloni, P. L. Ramazza, S. Residori, and F. T. Arecchi, “Two-dimensional crystals and quasicrystals in nonlinear optics,” Phys. Rev. Lett. 74, 258–261 (1995); B. Y. Rubinstein and L. M. Pismen, “Resonant two-dimensional patterns in optical cavities with rotated beam,” Phys. Rev. A 56, 4264–4272 (1997).
[CrossRef] [PubMed]

F. T. Arecchi, S. Boccaletti, G. Giacomelli, P. L. Ramazza, and S. Residori, “Pattern and vortex dynamics in photorefractive oscillators,” in Self-Organization in Optical Systems and Applications in Information Technology, M. A. Vorontsov and W. B. Miller, eds. (Springer, New York, 1995).

C. A. Mead, “Neuromorphic electronic systems,” Proc. IEEE 78, 1629–1640 (1990); R. P. Lippmann and D. S. Touretzky, eds., Neural Information Processing Systems, (Morgan Kaufmann, San Mateo, Calif., 1995), Vol. 3.
[CrossRef]

M. C. Wu, “Micromachining for optical and opto-electronic systems,” Proc. IEEE 85, 1833–1997 (1997); G. V. Vdovin and P. M. Sarro, “Flexible mirror micromachined in silicon,” Appl. Opt. 34, 2968–2972 (1995).
[CrossRef] [PubMed]

S. Serati, G. Sharp, R. Serati, D. McKnight, and J. Stookley, “128×128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent and T.-H. Chao, eds., Proc. SPIE 2490, 55–63 (1995); http://www.bnonlinear.com.

Currently developed arrays of turntable micromirrors as well as arrays of LC-on-silicon phase modulators may have 512×512 actuators with actuator (pixel) size less than 100 μm for MEMS and 15 μm for LC devices.

High-spatial resolution of both the micromirror and the photoarray allows the use of continuous-form mathematical models.

H. M. Gibbs, Optical Bistability—Controlling Light with Light (Academic, Orlando, Fla., 1985); I. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1991); L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” Adv. At. Mol. Opt. Phys. 40, 229–306 (1998); M. A. Vorontsov and W. B. Miller, eds., Self-Organization in Optical Systems and Applications in Information Technology (Springer, Berlin, 1995); C. O. Weiss, “Spatio-temporal structures. Part II,” Phys. Rep. 219, 311–328 (1992).
[CrossRef]

G. Hausler and M. Simon, “Generation of space and time picture oscillations by active incoherent feedback,” Opt. Acta 25, 327–336 (1978); J. P. Crutchfield, “Space-time dynamics in video-feedback,” Physica D 10, 229–338 (1984); G. Hausler, G. Seckmeyer, and T. Weiss, “Chaos and cooperation in nonlinear pictorial feedback systems,” Appl. Opt. APOPAI 25, 4656–4663 (1986).
[CrossRef]

U. Efron, ed., Spatial Light Modulator Technology: Materials, Devices, and Applications (Marcel Dekker, New York, 1995); V. G. Chigrinov, Liquid Crystal Devices: Physics and Applications (Artech House, Norwood, Mass., 1999).

M. A. Vorontsov, Yu. D. Dumarevsky, D. V. Pruidze, and V. I. Shmalhauzen, “Autowave processes in optical feedback systems,” 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, 611–614 (1988).

S. A. Akhmanov, M. A. Vorontsov, V. Yu. Ivanov, A. V. Larichev, and N. I. Zheleznykh, Controlling transverse-wave interactions in nonlinear optics: generation and interaction of spatiotemporal structures,” J. Opt. Soc. Am. B 9, 78–90 (1992); R. Neubecker, G. L. Oppo, B. Thuering, and T. Tschudi, “Pattern formation in a liquid-crystal light valve with feedback, including polarization, saturation, and internal threshold effects,” Phys. Rev. A 52, 791–808 (1995); P. L. Ramazza, E. Pampaloni, S. Residori, and F. T. Arecchi, “Optical pattern formation in a Kerr-like medium with feedback,” Physica D PDNPDT 96, 259–271 (1996).
[CrossRef] [PubMed]

H. Haken, Synergetics, an Introduction (Springer-Verlag, Berlin, 1997); G. Nicolis, Introduction to Nonlinear Science (Cambridge U. Press, Cambridge, UK, 1995).

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer-Verlag, Berlin, 1984); G. H. Gunaratne, Q. Ouyang, and H. L. Swinney, “Pattern formation in the presence of symmetries,” Phys. Rev. E 50, 2802–2820 (1994).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic for a pattern-forming system based on a large-scale array of optoelectronic feedback circuits.

Fig. 2
Fig. 2

Experimental setup of the pattern-forming system. The focal lengths corresponding to lenses L1 and L2 are f1=1000 mm and f2=125 mm.

Fig. 3
Fig. 3

Characteristic phase modulation of the LCTV panel.

Fig. 4
Fig. 4

Nonlinearity types used in the experiments: (1) linear dependence between intensity and phase corresponding to Kerr-type nonlinearity; (2) binary nonlinearity; (3) unimodal (Gaussian) nonlinearity; and (4) bimodal (sine-type) nonlinear function. Both the input diffractive intensity and the transformed signal Φ(Id) are measured in video signal gray levels.

Fig. 5
Fig. 5

Hexagonal patterns in optoelectronic Kerr-slice–feedback-mirror system corresponding to (a), (b) self-defocusing nonlinearity (nFB<0) and (c), (d) self-focusing nonlinearity (nFB>0); (a), (c) diffractive beam intensity patterns Id(r); and (b), (d) corresponding controlling images ν(r) sent to the LCTV panel (phase images). The laser beam diameter in the plane of the LCTV equals 20 mm. System parameters are L=8 mm, |K|1.2Kth,d=0.0001, α=0.1.

Fig. 6
Fig. 6

Chaotic alternation between different transversal quasi-stable patterns observed in the experiment with feedback field rotation. The diffractive intensity patterns (a), (c), and (e) and the corresponding phase images (b), (d), and (f) are shown in their order of consecutive appearance: (a), (b) n=400; (c), (d) n=640; (e), (f) n=720. The system parameters are Δ=30 deg., α=0.2, d=0,K=0.1,L=8 mm, I0=97. The input intensity I0 here and below corresponds to an aperture-averaged value measured in the CCD camera in video signal gray levels.

Fig. 7
Fig. 7

Intensity pattern of localized states (left-hand column) and phase patterns (right-hand column) obtained in the pattern-forming system with the nonlinearity types shown in Fig. 4: (a), (b) binary; (c), (d) unimodal (Gaussian); and (e), (f) bimodal (sine-type). Spatial filter cutoff frequencies are qcut=0.7q1 for binary, qcut=0.4q1 for unimodal, and qcut=0.5q1 for bimodal nonlinearities. Here and below q1=π3(λL)-1/2. The input intensities are (a), (b) I0=82; (c), (d) I0=111; and (e), (f) I0=120. System parameters are d=0, α=0.2, K=0.1, L=12 mm.

Fig. 8
Fig. 8

Self-organized array of localized states (a) in the system with unimodal nonlinearity in the presence of the external phase modulation (b) [seed pattern]. The amplitude of phase modulation in the seed pattern comprises 8% of the LCTV panel dynamical range. The system parameters are the same as in Figs. 7(c) and 7(d).

Fig. 9
Fig. 9

Pattern formation in the system with unimodal nonlinearity. Intensity (left-hand column) and phase (right-hand column) patterns for different input beam intensity I0 values measured in the CCD camera in video signal gray levels: (a), (b) disordered bright spots, I0=55; (c), (d) web pattern, I0=97; and (e), (f) black holes, I0=140. The system parameters are qcut=0.6q1, L=12 mm, d=0, α=0.2, and K=-0.1.

Fig. 10
Fig. 10

Pattern formation in the system with bimodal nonlinearity. Intensity (left-hand column) and phase (right-hand column) patterns for different input beam intensity I0 values measured in the CCD camera in video signal gray levels: (a), (b) black-eye array, I0=70; (c), (d) chaotic pattern, I0=134; and (e), (f) black-eye localized states, I0=200. The system parameters are qcut=4.0q1, L=12 mm, d=0, α=0.2, and K=-0.1.

Equations (13)

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τν(r, t)t+ν=d2ν+KwFB(r, t),
2=2x2+2y2,
wFB(r, t)=Φ[Id(r, t)]h(r-r)d2r,
Id(r, t)=O[A(r, t)],
A(r, t)=A0(r)exp[iγ ν(r, t)].
τu(r, t) t+u=d2u+nFB Id(r, t),
Id(r, t)=|A(r, L, t)|2,
-2ikA(r, z)z=2A(r, z),
ν(n+1)(r)=(1-α)ν(n)(r)+dΔ(n)ν(n)(r)+KwFB(n)(r),
wFB(n)(r)=Φ[Id(n)(r)]h(r-r)d2r, n=0,1 , ,
Id(r, t)=A(r, z=L, t)hopt(r-r)d2r2,
ν(n+1)(r)=(1-α)ν(n)(r)+dΔ(n)ν(n)(r)+KId(n)(r).
ν(n+1)(r)=(1-α)ν(n)(r)+dΔ(n)ν(n)(r)+KId(n)(rΔ).

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