Abstract

We investigate the temporal dynamics of transverse optical patterns spontaneously formed in a photorefractive single-feedback system with a virtual feedback mirror. The linear stability analysis for the system is reviewed and extended to the region of larger propagation lengths. The stationary patterns obtained experimentally are classified as a function of feedback reflectivity and feedback mirror position. Inserting masks into the feedback path permits pattern selection and control by Fourier filtering. When an asymmetry that is due to noncollinear pump beams is introduced, the otherwise stationary hexagons show several complex but periodic rotationlike motions. Furthermore, the competition of hexagonal and square patterns can be observed by the appropriate choice of feedback mirror position and coupling strength. The origin of this behavior is discussed. The temporal evolution of the patterns is illustrated by a method based on unfolding the angular distribution of the spots in the far field.

© 1998 Optical Society of America

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  1. J. Pender and L. Hesselink, “Degenerate conical emissions in atomic-sodium vapor,” J. Opt. Soc. Am. B 7, 1361 (1990).
    [Crossref]
  2. A. Petrossian, M. Pinard, A. Maı̂tre, J. Y. Courtois, and G. Grynberg, “Transverse pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689 (1992).
    [Crossref]
  3. R. Macdonald and H. J. Eichler, “Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror,” Opt. Commun. 89, 289 (1992).
    [Crossref]
  4. M. Tamburrini, M. Bonavita, S. Wabnitz, and E. Santamato, “Hexagonally patterned beam filamentation in a thin liquid-crystal film with single feedback mirror,” Opt. Lett. 18, 855 (1993).
    [Crossref]
  5. B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in an LCLV feedback system,” Opt. Commun. 102, 111 (1993).
    [Crossref]
  6. T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598 (1993).
    [Crossref]
  7. J. Glückstad and M. Saffman, “Spontaneous pattern formation in a thin film of bacteriorhodopsin with mixed absorptive dispersive nonlinearity,” Opt. Lett. 20, 551 (1995).
    [Crossref]
  8. M. A. Vorontsov and W. J. Firth, “Pattern formation and competition in nonlinear optical systems with two-dimensional feedback,” Phys. Rev. A 49, 2891 (1994).
    [Crossref] [PubMed]
  9. T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, and T. Tschudi, “Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror,” Opt. Commun. 133, 293 (1997).
    [Crossref]
  10. T. Honda, “Flow and controlled rotation of spontaneous optical hexagon in KNbO3,” Opt. Lett. 20, 851 (1995).
    [Crossref] [PubMed]
  11. A. V. Mamaev and M. Saffman, “Modulational instability and pattern formation in the field of noncollinear pump beams,” Opt. Lett. 22, 283 (1997).
    [Crossref] [PubMed]
  12. T. Honda and P. P. Banerjee, “Threshold for spontaneous pattern formation in reflection-grating-dominated photorefractive media with mirror feedback,” Opt. Lett. 21, 779 (1996).
    [Crossref] [PubMed]
  13. M. Saffman, A. A. Zozulya, and D. Z. Anderson, “Transverse instability of energy-exchanging counterpropagating waves in photorefractive media,” J. Opt. Soc. Am. B 11, 1409 (1994).
    [Crossref]
  14. N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H. L. Yu, and L. Hesselink, “Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261 (1995).
    [Crossref]
  15. A. I. Chernykh, B. I. Sturman, M. Aguilar, and F. Agulló-López, “Threshold for pattern formation in a medium with a local photorefractive response,” J. Opt. Soc. Am. B 14, 1754 (1997).
    [Crossref]
  16. E. V. Degtiarev and M. A. Vorontsov, “Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression,” J. Opt. Soc. Am. B 12, 1238 (1995).
    [Crossref]
  17. 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 (1996).
    [Crossref] [PubMed]
  18. R. Martin, G.-L. Oppo, G. K. Harkness, A. J. Scroggie, and W. J. Firth, “Controlling pattern formation and spatio-temporal disorder in nonlinear optics,” Opt. Expr. 1, 39 (1997).
    [Crossref]
  19. A. V. Mamaev and M. Saffman, “Selection of optical patterns by Fourier filtering,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.
  20. B. Thüring, A. Schreiber, M. Kreuzer, and T. Tschudi, “Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system,” Physica D 96, 282 (1996).
    [Crossref]
  21. A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
    [Crossref]
  22. M. Sedlatschek, C. Denz, M. Schwab, B. Thüring, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.
  23. O. Sandfuchs, J. Leonardy, F. Kaiser, and M. R. Belić, “Transverse instabilities in photorefractive counterpropagating two-wave mixing,” Opt. Lett. 22, 498 (1997).
    [Crossref] [PubMed]
  24. O. Sandfuchs, F. Kaiser, and M. R. Belić, “Spatiotemporal pattern formation in counterpropagating two-wave mixing with an externally applied field,” J. Opt. Soc. Am. B 15, 2070 (1998).
    [Crossref]

1998 (1)

1997 (5)

O. Sandfuchs, J. Leonardy, F. Kaiser, and M. R. Belić, “Transverse instabilities in photorefractive counterpropagating two-wave mixing,” Opt. Lett. 22, 498 (1997).
[Crossref] [PubMed]

T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, and T. Tschudi, “Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror,” Opt. Commun. 133, 293 (1997).
[Crossref]

A. V. Mamaev and M. Saffman, “Modulational instability and pattern formation in the field of noncollinear pump beams,” Opt. Lett. 22, 283 (1997).
[Crossref] [PubMed]

A. I. Chernykh, B. I. Sturman, M. Aguilar, and F. Agulló-López, “Threshold for pattern formation in a medium with a local photorefractive response,” J. Opt. Soc. Am. B 14, 1754 (1997).
[Crossref]

R. Martin, G.-L. Oppo, G. K. Harkness, A. J. Scroggie, and W. J. Firth, “Controlling pattern formation and spatio-temporal disorder in nonlinear optics,” Opt. Expr. 1, 39 (1997).
[Crossref]

1996 (3)

B. Thüring, A. Schreiber, M. Kreuzer, and T. Tschudi, “Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system,” Physica D 96, 282 (1996).
[Crossref]

T. Honda and P. P. Banerjee, “Threshold for spontaneous pattern formation in reflection-grating-dominated photorefractive media with mirror feedback,” Opt. Lett. 21, 779 (1996).
[Crossref] [PubMed]

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 (1996).
[Crossref] [PubMed]

1995 (5)

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H. L. Yu, and L. Hesselink, “Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261 (1995).
[Crossref]

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
[Crossref]

E. V. Degtiarev and M. A. Vorontsov, “Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression,” J. Opt. Soc. Am. B 12, 1238 (1995).
[Crossref]

T. Honda, “Flow and controlled rotation of spontaneous optical hexagon in KNbO3,” Opt. Lett. 20, 851 (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 (1995).
[Crossref]

1994 (2)

M. A. Vorontsov and W. J. Firth, “Pattern formation and competition in nonlinear optical systems with two-dimensional feedback,” Phys. Rev. A 49, 2891 (1994).
[Crossref] [PubMed]

M. Saffman, A. A. Zozulya, and D. Z. Anderson, “Transverse instability of energy-exchanging counterpropagating waves in photorefractive media,” J. Opt. Soc. Am. B 11, 1409 (1994).
[Crossref]

1993 (3)

1992 (2)

A. Petrossian, M. Pinard, A. Maı̂tre, J. Y. Courtois, and G. Grynberg, “Transverse pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689 (1992).
[Crossref]

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

1990 (1)

Aguilar, M.

Agulló-López, F.

Anderson, D. Z.

Banerjee, P. P.

T. Honda and P. P. Banerjee, “Threshold for spontaneous pattern formation in reflection-grating-dominated photorefractive media with mirror feedback,” Opt. Lett. 21, 779 (1996).
[Crossref] [PubMed]

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H. L. Yu, and L. Hesselink, “Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261 (1995).
[Crossref]

Belic, M. R.

Bonavita, M.

Caulfield, H. J.

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H. L. Yu, and L. Hesselink, “Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261 (1995).
[Crossref]

Chernykh, A. I.

Courtois, J. Y.

A. Petrossian, M. Pinard, A. Maı̂tre, J. Y. Courtois, and G. Grynberg, “Transverse pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689 (1992).
[Crossref]

Dambly, L.

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
[Crossref]

Degtiarev, E. V.

Denz, C.

T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, and T. Tschudi, “Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror,” Opt. Commun. 133, 293 (1997).
[Crossref]

M. Sedlatschek, C. Denz, M. Schwab, B. Thüring, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.

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 (1992).
[Crossref]

Firth, W. J.

R. Martin, G.-L. Oppo, G. K. Harkness, A. J. Scroggie, and W. J. Firth, “Controlling pattern formation and spatio-temporal disorder in nonlinear optics,” Opt. Expr. 1, 39 (1997).
[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 (1996).
[Crossref] [PubMed]

M. A. Vorontsov and W. J. Firth, “Pattern formation and competition in nonlinear optical systems with two-dimensional feedback,” Phys. Rev. A 49, 2891 (1994).
[Crossref] [PubMed]

Glückstad, J.

Grynberg, G.

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
[Crossref]

A. Petrossian, M. Pinard, A. Maı̂tre, J. Y. Courtois, and G. Grynberg, “Transverse pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689 (1992).
[Crossref]

Harkness, G. K.

R. Martin, G.-L. Oppo, G. K. Harkness, A. J. Scroggie, and W. J. Firth, “Controlling pattern formation and spatio-temporal disorder in nonlinear optics,” Opt. Expr. 1, 39 (1997).
[Crossref]

Hesselink, L.

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H. L. Yu, and L. Hesselink, “Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261 (1995).
[Crossref]

J. Pender and L. Hesselink, “Degenerate conical emissions in atomic-sodium vapor,” J. Opt. Soc. Am. B 7, 1361 (1990).
[Crossref]

Honda, T.

T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, and T. Tschudi, “Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror,” Opt. Commun. 133, 293 (1997).
[Crossref]

T. Honda and P. P. Banerjee, “Threshold for spontaneous pattern formation in reflection-grating-dominated photorefractive media with mirror feedback,” Opt. Lett. 21, 779 (1996).
[Crossref] [PubMed]

T. Honda, “Flow and controlled rotation of spontaneous optical hexagon in KNbO3,” Opt. Lett. 20, 851 (1995).
[Crossref] [PubMed]

T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598 (1993).
[Crossref]

M. Sedlatschek, C. Denz, M. Schwab, B. Thüring, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.

Kaiser, F.

Kreuzer, M.

B. Thüring, A. Schreiber, M. Kreuzer, and T. Tschudi, “Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system,” Physica D 96, 282 (1996).
[Crossref]

Kukhtarev, N. V.

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H. L. Yu, and L. Hesselink, “Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261 (1995).
[Crossref]

Kukhtareva, T.

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H. L. Yu, and L. Hesselink, “Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261 (1995).
[Crossref]

Leonardy, J.

Macdonald, R.

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

Mai^tre, A.

A. Petrossian, M. Pinard, A. Maı̂tre, J. Y. Courtois, and G. Grynberg, “Transverse pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689 (1992).
[Crossref]

Mamaev, A. V.

A. V. Mamaev and M. Saffman, “Modulational instability and pattern formation in the field of noncollinear pump beams,” Opt. Lett. 22, 283 (1997).
[Crossref] [PubMed]

A. V. Mamaev and M. Saffman, “Selection of optical patterns by Fourier filtering,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.

Martin, R.

R. Martin, G.-L. Oppo, G. K. Harkness, A. J. Scroggie, and W. J. Firth, “Controlling pattern formation and spatio-temporal disorder in nonlinear optics,” Opt. Expr. 1, 39 (1997).
[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 (1996).
[Crossref] [PubMed]

Matsumoto, H.

T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, and T. Tschudi, “Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror,” Opt. Commun. 133, 293 (1997).
[Crossref]

Neubecker, R.

B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in an LCLV feedback system,” Opt. Commun. 102, 111 (1993).
[Crossref]

Oppo, G.-L.

R. Martin, G.-L. Oppo, G. K. Harkness, A. J. Scroggie, and W. J. Firth, “Controlling pattern formation and spatio-temporal disorder in nonlinear optics,” Opt. Expr. 1, 39 (1997).
[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 (1996).
[Crossref] [PubMed]

Pender, J.

Petrossian, A.

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
[Crossref]

A. Petrossian, M. Pinard, A. Maı̂tre, J. Y. Courtois, and G. Grynberg, “Transverse pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689 (1992).
[Crossref]

Pinard, M.

A. Petrossian, M. Pinard, A. Maı̂tre, J. Y. Courtois, and G. Grynberg, “Transverse pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689 (1992).
[Crossref]

Saffman, M.

Sandfuchs, O.

Santamato, E.

Schreiber, A.

B. Thüring, A. Schreiber, M. Kreuzer, and T. Tschudi, “Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system,” Physica D 96, 282 (1996).
[Crossref]

Schwab, M.

M. Sedlatschek, C. Denz, M. Schwab, B. Thüring, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.

Scroggie, A. J.

R. Martin, G.-L. Oppo, G. K. Harkness, A. J. Scroggie, and W. J. Firth, “Controlling pattern formation and spatio-temporal disorder in nonlinear optics,” Opt. Expr. 1, 39 (1997).
[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 (1996).
[Crossref] [PubMed]

Sedlatschek, M.

T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, and T. Tschudi, “Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror,” Opt. Commun. 133, 293 (1997).
[Crossref]

M. Sedlatschek, C. Denz, M. Schwab, B. Thüring, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.

Sturman, B. I.

Tamburrini, M.

Thüring, B.

B. Thüring, A. Schreiber, M. Kreuzer, and T. Tschudi, “Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system,” Physica D 96, 282 (1996).
[Crossref]

B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in an LCLV feedback system,” Opt. Commun. 102, 111 (1993).
[Crossref]

M. Sedlatschek, C. Denz, M. Schwab, B. Thüring, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.

Tschudi, T.

T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, and T. Tschudi, “Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror,” Opt. Commun. 133, 293 (1997).
[Crossref]

B. Thüring, A. Schreiber, M. Kreuzer, and T. Tschudi, “Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system,” Physica D 96, 282 (1996).
[Crossref]

B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in an LCLV feedback system,” Opt. Commun. 102, 111 (1993).
[Crossref]

M. Sedlatschek, C. Denz, M. Schwab, B. Thüring, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.

Vorontsov, M. A.

E. V. Degtiarev and M. A. Vorontsov, “Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression,” J. Opt. Soc. Am. B 12, 1238 (1995).
[Crossref]

M. A. Vorontsov and W. J. Firth, “Pattern formation and competition in nonlinear optical systems with two-dimensional feedback,” Phys. Rev. A 49, 2891 (1994).
[Crossref] [PubMed]

Wabnitz, S.

Yu, H. L.

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H. L. Yu, and L. Hesselink, “Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261 (1995).
[Crossref]

Zozulya, A. A.

Europhys. Lett. (2)

A. Petrossian, M. Pinard, A. Maı̂tre, J. Y. Courtois, and G. Grynberg, “Transverse pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689 (1992).
[Crossref]

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
[Crossref]

J. Opt. Soc. Am. B (5)

Opt. Commun. (3)

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

B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in an LCLV feedback system,” Opt. Commun. 102, 111 (1993).
[Crossref]

T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, and T. Tschudi, “Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror,” Opt. Commun. 133, 293 (1997).
[Crossref]

Opt. Eng. (1)

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H. L. Yu, and L. Hesselink, “Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261 (1995).
[Crossref]

Opt. Expr. (1)

R. Martin, G.-L. Oppo, G. K. Harkness, A. J. Scroggie, and W. J. Firth, “Controlling pattern formation and spatio-temporal disorder in nonlinear optics,” Opt. Expr. 1, 39 (1997).
[Crossref]

Opt. Lett. (7)

Phys. Rev. A (1)

M. A. Vorontsov and W. J. Firth, “Pattern formation and competition in nonlinear optical systems with two-dimensional feedback,” Phys. Rev. A 49, 2891 (1994).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

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 (1996).
[Crossref] [PubMed]

Physica D (1)

B. Thüring, A. Schreiber, M. Kreuzer, and T. Tschudi, “Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system,” Physica D 96, 282 (1996).
[Crossref]

Other (2)

M. Sedlatschek, C. Denz, M. Schwab, B. Thüring, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.

A. V. Mamaev and M. Saffman, “Selection of optical patterns by Fourier filtering,” presented at the Topical Meeting on Photorefractive Materials, Effects and Devices, Cosponsored by the Optical Society of Japan and the Optical Society of America, June 11–13, 1997, Chiba, Japan.

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Figures (13)

Fig. 1
Fig. 1

Principle of the interaction geometry: SB’s, spatial sidebands; M, mirror; v.M., virtual mirror; L, propagation length; l, crystal length; θ’s, sideband angles; p.r., photorefractive.

Fig. 2
Fig. 2

Sideband angle θ as a function of normalized virtual mirror position n0L/l according to the linear stability analysis.

Fig. 3
Fig. 3

Threshold curve for n0L/l=1.4, indicating the leap in the sideband angle. The minimum of the threshold curve is passed over from the first to the second branch, resulting in a leap of Δθ of the sideband angle.

Fig. 4
Fig. 4

Sideband angle θ as a function of virtual mirror position for the case of a virtual mirror position inside the crystal. Filled circles, experimental points; solid curve, theoretical curve. Inset, experimentally obtained hexagonal pattern in the far field, including second- and third-order spots.

Fig. 5
Fig. 5

Minima of threshold curves as a function of virtual mirror position for the parameter region -1n0L/l1. Selected parameter values are n0L/l=-1, 0 (point A), n0L/l=-0.7, -0.3 (point B), n0L/l=-0.5 (point C), and n0L/l=1 (point D). The largest observable sideband angle θmax is indicated, corresponding to a coupling strength of γl5.5.

Fig. 6
Fig. 6

Experimental setup: OD, optical diode; L’s, lenses; M’s, mirrors; BS’s, beam splitters; MLS, microscope lens system.

Fig. 7
Fig. 7

Dependence of pattern type on feedback reflectivity and mirror position. I, no observable pattern; II, weak hexagonal pattern; III, pattern without geometric symmetry; IV, washed-out hexagonal pattern with emphasis on two spots opposite each other; V, static hexagonal pattern.

Fig. 8
Fig. 8

Method for illustrating pattern dynamics based on the unfolding of the angular spectrum of the far-field pattern.

Fig. 9
Fig. 9

Rock ’n’ roll motion. Two spots, opposite each other, are nearly stable; the other four spots exhibit a rotation and a fast leap back to the initial position.

Fig. 10
Fig. 10

Frequency of the rock ’n’ roll motion as a function of the tilt angle (full angle between the incoming and the feedback beam outside the crystal).

Fig. 11
Fig. 11

Rocking motion with two time scales. All six spots show a rotation and a fast leap back on a long time scale, whereas on a shorter time scale the spots oscillate periodically.  

Fig. 12
Fig. 12

Experimentally observed far-field pattern of (left) hexagonal and (right) square symmetry. For strong coupling of γl11.5 and an appropriate choice of the virtual mirror position (e.g., n0L/l-0.25), the patterns alternate in time.

Fig. 13
Fig. 13

Angular distribution of spots during competition of hexagons and squares. Top, radius k=k1; bottom, radius k=2k1.

Equations (9)

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Fz-i2k0n0 2F=iγ |B|2|F|2+|B|2 F,
Bz+i2k0n0 2B=-iγ* |F|2|F|2+|B|2 B,
F(r)=F0(z)[1+F+1(z)exp(ikr)+F-1(z)exp(-ikr)],
B(r)=B0(z)[1+B+1(z)exp(ikr)+B-1(z)exp(-ikr)],
F±1(0)=0,
B±1(l)=exp(2ikdn0L)F±1(l),
cos wl cos kdl+γI2w sin wl cos kd(l+2n0L)
+γR+2kd2w sin wl sin kdl
-γR2w sin wl sin kd(l+2n0L)=0,

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