Abstract

We experimentally investigate pump beam frequency detuning as a control technique in a photorefractive single-mirror feedback system exhibiting spontaneous self-organized transverse pattern formation with a regime of multistability. Deterministic switching between bistable patterns is demonstrated experimentally. An experimental stability analysis uncovers the effect of frequency detuning on the thresholds of unstable modes. The interaction of independent modes is found to be responsible for the different pattern symmetries observed in this system.

© 2007 Optical Society of America

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  1. M. C. Cross and P. C. Hohenberg, "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112 (1993).
    [CrossRef]
  2. F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
    [CrossRef]
  3. G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
    [CrossRef]
  4. J. Pender and L. Hesselink, "Degenerate conical emissions in atomic-sodium vapor," J. Opt. Soc. Am. B 7, 1361-1373 (1990).
    [CrossRef]
  5. R. Macdonald and H. J. Eichler, "Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror," Opt. Commun. 89, 289-295 (1992).
    [CrossRef]
  6. R. Neubecker, G. L. Oppo, B. Thüring, 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).
    [CrossRef] [PubMed]
  7. J. Glückstad and M. Saffman, "Spontaneous pattern formation in a thin film of bacteriorhodopsin with mixed absorptive-dispersive nonlinearity," Opt. Lett. 20, 551-553 (1995).
    [CrossRef] [PubMed]
  8. T. Honda, "Hexagonal pattern formation due to counterpropagation in KNbO3," Opt. Lett. 18, 598-600 (1993).
    [CrossRef] [PubMed]
  9. J. V. Moloney, "Spontaneous generation of patterns and their control in nonlinear optics," J. Opt. B 1, 183-190 (1999).
    [CrossRef]
  10. R. Martin, A. J. Scroggie, G.-L. Oppo, and W. J. Firth, "Stabilization, selection, and tracking of unstable patterns by Fourier space techniques," Phys. Rev. Lett. 77, 4007-4010 (1996).
    [CrossRef] [PubMed]
  11. A. V. Mamaev and M. Saffman, "Selection of unstable patterns and control of optical turbulence by Fourier plane filtering," Phys. Rev. Lett. 80, 3499-3502 (1998).
    [CrossRef]
  12. L. Pastur, L. Gostiaux, U. Bortolozzo, S. Boccaletti, and P. L. Ramazza, "Experimental targeting and control of spatiotemporal chaos in nonlinear optics," Phys. Rev. Lett. 93, 063902 (2004).
    [CrossRef] [PubMed]
  13. G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, "Elimination of spatiotemporal disorder by Fourier space techniques," Phys. Rev. A 58, 2577-2586 (1998).
    [CrossRef]
  14. M. Schwab, C. Denz, and M. Saffman, "Multiple-pattern stability in a photorefractive feedback system," Appl. Phys. B 69, 429-433 (1999).
    [CrossRef]
  15. J. P. Huignard and A. Marrakchi, "Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals," Opt. Commun. 38, 249-254 (1981).
    [CrossRef]
  16. M. Schwab, C. Denz, A. V. Mamaev, and M. Saffman, "Manipulation of optical patterns by frequency detuning of the pump beams," J. Opt. B 3, 318-327 (2001).
    [CrossRef]
  17. U. Bortolozzo, P. Villoresi, and P. L. Ramazza, "Experimental evidence for detuning induced pattern selection in nonlinear optics," Phys. Rev. Lett. 87, 274102 (2001).
    [CrossRef]
  18. P. Yeh, Introduction to Photorefractive Nonlinear Optics, Series in Pure and Applied Optics (Wiley, 1993).
  19. M. Pesch, E. Große Westhoff, T. Ackemann, and W. Lange, "Direct measurement of multiple instability regions via a Fourier filtering method in an optical pattern forming system," Phys. Rev. E 68, 016209 (2003).
    [CrossRef]
  20. O. Kamps, Ph. Jander, and C. Denz, "Instability threshold of a photorefractive pattern-forming system," Phys. Rev. E 72, 016215 (2005).
    [CrossRef]
  21. P. M. Lushnikov, "Hexagonal optical structures in photorefractive crystals with a feedback mirror," J. Exp. Theor. Phys. 86, 614-627 (1998).
    [CrossRef]

2005

O. Kamps, Ph. Jander, and C. Denz, "Instability threshold of a photorefractive pattern-forming system," Phys. Rev. E 72, 016215 (2005).
[CrossRef]

2004

L. Pastur, L. Gostiaux, U. Bortolozzo, S. Boccaletti, and P. L. Ramazza, "Experimental targeting and control of spatiotemporal chaos in nonlinear optics," Phys. Rev. Lett. 93, 063902 (2004).
[CrossRef] [PubMed]

2003

M. Pesch, E. Große Westhoff, T. Ackemann, and W. Lange, "Direct measurement of multiple instability regions via a Fourier filtering method in an optical pattern forming system," Phys. Rev. E 68, 016209 (2003).
[CrossRef]

2001

M. Schwab, C. Denz, A. V. Mamaev, and M. Saffman, "Manipulation of optical patterns by frequency detuning of the pump beams," J. Opt. B 3, 318-327 (2001).
[CrossRef]

U. Bortolozzo, P. Villoresi, and P. L. Ramazza, "Experimental evidence for detuning induced pattern selection in nonlinear optics," Phys. Rev. Lett. 87, 274102 (2001).
[CrossRef]

1999

M. Schwab, C. Denz, and M. Saffman, "Multiple-pattern stability in a photorefractive feedback system," Appl. Phys. B 69, 429-433 (1999).
[CrossRef]

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

J. V. Moloney, "Spontaneous generation of patterns and their control in nonlinear optics," J. Opt. B 1, 183-190 (1999).
[CrossRef]

1998

G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
[CrossRef]

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, "Elimination of spatiotemporal disorder by Fourier space techniques," Phys. Rev. A 58, 2577-2586 (1998).
[CrossRef]

A. V. Mamaev and M. Saffman, "Selection of unstable patterns and control of optical turbulence by Fourier plane filtering," Phys. Rev. Lett. 80, 3499-3502 (1998).
[CrossRef]

P. M. Lushnikov, "Hexagonal optical structures in photorefractive crystals with a feedback mirror," J. Exp. Theor. Phys. 86, 614-627 (1998).
[CrossRef]

1996

R. Martin, A. J. Scroggie, G.-L. Oppo, and W. J. Firth, "Stabilization, selection, and tracking of unstable patterns by Fourier space techniques," Phys. Rev. Lett. 77, 4007-4010 (1996).
[CrossRef] [PubMed]

1995

R. Neubecker, G. L. Oppo, B. Thüring, 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).
[CrossRef] [PubMed]

J. Glückstad and M. Saffman, "Spontaneous pattern formation in a thin film of bacteriorhodopsin with mixed absorptive-dispersive nonlinearity," Opt. Lett. 20, 551-553 (1995).
[CrossRef] [PubMed]

1993

T. Honda, "Hexagonal pattern formation due to counterpropagation in KNbO3," Opt. Lett. 18, 598-600 (1993).
[CrossRef] [PubMed]

M. C. Cross and P. C. Hohenberg, "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112 (1993).
[CrossRef]

1992

R. Macdonald and H. J. Eichler, "Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror," Opt. Commun. 89, 289-295 (1992).
[CrossRef]

1990

1981

J. P. Huignard and A. Marrakchi, "Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals," Opt. Commun. 38, 249-254 (1981).
[CrossRef]

Ackemann, T.

M. Pesch, E. Große Westhoff, T. Ackemann, and W. Lange, "Direct measurement of multiple instability regions via a Fourier filtering method in an optical pattern forming system," Phys. Rev. E 68, 016209 (2003).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Bloch, D.

G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
[CrossRef]

Boccaletti, S.

L. Pastur, L. Gostiaux, U. Bortolozzo, S. Boccaletti, and P. L. Ramazza, "Experimental targeting and control of spatiotemporal chaos in nonlinear optics," Phys. Rev. Lett. 93, 063902 (2004).
[CrossRef] [PubMed]

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Bortolozzo, U.

L. Pastur, L. Gostiaux, U. Bortolozzo, S. Boccaletti, and P. L. Ramazza, "Experimental targeting and control of spatiotemporal chaos in nonlinear optics," Phys. Rev. Lett. 93, 063902 (2004).
[CrossRef] [PubMed]

U. Bortolozzo, P. Villoresi, and P. L. Ramazza, "Experimental evidence for detuning induced pattern selection in nonlinear optics," Phys. Rev. Lett. 87, 274102 (2001).
[CrossRef]

Cross, M. C.

M. C. Cross and P. C. Hohenberg, "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112 (1993).
[CrossRef]

Denz, C.

O. Kamps, Ph. Jander, and C. Denz, "Instability threshold of a photorefractive pattern-forming system," Phys. Rev. E 72, 016215 (2005).
[CrossRef]

M. Schwab, C. Denz, A. V. Mamaev, and M. Saffman, "Manipulation of optical patterns by frequency detuning of the pump beams," J. Opt. B 3, 318-327 (2001).
[CrossRef]

M. Schwab, C. Denz, and M. Saffman, "Multiple-pattern stability in a photorefractive feedback system," Appl. Phys. B 69, 429-433 (1999).
[CrossRef]

Ducloy, M.

G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
[CrossRef]

Eichler, H. J.

R. Macdonald and H. J. Eichler, "Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror," Opt. Commun. 89, 289-295 (1992).
[CrossRef]

Firth, W. J.

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, "Elimination of spatiotemporal disorder by Fourier space techniques," Phys. Rev. A 58, 2577-2586 (1998).
[CrossRef]

R. Martin, A. J. Scroggie, G.-L. Oppo, and W. J. Firth, "Stabilization, selection, and tracking of unstable patterns by Fourier space techniques," Phys. Rev. Lett. 77, 4007-4010 (1996).
[CrossRef] [PubMed]

Glückstad, J.

Gostiaux, L.

L. Pastur, L. Gostiaux, U. Bortolozzo, S. Boccaletti, and P. L. Ramazza, "Experimental targeting and control of spatiotemporal chaos in nonlinear optics," Phys. Rev. Lett. 93, 063902 (2004).
[CrossRef] [PubMed]

Große Westhoff, E.

M. Pesch, E. Große Westhoff, T. Ackemann, and W. Lange, "Direct measurement of multiple instability regions via a Fourier filtering method in an optical pattern forming system," Phys. Rev. E 68, 016209 (2003).
[CrossRef]

Grynberg, G.

G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
[CrossRef]

Harkness, G. K.

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, "Elimination of spatiotemporal disorder by Fourier space techniques," Phys. Rev. A 58, 2577-2586 (1998).
[CrossRef]

Hesselink, L.

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112 (1993).
[CrossRef]

Honda, T.

Huignard, J. P.

J. P. Huignard and A. Marrakchi, "Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals," Opt. Commun. 38, 249-254 (1981).
[CrossRef]

Jander, Ph.

O. Kamps, Ph. Jander, and C. Denz, "Instability threshold of a photorefractive pattern-forming system," Phys. Rev. E 72, 016215 (2005).
[CrossRef]

Kamps, O.

O. Kamps, Ph. Jander, and C. Denz, "Instability threshold of a photorefractive pattern-forming system," Phys. Rev. E 72, 016215 (2005).
[CrossRef]

Lange, W.

M. Pesch, E. Große Westhoff, T. Ackemann, and W. Lange, "Direct measurement of multiple instability regions via a Fourier filtering method in an optical pattern forming system," Phys. Rev. E 68, 016209 (2003).
[CrossRef]

LeBihan, E.

G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
[CrossRef]

LeBioteux, S.

G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
[CrossRef]

Leite, J. R. R.

G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
[CrossRef]

Lushnikov, P. M.

P. M. Lushnikov, "Hexagonal optical structures in photorefractive crystals with a feedback mirror," J. Exp. Theor. Phys. 86, 614-627 (1998).
[CrossRef]

Macdonald, R.

R. Macdonald and H. J. Eichler, "Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror," Opt. Commun. 89, 289-295 (1992).
[CrossRef]

Mamaev, A. V.

M. Schwab, C. Denz, A. V. Mamaev, and M. Saffman, "Manipulation of optical patterns by frequency detuning of the pump beams," J. Opt. B 3, 318-327 (2001).
[CrossRef]

A. V. Mamaev and M. Saffman, "Selection of unstable patterns and control of optical turbulence by Fourier plane filtering," Phys. Rev. Lett. 80, 3499-3502 (1998).
[CrossRef]

Marrakchi, A.

J. P. Huignard and A. Marrakchi, "Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals," Opt. Commun. 38, 249-254 (1981).
[CrossRef]

Martin, R.

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, "Elimination of spatiotemporal disorder by Fourier space techniques," Phys. Rev. A 58, 2577-2586 (1998).
[CrossRef]

R. Martin, A. J. Scroggie, G.-L. Oppo, and W. J. Firth, "Stabilization, selection, and tracking of unstable patterns by Fourier space techniques," Phys. Rev. Lett. 77, 4007-4010 (1996).
[CrossRef] [PubMed]

Moloney, J. V.

J. V. Moloney, "Spontaneous generation of patterns and their control in nonlinear optics," J. Opt. B 1, 183-190 (1999).
[CrossRef]

Neubecker, R.

R. Neubecker, G. L. Oppo, B. Thüring, 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).
[CrossRef] [PubMed]

Oppo, G. L.

R. Neubecker, G. L. Oppo, B. Thüring, 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).
[CrossRef] [PubMed]

Oppo, G.-L.

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, "Elimination of spatiotemporal disorder by Fourier space techniques," Phys. Rev. A 58, 2577-2586 (1998).
[CrossRef]

R. Martin, A. J. Scroggie, G.-L. Oppo, and W. J. Firth, "Stabilization, selection, and tracking of unstable patterns by Fourier space techniques," Phys. Rev. Lett. 77, 4007-4010 (1996).
[CrossRef] [PubMed]

Pastur, L.

L. Pastur, L. Gostiaux, U. Bortolozzo, S. Boccaletti, and P. L. Ramazza, "Experimental targeting and control of spatiotemporal chaos in nonlinear optics," Phys. Rev. Lett. 93, 063902 (2004).
[CrossRef] [PubMed]

Pender, J.

Pesch, M.

M. Pesch, E. Große Westhoff, T. Ackemann, and W. Lange, "Direct measurement of multiple instability regions via a Fourier filtering method in an optical pattern forming system," Phys. Rev. E 68, 016209 (2003).
[CrossRef]

Ramazza, P.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Ramazza, P. L.

L. Pastur, L. Gostiaux, U. Bortolozzo, S. Boccaletti, and P. L. Ramazza, "Experimental targeting and control of spatiotemporal chaos in nonlinear optics," Phys. Rev. Lett. 93, 063902 (2004).
[CrossRef] [PubMed]

U. Bortolozzo, P. Villoresi, and P. L. Ramazza, "Experimental evidence for detuning induced pattern selection in nonlinear optics," Phys. Rev. Lett. 87, 274102 (2001).
[CrossRef]

Saffman, M.

M. Schwab, C. Denz, A. V. Mamaev, and M. Saffman, "Manipulation of optical patterns by frequency detuning of the pump beams," J. Opt. B 3, 318-327 (2001).
[CrossRef]

M. Schwab, C. Denz, and M. Saffman, "Multiple-pattern stability in a photorefractive feedback system," Appl. Phys. B 69, 429-433 (1999).
[CrossRef]

A. V. Mamaev and M. Saffman, "Selection of unstable patterns and control of optical turbulence by Fourier plane filtering," Phys. Rev. Lett. 80, 3499-3502 (1998).
[CrossRef]

J. Glückstad and M. Saffman, "Spontaneous pattern formation in a thin film of bacteriorhodopsin with mixed absorptive-dispersive nonlinearity," Opt. Lett. 20, 551-553 (1995).
[CrossRef] [PubMed]

Schwab, M.

M. Schwab, C. Denz, A. V. Mamaev, and M. Saffman, "Manipulation of optical patterns by frequency detuning of the pump beams," J. Opt. B 3, 318-327 (2001).
[CrossRef]

M. Schwab, C. Denz, and M. Saffman, "Multiple-pattern stability in a photorefractive feedback system," Appl. Phys. B 69, 429-433 (1999).
[CrossRef]

Scroggie, A. J.

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, "Elimination of spatiotemporal disorder by Fourier space techniques," Phys. Rev. A 58, 2577-2586 (1998).
[CrossRef]

R. Martin, A. J. Scroggie, G.-L. Oppo, and W. J. Firth, "Stabilization, selection, and tracking of unstable patterns by Fourier space techniques," Phys. Rev. Lett. 77, 4007-4010 (1996).
[CrossRef] [PubMed]

Simoneau, P.

G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
[CrossRef]

Thüring, B.

R. Neubecker, G. L. Oppo, B. Thüring, 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).
[CrossRef] [PubMed]

Tschudi, T.

R. Neubecker, G. L. Oppo, B. Thüring, 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).
[CrossRef] [PubMed]

Verkerk, P.

G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
[CrossRef]

Villoresi, P.

U. Bortolozzo, P. Villoresi, and P. L. Ramazza, "Experimental evidence for detuning induced pattern selection in nonlinear optics," Phys. Rev. Lett. 87, 274102 (2001).
[CrossRef]

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics, Series in Pure and Applied Optics (Wiley, 1993).

Appl. Phys. B

M. Schwab, C. Denz, and M. Saffman, "Multiple-pattern stability in a photorefractive feedback system," Appl. Phys. B 69, 429-433 (1999).
[CrossRef]

J. Exp. Theor. Phys.

P. M. Lushnikov, "Hexagonal optical structures in photorefractive crystals with a feedback mirror," J. Exp. Theor. Phys. 86, 614-627 (1998).
[CrossRef]

J. Opt. B

M. Schwab, C. Denz, A. V. Mamaev, and M. Saffman, "Manipulation of optical patterns by frequency detuning of the pump beams," J. Opt. B 3, 318-327 (2001).
[CrossRef]

J. V. Moloney, "Spontaneous generation of patterns and their control in nonlinear optics," J. Opt. B 1, 183-190 (1999).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

R. Macdonald and H. J. Eichler, "Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror," Opt. Commun. 89, 289-295 (1992).
[CrossRef]

J. P. Huignard and A. Marrakchi, "Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals," Opt. Commun. 38, 249-254 (1981).
[CrossRef]

G. Grynberg, E. LeBihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. LeBioteux, and M. Ducloy, "Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium," Opt. Commun. 67, 363-366 (1998).
[CrossRef]

Opt. Lett.

Phys. Rep.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Phys. Rev. A

R. Neubecker, G. L. Oppo, B. Thüring, 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).
[CrossRef] [PubMed]

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, "Elimination of spatiotemporal disorder by Fourier space techniques," Phys. Rev. A 58, 2577-2586 (1998).
[CrossRef]

Phys. Rev. E

M. Pesch, E. Große Westhoff, T. Ackemann, and W. Lange, "Direct measurement of multiple instability regions via a Fourier filtering method in an optical pattern forming system," Phys. Rev. E 68, 016209 (2003).
[CrossRef]

O. Kamps, Ph. Jander, and C. Denz, "Instability threshold of a photorefractive pattern-forming system," Phys. Rev. E 72, 016215 (2005).
[CrossRef]

Phys. Rev. Lett.

U. Bortolozzo, P. Villoresi, and P. L. Ramazza, "Experimental evidence for detuning induced pattern selection in nonlinear optics," Phys. Rev. Lett. 87, 274102 (2001).
[CrossRef]

R. Martin, A. J. Scroggie, G.-L. Oppo, and W. J. Firth, "Stabilization, selection, and tracking of unstable patterns by Fourier space techniques," Phys. Rev. Lett. 77, 4007-4010 (1996).
[CrossRef] [PubMed]

A. V. Mamaev and M. Saffman, "Selection of unstable patterns and control of optical turbulence by Fourier plane filtering," Phys. Rev. Lett. 80, 3499-3502 (1998).
[CrossRef]

L. Pastur, L. Gostiaux, U. Bortolozzo, S. Boccaletti, and P. L. Ramazza, "Experimental targeting and control of spatiotemporal chaos in nonlinear optics," Phys. Rev. Lett. 93, 063902 (2004).
[CrossRef] [PubMed]

Rev. Mod. Phys.

M. C. Cross and P. C. Hohenberg, "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112 (1993).
[CrossRef]

Other

P. Yeh, Introduction to Photorefractive Nonlinear Optics, Series in Pure and Applied Optics (Wiley, 1993).

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Figures (8)

Fig. 1
Fig. 1

Experimental setup for investigation transverse pattern formation. L, lenses; (p)BS, (polarizing) beam splitter; M1, mirror; M2, piezomounted mirrorr; vm, virtual mirror plane; S, shutter; P, pinhole; F, Fourier plane of the feedback arm.

Fig. 2
Fig. 2

Far-field patterns observed without control. Central in each image is the output pump beam while the lateral spots correspond to the modes constituting the transverse modulation of the actual near-field beams. (a) Common hexagonal pattern, (b) square pattern, (c) squeezed hexagonal pattern.

Fig. 3
Fig. 3

Relative stability of two particular patterns, measured by their frequency to be generated from noise, depending on the location of the feedback mirror. Open triangles, squeezed hexagonal pattern; filled triangles, square pattern. Exemplary error bar gives the expected statistical error. Squeezed hexagon is practically always observed at mirror positions around d = 0.75 . Both patterns have about equal probability of being generated at d = 0.67 .

Fig. 4
Fig. 4

Pattern transition induced by negative frequency detuning. Azimuth angle of the camera image (right) is projected onto the Cartesian x axis while the temporal evolution is plotted on the y axis. After activation of the control signal, the square pattern changes into a squeezed hexagon via a transient state in which the old and new pattern share a single mode.

Fig. 5
Fig. 5

System state curves tracking the response to an oscillating control signal. Intensity of far-field spots characteristic of two pattern classes is filtered by masks (a), (b) in the transverse Fourier space and detected. Intensities of two patterns, (c) squeezed hexagon, (d) square, during multiple transitions induced by a triangular control signal. Pattern selected by large control signals remains as the control diminished, showing hysteretic behavior and indicating a symmetric bistability. Paths seemingly crossing through the center of the hystereses stem from slightly sheared square patterns as well as occasional lack of one pair of the square pattern spots. As a result of some spots lying outside of the discrimination window or having lower than normal intensity, only half of the intensity is registered.

Fig. 6
Fig. 6

Hystereses for several feedback mirror positions: (a) d = 0.66 , (b) d = 0.68 , (c) d = 0.70 , (d) d = 0.72 . In contrast to Fig. 5, the y axis now denotes the difference of squeezed hexagon and square pattern intensities, yielding a direct measure for the actual system state. As the squeezed hexagonal pattern grows in stability, the hysteresis is moved to farther positive control signal values and finally no square pattern is observable without control signal. Paths seemingly crossing the hystereses’ centers again stem from variations in the square pattern (compare Fig. 5).

Fig. 7
Fig. 7

Center of gravity of the hystereses as a function of the feedback mirror position (normalized diffraction length), which changes the relative stability of the two patterns involved.

Fig. 8
Fig. 8

Experimental threshold curves for zero, positive, and negative control signals at the squeezed hexagon–square pattern bistability. A positive control signal inhibits most modes except for the one at k d l = 7.5 . A negative control signal weakens the broad mode around k d l = 7.5 but leaves the mode at k d l = 3 unchanged, possibly slightly strengthened. Lines are drawn as a guide to the eye. Quantitative coupling strength values are derived from two-beam coupling experiments (maximum value) and the beam polarization angle (effective electro-optic coefficient).

Equations (2)

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d = n 0 L l ,
k d l = k 0 θ 2 2 n 0 l

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