Abstract

The spontaneous appearance of hexagon patterns in photorefractive crystals is treated as mirrorless coherent four-wave mixing oscillation. It is shown experimentally that the system has a well-defined coupling-strength threshold and behaves as other coherent optical oscillators near the threshold. A simple relation for the angle of the hexagon sideband as a function of the distance between the end mirror and the sample is derived and verified experimentally.

© 1998 Optical Society of America

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  1. T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598–560 (1993).
    [CrossRef]
  2. P. B. Banerjee, H.-L. Yu, D. A. Gregory, N. Kukhtarev, and H. J. Caulfield, “Self-organization of scattering in photorefractive KNbO3 into a reconfigurable hexagonal spot array,” Opt. Lett. 20, 10–12 (1995).
    [CrossRef] [PubMed]
  3. T. Honda, “Flow and controlled rotation of the spontaneous optical hexagons in KNbO3,” Opt. Lett. 20, 851–853 (1995).
    [CrossRef] [PubMed]
  4. T. Honda and H. Matsumoto, “Buildup of spontaneous hexagonal patterns in photorefractive BaTiO3 with a feedback mirror,” Opt. Lett. 20, 1755–1757 (1995).
    [CrossRef]
  5. N. Kukhtarev, N. Kukhtareva, H. J. Caulfield, P. B. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261–2265 (1995).
    [CrossRef]
  6. 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, 293–299 (1997).
    [CrossRef]
  7. A. Mamaev and M. Saffman, “Modulation instability and pattern formation in the field of noncollinear pump waves,” Opt. Lett. 22, 283–285 (1997).
    [CrossRef] [PubMed]
  8. M. Sedlatschek, C. Denz, M. Schwab, B. Thuering, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 379–382.
  9. M. Saffman and A. Mamaev, “Selection of optical patterns by optical filtering,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 555–557.
  10. A. Ueno, M. Kobayashi, and Y. Uesu, “Observation of growth of the hexagon pattern in Co:BaTiO3,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 250–253.
  11. M. Saffman, A. A. Zozulya, and D. Anderson, “Transverse instability of energy-exchanging counterpropagating waves in photorefractive media,” J. Opt. Soc. Am. B 11, 1409–1417 (1994).
    [CrossRef]
  12. B. Sturman and A. Chernykh, “Mechanism of transverse instability of counterpropagation in photorefractive media,” J. Opt. Soc. Am. B 12, 1384–1386 (1995).
    [CrossRef]
  13. T. Honda and P. B. Banerjee, “Threshold for spontaneous pattern formation in reflection-grating dominated photorefractive media with mirror feedback,” Opt. Lett. 21, 779–781 (1996).
    [CrossRef] [PubMed]
  14. A. Chernykh, B. Sturman, M. Aguilar, and F. Agullo-Lopez, “Threshold for pattern formation in a medium with a local photorefractive response,” J. Opt. Soc. Am. B 14, 1754–1760 (1997).
    [CrossRef]
  15. A. Yariv and D. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
    [CrossRef] [PubMed]
  16. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
    [CrossRef]
  17. S. Odoulov, M. Soskin, and A. Khyznyak, Coherent Oscillators with Degenerate Four-Wave Mixing (Harwood, London, 1991).
  18. S. Odoulov, B. Sturman, and M. Goulkov, “Parametric four-wave processes in photorefractive crystals,” Phys. Rep. 275, 197–254 (1997).
  19. V. Lemeshko and V. Obukhovski, “Autowaves of photoinduced light scattering,” Sov. Tech. Phys. Lett. 11, 573–574 (1985).
  20. V. Bazenov, S. Lyuksyutov, R. Jungen, S. Odoulov, and M. Soskin, in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., “Copper-vapour laser with adaptive holographic mirror based on photorefractive crystal,” Proc. SPIE1237, 48–53 (1990).
    [CrossRef]
  21. G. Grynberg, A. Maistre, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72, 2379–2382 (1994).
    [CrossRef] [PubMed]
  22. T. Ackemann, Yu. A. Logvin, A. Heuer, and W. Lange, “Transitions between positive and negative hexagons in optical pattern formation,” Phys. Rev. Lett. 75, 3450–3454 (1995).
    [CrossRef] [PubMed]
  23. M. Goulkov, S. Odoulov, and R. Trott, “Dynamics of coherent oscillation in photorrefractive oscillator with BaTiO3,” Ukr. Fiz. Zh. 36, 402–406 (1991).
  24. A. Zozulya, “Fanning and photorefractive self-pumped four-wave mixing geometries,” IEEE J. Quantum Electron. 29, 538–555 (1993).
    [CrossRef]
  25. D. Engin, S. Orlov, M. Segev, G. Valley, and A. Yariv, “Order-disorder phase transition and critical slowing down in photorefractive self-oscillators,” Phys. Rev. Lett. 74, 1743–1746 (1995).
    [CrossRef] [PubMed]

1997 (4)

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, 293–299 (1997).
[CrossRef]

S. Odoulov, B. Sturman, and M. Goulkov, “Parametric four-wave processes in photorefractive crystals,” Phys. Rep. 275, 197–254 (1997).

A. Chernykh, B. Sturman, M. Aguilar, and F. Agullo-Lopez, “Threshold for pattern formation in a medium with a local photorefractive response,” J. Opt. Soc. Am. B 14, 1754–1760 (1997).
[CrossRef]

A. Mamaev and M. Saffman, “Modulation instability and pattern formation in the field of noncollinear pump waves,” Opt. Lett. 22, 283–285 (1997).
[CrossRef] [PubMed]

1996 (1)

1995 (7)

N. Kukhtarev, N. Kukhtareva, H. J. Caulfield, P. B. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261–2265 (1995).
[CrossRef]

T. Ackemann, Yu. A. Logvin, A. Heuer, and W. Lange, “Transitions between positive and negative hexagons in optical pattern formation,” Phys. Rev. Lett. 75, 3450–3454 (1995).
[CrossRef] [PubMed]

D. Engin, S. Orlov, M. Segev, G. Valley, and A. Yariv, “Order-disorder phase transition and critical slowing down in photorefractive self-oscillators,” Phys. Rev. Lett. 74, 1743–1746 (1995).
[CrossRef] [PubMed]

B. Sturman and A. Chernykh, “Mechanism of transverse instability of counterpropagation in photorefractive media,” J. Opt. Soc. Am. B 12, 1384–1386 (1995).
[CrossRef]

P. B. Banerjee, H.-L. Yu, D. A. Gregory, N. Kukhtarev, and H. J. Caulfield, “Self-organization of scattering in photorefractive KNbO3 into a reconfigurable hexagonal spot array,” Opt. Lett. 20, 10–12 (1995).
[CrossRef] [PubMed]

T. Honda, “Flow and controlled rotation of the spontaneous optical hexagons in KNbO3,” Opt. Lett. 20, 851–853 (1995).
[CrossRef] [PubMed]

T. Honda and H. Matsumoto, “Buildup of spontaneous hexagonal patterns in photorefractive BaTiO3 with a feedback mirror,” Opt. Lett. 20, 1755–1757 (1995).
[CrossRef]

1994 (2)

G. Grynberg, A. Maistre, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72, 2379–2382 (1994).
[CrossRef] [PubMed]

M. Saffman, A. A. Zozulya, and D. Anderson, “Transverse instability of energy-exchanging counterpropagating waves in photorefractive media,” J. Opt. Soc. Am. B 11, 1409–1417 (1994).
[CrossRef]

1993 (2)

A. Zozulya, “Fanning and photorefractive self-pumped four-wave mixing geometries,” IEEE J. Quantum Electron. 29, 538–555 (1993).
[CrossRef]

T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598–560 (1993).
[CrossRef]

1991 (1)

M. Goulkov, S. Odoulov, and R. Trott, “Dynamics of coherent oscillation in photorrefractive oscillator with BaTiO3,” Ukr. Fiz. Zh. 36, 402–406 (1991).

1985 (1)

V. Lemeshko and V. Obukhovski, “Autowaves of photoinduced light scattering,” Sov. Tech. Phys. Lett. 11, 573–574 (1985).

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

1977 (1)

Ackemann, T.

T. Ackemann, Yu. A. Logvin, A. Heuer, and W. Lange, “Transitions between positive and negative hexagons in optical pattern formation,” Phys. Rev. Lett. 75, 3450–3454 (1995).
[CrossRef] [PubMed]

Aguilar, M.

Agullo-Lopez, F.

Anderson, D.

Banerjee, P. B.

Bazenov, V.

V. Bazenov, S. Lyuksyutov, R. Jungen, S. Odoulov, and M. Soskin, in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., “Copper-vapour laser with adaptive holographic mirror based on photorefractive crystal,” Proc. SPIE1237, 48–53 (1990).
[CrossRef]

Caulfield, H. J.

P. B. Banerjee, H.-L. Yu, D. A. Gregory, N. Kukhtarev, and H. J. Caulfield, “Self-organization of scattering in photorefractive KNbO3 into a reconfigurable hexagonal spot array,” Opt. Lett. 20, 10–12 (1995).
[CrossRef] [PubMed]

N. Kukhtarev, N. Kukhtareva, H. J. Caulfield, P. B. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261–2265 (1995).
[CrossRef]

Chernykh, A.

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

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, 293–299 (1997).
[CrossRef]

M. Sedlatschek, C. Denz, M. Schwab, B. Thuering, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 379–382.

Engin, D.

D. Engin, S. Orlov, M. Segev, G. Valley, and A. Yariv, “Order-disorder phase transition and critical slowing down in photorefractive self-oscillators,” Phys. Rev. Lett. 74, 1743–1746 (1995).
[CrossRef] [PubMed]

Fischer, B.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

Goulkov, M.

S. Odoulov, B. Sturman, and M. Goulkov, “Parametric four-wave processes in photorefractive crystals,” Phys. Rep. 275, 197–254 (1997).

M. Goulkov, S. Odoulov, and R. Trott, “Dynamics of coherent oscillation in photorrefractive oscillator with BaTiO3,” Ukr. Fiz. Zh. 36, 402–406 (1991).

Gregory, D. A.

Grynberg, G.

G. Grynberg, A. Maistre, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72, 2379–2382 (1994).
[CrossRef] [PubMed]

Hesselink, L.

N. Kukhtarev, N. Kukhtareva, H. J. Caulfield, P. B. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261–2265 (1995).
[CrossRef]

Heuer, A.

T. Ackemann, Yu. A. Logvin, A. Heuer, and W. Lange, “Transitions between positive and negative hexagons in optical pattern formation,” Phys. Rev. Lett. 75, 3450–3454 (1995).
[CrossRef] [PubMed]

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, 293–299 (1997).
[CrossRef]

T. Honda and P. B. Banerjee, “Threshold for spontaneous pattern formation in reflection-grating dominated photorefractive media with mirror feedback,” Opt. Lett. 21, 779–781 (1996).
[CrossRef] [PubMed]

T. Honda, “Flow and controlled rotation of the spontaneous optical hexagons in KNbO3,” Opt. Lett. 20, 851–853 (1995).
[CrossRef] [PubMed]

T. Honda and H. Matsumoto, “Buildup of spontaneous hexagonal patterns in photorefractive BaTiO3 with a feedback mirror,” Opt. Lett. 20, 1755–1757 (1995).
[CrossRef]

T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598–560 (1993).
[CrossRef]

M. Sedlatschek, C. Denz, M. Schwab, B. Thuering, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 379–382.

Jungen, R.

V. Bazenov, S. Lyuksyutov, R. Jungen, S. Odoulov, and M. Soskin, in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., “Copper-vapour laser with adaptive holographic mirror based on photorefractive crystal,” Proc. SPIE1237, 48–53 (1990).
[CrossRef]

Khyznyak, A.

S. Odoulov, M. Soskin, and A. Khyznyak, Coherent Oscillators with Degenerate Four-Wave Mixing (Harwood, London, 1991).

Kobayashi, M.

A. Ueno, M. Kobayashi, and Y. Uesu, “Observation of growth of the hexagon pattern in Co:BaTiO3,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 250–253.

Kukhtarev, N.

N. Kukhtarev, N. Kukhtareva, H. J. Caulfield, P. B. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261–2265 (1995).
[CrossRef]

P. B. Banerjee, H.-L. Yu, D. A. Gregory, N. Kukhtarev, and H. J. Caulfield, “Self-organization of scattering in photorefractive KNbO3 into a reconfigurable hexagonal spot array,” Opt. Lett. 20, 10–12 (1995).
[CrossRef] [PubMed]

Kukhtareva, N.

N. Kukhtarev, N. Kukhtareva, H. J. Caulfield, P. B. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261–2265 (1995).
[CrossRef]

Lange, W.

T. Ackemann, Yu. A. Logvin, A. Heuer, and W. Lange, “Transitions between positive and negative hexagons in optical pattern formation,” Phys. Rev. Lett. 75, 3450–3454 (1995).
[CrossRef] [PubMed]

Lemeshko, V.

V. Lemeshko and V. Obukhovski, “Autowaves of photoinduced light scattering,” Sov. Tech. Phys. Lett. 11, 573–574 (1985).

Logvin, Yu. A.

T. Ackemann, Yu. A. Logvin, A. Heuer, and W. Lange, “Transitions between positive and negative hexagons in optical pattern formation,” Phys. Rev. Lett. 75, 3450–3454 (1995).
[CrossRef] [PubMed]

Lyuksyutov, S.

V. Bazenov, S. Lyuksyutov, R. Jungen, S. Odoulov, and M. Soskin, in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., “Copper-vapour laser with adaptive holographic mirror based on photorefractive crystal,” Proc. SPIE1237, 48–53 (1990).
[CrossRef]

Maistre, A.

G. Grynberg, A. Maistre, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72, 2379–2382 (1994).
[CrossRef] [PubMed]

Mamaev, A.

A. Mamaev and M. Saffman, “Modulation instability and pattern formation in the field of noncollinear pump waves,” Opt. Lett. 22, 283–285 (1997).
[CrossRef] [PubMed]

M. Saffman and A. Mamaev, “Selection of optical patterns by optical filtering,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 555–557.

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, 293–299 (1997).
[CrossRef]

T. Honda and H. Matsumoto, “Buildup of spontaneous hexagonal patterns in photorefractive BaTiO3 with a feedback mirror,” Opt. Lett. 20, 1755–1757 (1995).
[CrossRef]

Obukhovski, V.

V. Lemeshko and V. Obukhovski, “Autowaves of photoinduced light scattering,” Sov. Tech. Phys. Lett. 11, 573–574 (1985).

Odoulov, S.

S. Odoulov, B. Sturman, and M. Goulkov, “Parametric four-wave processes in photorefractive crystals,” Phys. Rep. 275, 197–254 (1997).

M. Goulkov, S. Odoulov, and R. Trott, “Dynamics of coherent oscillation in photorrefractive oscillator with BaTiO3,” Ukr. Fiz. Zh. 36, 402–406 (1991).

V. Bazenov, S. Lyuksyutov, R. Jungen, S. Odoulov, and M. Soskin, in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., “Copper-vapour laser with adaptive holographic mirror based on photorefractive crystal,” Proc. SPIE1237, 48–53 (1990).
[CrossRef]

S. Odoulov, M. Soskin, and A. Khyznyak, Coherent Oscillators with Degenerate Four-Wave Mixing (Harwood, London, 1991).

Orlov, S.

D. Engin, S. Orlov, M. Segev, G. Valley, and A. Yariv, “Order-disorder phase transition and critical slowing down in photorefractive self-oscillators,” Phys. Rev. Lett. 74, 1743–1746 (1995).
[CrossRef] [PubMed]

Pepper, D.

Petrossian, A.

G. Grynberg, A. Maistre, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72, 2379–2382 (1994).
[CrossRef] [PubMed]

Saffman, M.

Schwab, M.

M. Sedlatschek, C. Denz, M. Schwab, B. Thuering, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 379–382.

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, 293–299 (1997).
[CrossRef]

M. Sedlatschek, C. Denz, M. Schwab, B. Thuering, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 379–382.

Segev, M.

D. Engin, S. Orlov, M. Segev, G. Valley, and A. Yariv, “Order-disorder phase transition and critical slowing down in photorefractive self-oscillators,” Phys. Rev. Lett. 74, 1743–1746 (1995).
[CrossRef] [PubMed]

Soskin, M.

V. Bazenov, S. Lyuksyutov, R. Jungen, S. Odoulov, and M. Soskin, in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., “Copper-vapour laser with adaptive holographic mirror based on photorefractive crystal,” Proc. SPIE1237, 48–53 (1990).
[CrossRef]

S. Odoulov, M. Soskin, and A. Khyznyak, Coherent Oscillators with Degenerate Four-Wave Mixing (Harwood, London, 1991).

Sturman, B.

Thuering, B.

M. Sedlatschek, C. Denz, M. Schwab, B. Thuering, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 379–382.

Trott, R.

M. Goulkov, S. Odoulov, and R. Trott, “Dynamics of coherent oscillation in photorrefractive oscillator with BaTiO3,” Ukr. Fiz. Zh. 36, 402–406 (1991).

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, 293–299 (1997).
[CrossRef]

M. Sedlatschek, C. Denz, M. Schwab, B. Thuering, T. Tschudi, and T. Honda, “Dynamics, symmetries and competition in hexagonal and square pattern formation in a photorefractive single-feedback system,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 379–382.

Ueno, A.

A. Ueno, M. Kobayashi, and Y. Uesu, “Observation of growth of the hexagon pattern in Co:BaTiO3,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 250–253.

Uesu, Y.

A. Ueno, M. Kobayashi, and Y. Uesu, “Observation of growth of the hexagon pattern in Co:BaTiO3,” in Proceedings of Topical Meeting on Photorefractive Effects, Materials and Devices (Waseda University, Tokyo, 1997), pp. 250–253.

Valley, G.

D. Engin, S. Orlov, M. Segev, G. Valley, and A. Yariv, “Order-disorder phase transition and critical slowing down in photorefractive self-oscillators,” Phys. Rev. Lett. 74, 1743–1746 (1995).
[CrossRef] [PubMed]

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

Yariv, A.

D. Engin, S. Orlov, M. Segev, G. Valley, and A. Yariv, “Order-disorder phase transition and critical slowing down in photorefractive self-oscillators,” Phys. Rev. Lett. 74, 1743–1746 (1995).
[CrossRef] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

A. Yariv and D. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
[CrossRef] [PubMed]

Yu, H.-L.

P. B. Banerjee, H.-L. Yu, D. A. Gregory, N. Kukhtarev, and H. J. Caulfield, “Self-organization of scattering in photorefractive KNbO3 into a reconfigurable hexagonal spot array,” Opt. Lett. 20, 10–12 (1995).
[CrossRef] [PubMed]

N. Kukhtarev, N. Kukhtareva, H. J. Caulfield, P. B. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261–2265 (1995).
[CrossRef]

Zozulya, A.

A. Zozulya, “Fanning and photorefractive self-pumped four-wave mixing geometries,” IEEE J. Quantum Electron. 29, 538–555 (1993).
[CrossRef]

Zozulya, A. A.

IEEE J. Quantum Electron. (2)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

A. Zozulya, “Fanning and photorefractive self-pumped four-wave mixing geometries,” IEEE J. Quantum Electron. 29, 538–555 (1993).
[CrossRef]

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

Opt. Commun. (1)

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, 293–299 (1997).
[CrossRef]

Opt. Eng. (1)

N. Kukhtarev, N. Kukhtareva, H. J. Caulfield, P. B. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive KNbO3:Fe,” Opt. Eng. 34, 2261–2265 (1995).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

A, Orientation of the incident and generated light beams in hexagon formation. B, C, Elementary backward-wave four-wave mixing processes, and D, E, diffraction from the principal grating. The wave vectors k are shown as the solid arrows in B–E; the grating vectors are shown by dotted lines.

Fig. 2
Fig. 2

Schematic of the experimental setup: VBA, variable beam attenuator; λ/4, quarter-wave plate; P, O, objective; BS1, BS2, beam splitters; PRC, BaTiO3:Co sample; M, highly reflecting mirror; S, screen. Shutter Sh is used to open the erasing beam. The vectors e and c define the polarization of the pump wave and the orientation of the crystal polar axis, respectively. In some experiments, instead of a screen a high-pass spatial filter and a lens focusing the radiation of six hexagon spots to the photodetector (not shown in the figure) are installed.

Fig. 3
Fig. 3

Temporal development of the sideband intensity. At t=0 the pump waves start to illuminate the sample in which all photorefractive gratings were previously erased by an additional coherent light wave. Dashed line, 10% level from the saturation intensity of the hexagon sideband; Δt, delay time of the oscillation switch-on.

Fig. 4
Fig. 4

Polarization angle of hexagon sidebands ψh versus polarization angle of incident pump wave ψp.

Fig. 5
Fig. 5

A, Hexagon sideband intensity, and B, oscillation switch-on time versus polarization angle ψp of the incident pump wave. Dashed lines are guides for the eye.

Fig. 6
Fig. 6

Angle θ between the hexagon sideband and the pump beam versus the distance L between the sample and the backreflecting mirror. The solid curve is the best fit to relation (2) with N=1.

Equations (2)

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LN=Λ2N/λ,
θ(λN/2L)1/2,

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