Abstract

We report theoretical and experimental evidence for transverse modulational instability of two counterpropagating beams in a photorefractive medium with no external feedback. A frequency detuning is applied to one of the beams in order to drive the system to instability. We perform a linear-stability analysis that allows for detuning of the counterpropagating pump beams in addition to an additional frequency detuning of the generated sidebands relative to the main beams. The threshold condition for the general case of a complex photorefractive coupling constant is found, and instability is predicted for diffusion-dominated, drift-dominated, and mixed charge transport. We show that for the specific case of diffusion-dominated charge transport, transverse instability is always accompanied by a frequency shift of the sidebands. For frequency-degenerate pump beams the instability threshold is reached at a coupling-constant times interaction-length product of γl=5.25i. The threshold is lowered (raised) for small positive (negative) frequency shifts of one of the pump beams. The theoretical predictions were verified experimentally with a photorefractive crystal of KNbO3. A modulational instability resulting in a spatially periodic roll pattern was observed for a certain range of positive frequency detunings. Measurements of the transverse scale of the structures and the relative sideband intensities were in agreement with the theoretical analysis.

© 2001 Optical Society of America

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  13. J. Glückstad and M. Saffman, “Spontaneous pattern formation in a thin film of bacteriorhodopsin with mixed absorptive-dispersive nonlinearity,” Opt. Lett. 20, 551–551 (1995).
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  17. A. V. Mamaev and M. Saffman, “Hexagonal optical patterns in anisotropic nonlinear media,” Europhys. Lett. 34, 669–674 (1996).
    [CrossRef]
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    [CrossRef]
  19. C. Denz, M. Schwab, M. Sedlatschek, T. Tschudi, and T. Honda, “Pattern dynamics and competition in a photorefractive feedback system,” J. Opt. Soc. Am. B 15, 2057–2064 (1998).
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    [CrossRef]
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1999 (3)

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

M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, and T. Tschudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos, Solitons Fractals 10, 701–707 (1999).
[CrossRef]

P. M. Lushnikov and A. V. Mamaev, “Spontaneous hexagon formation in photorefractive crystal with a single pump wave,” Opt. Lett. 24, 1511–1513 (1999).
[CrossRef]

1998 (2)

1997 (3)

1996 (2)

1995 (2)

1994 (2)

R. Neubecker, B. Thüring, and T. Tschudi, “Formation and characterization of hexagonal patterns in a single feedback experiment,” Chaos, Solitons Fractals 4, 1307–1322 (1994).
[CrossRef]

M. Saffman, A. A. Zozulya, and D. Z. Anderson, “Trans-verse instability of energy exchanging counterpropagating waves in photorefractive media,” J. Opt. Soc. Am. B 14, 1754–1760 (1994).

1993 (5)

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[CrossRef] [PubMed]

T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598–600 (1993).
[CrossRef]

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1113 (1993).
[CrossRef]

M. Tamburrini, M. Bonavita, S. Wabnitz, and E. Santamato, “Hexagonally patterned beam filamentation in a thin liquid-crystal film with a single feedback mirror,” Opt. Lett. 18, 855–857 (1993).
[CrossRef] [PubMed]

B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in liquid crystal light valve feedback system,” Opt. Commun. 102, 111–115 (1993).
[CrossRef]

1992 (2)

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]

A. Petrossian, M. Pinard, A. Mai⁁tre, J.-Y. Courtois, and G. Grynberg, “Transverse-pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689–695 (1992).
[CrossRef]

1990 (2)

1989 (1)

C. Denz, J. Goltz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
[CrossRef]

1988 (1)

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteaux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[CrossRef]

1985 (1)

K. R. MacDonald and J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in photorefractive crystals I+II,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Aguilar, M.

Agulló-López, F.

Anderson, D. Z.

M. Saffman, A. A. Zozulya, and D. Z. Anderson, “Trans-verse instability of energy exchanging counterpropagating waves in photorefractive media,” J. Opt. Soc. Am. B 14, 1754–1760 (1994).

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[CrossRef] [PubMed]

Banerjee, P. P.

Belic, M. R.

Bloch, D.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteaux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[CrossRef]

Bonavita, M.

Chernykh, A. I.

Courtois, J.-Y.

A. Petrossian, M. Pinard, A. Mai⁁tre, J.-Y. Courtois, and G. Grynberg, “Transverse-pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689–695 (1992).
[CrossRef]

Cross, M. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1113 (1993).
[CrossRef]

Denz, C.

M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, and T. Tschudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos, Solitons Fractals 10, 701–707 (1999).
[CrossRef]

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

C. Denz, M. Schwab, M. Sedlatschek, T. Tschudi, and T. Honda, “Pattern dynamics and competition in a photorefractive feedback system,” J. Opt. Soc. Am. B 15, 2057–2064 (1998).
[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, 293–299 (1997).
[CrossRef]

C. Denz, J. Goltz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
[CrossRef]

Ducloy, M.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteaux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[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]

Feinberg, J.

K. R. MacDonald and J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

Firth, W. J.

Fitzgerald, A.

Glückstad, J.

Goltz, J.

C. Denz, J. Goltz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
[CrossRef]

Grynberg, G.

A. Petrossian, M. Pinard, A. Mai⁁tre, J.-Y. Courtois, and G. Grynberg, “Transverse-pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689–695 (1992).
[CrossRef]

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteaux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[CrossRef]

Hesselink, L.

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1113 (1993).
[CrossRef]

Honda, T.

Kaiser, F.

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in photorefractive crystals I+II,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Kuroda, K.

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[CrossRef] [PubMed]

Le Bihan, E.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteaux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[CrossRef]

Le Boiteaux, S.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteaux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[CrossRef]

Leite, J. R. R.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteaux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[CrossRef]

Leonardy, J.

Lushnikov, P. M.

MacDonald, K. R.

K. R. MacDonald and J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

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]

Mai?tre, A.

A. Petrossian, M. Pinard, A. Mai⁁tre, J.-Y. Courtois, and G. Grynberg, “Transverse-pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689–695 (1992).
[CrossRef]

Mamaev, A. V.

P. M. Lushnikov and A. V. Mamaev, “Spontaneous hexagon formation in photorefractive crystal with a single pump wave,” Opt. Lett. 24, 1511–1513 (1999).
[CrossRef]

A. V. Mamaev and M. Saffman, “Hexagonal optical patterns in anisotropic nonlinear media,” Europhys. Lett. 34, 669–674 (1996).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in photorefractive crystals I+II,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

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]

Montgomery, D.

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[CrossRef] [PubMed]

Neubecker, R.

R. Neubecker, B. Thüring, and T. Tschudi, “Formation and characterization of hexagonal patterns in a single feedback experiment,” Chaos, Solitons Fractals 4, 1307–1322 (1994).
[CrossRef]

B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in liquid crystal light valve feedback system,” Opt. Commun. 102, 111–115 (1993).
[CrossRef]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in photorefractive crystals I+II,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Pare, C.

Pender, J.

Petrossian, A.

A. Petrossian, M. Pinard, A. Mai⁁tre, J.-Y. Courtois, and G. Grynberg, “Transverse-pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689–695 (1992).
[CrossRef]

Pinard, M.

A. Petrossian, M. Pinard, A. Mai⁁tre, J.-Y. Courtois, and G. Grynberg, “Transverse-pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689–695 (1992).
[CrossRef]

Saffman, M.

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, “Hexagonal optical patterns in anisotropic nonlinear media,” Europhys. Lett. 34, 669–674 (1996).
[CrossRef]

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

M. Saffman, A. A. Zozulya, and D. Z. Anderson, “Trans-verse instability of energy exchanging counterpropagating waves in photorefractive media,” J. Opt. Soc. Am. B 14, 1754–1760 (1994).

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[CrossRef] [PubMed]

Sandfuchs, O.

Santamato, E.

Schwab, M.

M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, and T. Tschudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos, Solitons Fractals 10, 701–707 (1999).
[CrossRef]

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

C. Denz, M. Schwab, M. Sedlatschek, T. Tschudi, and T. Honda, “Pattern dynamics and competition in a photorefractive feedback system,” J. Opt. Soc. Am. B 15, 2057–2064 (1998).
[CrossRef]

Sedlatschek, M.

M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, and T. Tschudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos, Solitons Fractals 10, 701–707 (1999).
[CrossRef]

C. Denz, M. Schwab, M. Sedlatschek, T. Tschudi, and T. Honda, “Pattern dynamics and competition in a photorefractive feedback system,” J. Opt. Soc. Am. B 15, 2057–2064 (1998).
[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, 293–299 (1997).
[CrossRef]

Simoneau, P.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteaux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in photorefractive crystals I+II,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Sturman, B. I.

Tamburrini, M.

Thüring, B.

M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, and T. Tschudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos, Solitons Fractals 10, 701–707 (1999).
[CrossRef]

R. Neubecker, B. Thüring, and T. Tschudi, “Formation and characterization of hexagonal patterns in a single feedback experiment,” Chaos, Solitons Fractals 4, 1307–1322 (1994).
[CrossRef]

B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in liquid crystal light valve feedback system,” Opt. Commun. 102, 111–115 (1993).
[CrossRef]

Tschudi, T.

M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, and T. Tschudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos, Solitons Fractals 10, 701–707 (1999).
[CrossRef]

C. Denz, M. Schwab, M. Sedlatschek, T. Tschudi, and T. Honda, “Pattern dynamics and competition in a photorefractive feedback system,” J. Opt. Soc. Am. B 15, 2057–2064 (1998).
[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, 293–299 (1997).
[CrossRef]

R. Neubecker, B. Thüring, and T. Tschudi, “Formation and characterization of hexagonal patterns in a single feedback experiment,” Chaos, Solitons Fractals 4, 1307–1322 (1994).
[CrossRef]

B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in liquid crystal light valve feedback system,” Opt. Commun. 102, 111–115 (1993).
[CrossRef]

C. Denz, J. Goltz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
[CrossRef]

Verkerk, P.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteaux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[CrossRef]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in photorefractive crystals I+II,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Wabnitz, S.

Zozulya, A. A.

M. Saffman, A. A. Zozulya, and D. Z. Anderson, “Trans-verse instability of energy exchanging counterpropagating waves in photorefractive media,” J. Opt. Soc. Am. B 14, 1754–1760 (1994).

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[CrossRef] [PubMed]

Appl. Phys. B (1)

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

Chaos, Solitons Fractals (2)

M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, and T. Tschudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos, Solitons Fractals 10, 701–707 (1999).
[CrossRef]

R. Neubecker, B. Thüring, and T. Tschudi, “Formation and characterization of hexagonal patterns in a single feedback experiment,” Chaos, Solitons Fractals 4, 1307–1322 (1994).
[CrossRef]

Europhys. Lett. (2)

A. V. Mamaev and M. Saffman, “Hexagonal optical patterns in anisotropic nonlinear media,” Europhys. Lett. 34, 669–674 (1996).
[CrossRef]

A. Petrossian, M. Pinard, A. Mai⁁tre, J.-Y. Courtois, and G. Grynberg, “Transverse-pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689–695 (1992).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in photorefractive crystals I+II,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

J. Opt. Soc. Am. B (6)

Opt. Commun. (5)

B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in liquid crystal light valve feedback system,” Opt. Commun. 102, 111–115 (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, 293–299 (1997).
[CrossRef]

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]

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteaux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
[CrossRef]

C. Denz, J. Goltz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. A (1)

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

K. R. MacDonald and J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1113 (1993).
[CrossRef]

Other (7)

M. A. Vorontsov and W. B. Miller, eds., Self-Organization in Optical Systems and Applications in Information Technology (Springer, Berlin, 1995).

D. Walgraef, Spatio-Temporal Pattern Formation (Springer, New York, 1995).

S. N. Vlasov and E. V. Sheinina, “On the theory of interaction of counterpropagating waves in a nonlinear cubic medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 26, 20 (1983) [Radiophys. Quantum Electron. 27, 15 (1983)].

B. Ya. Zel’dovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, Boca Raton, Fla., 1995), p. 256.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, Oxford, 1996).

In other experiments, 4 it was shown that introduction of an angular misalignment breaks the symmetry and collapses a hexagonal pattern to a roll pattern. Here we see the opposite effect of a hexagonal pattern occurring only in the presence of a misalignment. Since the angular misalignment needed for a sufficient Bragg mismatch for hexagon formation (~0.1 °) is much smaller than the angular scale of the pattern (~1 °), the hexagonal symmetry is not broken.

P. M. Lushnikov, “Hexagonal optical structures in photorefractive crystals with a feedback mirror,” Zh. Eksp. Teor. Fiz. 113, 1122–1135 (1998) [JETP 86, 614–627 (1998)].
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Basic interaction geometry. Two externally supplied counterpropagating beams with a relative frequency detuning Ω write a reflection grating inside the crystal. The created sidebands with an additional frequency shift δ and transverse k vector k are also indicated.

Fig. 2
Fig. 2

Threshold curves obtained with Eq. (28) for a parameter value of Ωτ=1 and pure energy coupling (γR=0): (a) Coupling strength γl (normalized coupling constant) as a function of the normalized transverse wave number kdl; (b) corresponding threshold curve for the frequency detuning of the sidebands δτ (normalized to the relaxation constant τ) as a function of the normalized transverse wave number kdl.

Fig. 3
Fig. 3

Threshold coupling strength as a function of the frequency detuning Ωτ for pure energy coupling (γR=0). The minimum values of the coupling strength in the curves similar to Fig. 2 are shown. Solid curve, δ- instability; dashed curve, δ+ instability; horizontal line, assumed coupling strength of γli4.4.

Fig. 4
Fig. 4

Transverse pattern scale kdl as a function of the frequency detuning Ωτ corresponding to the coupling strength in Fig. 3. Solid curve, δ- instability; dashed curve, δ+ instability.

Fig. 5
Fig. 5

Frequency detuning of the sidebands with respect to the carrier as a function of the frequency detuning Ωτ of the carriers corresponding to Figs. 3 and 4. Solid curve, δ- instability; dashed curve, δ+ instability.

Fig. 6
Fig. 6

Threshold curves for Ω=0: (a) as a function of the imaginary part of the coupling strength and various values of γR; (b), (c) as a function of the real part of the coupling strength and various values of γI.

Fig. 7
Fig. 7

Experimental setup: o.d., optical diode; v.a.’s, variable attenuators; L, lens; W, wedge; M, mirror; Piezo, piezo-controlled mirror; BS’s, beam splitters. The c axis of the crystal indicates the direction of energy transfer.

Fig. 8
Fig. 8

Experimentally found transverse instabilities: (a) Ω=18 Hz, roll pattern and underlying instabilities; (b) Ω=34 Hz, roll pattern with weak coexisting hexagon; (c) nonstationary hexagon for Ω=0 and slight externally applied misalignment of 0.1 °.

Fig. 9
Fig. 9

Relative sideband intensity (integrated sideband intensity with respect to the central beam) versus frequency detuning Ω for τ=36 ms (solid curve), τ=31 ms (dotted curve), and τ=41 ms (dashed curve) as predicted by the linear-stability analysis. Circles represent experimental values normalized to the maximum sideband intensity.

Fig. 10
Fig. 10

Transverse wave number as a function of the frequency detuning Ωτ for τ=36 ms (solid curve), τ=10 ms (dotted curve), and τ=60 ms (dashed–dotted curve) as predicted by the linear-stability analysis. Circles represent experimental values.

Equations (50)

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Fz-i2k0n02F=gB,
Bz+i2k0n02B=g*F,
τ t+1g=iγ FB*|F|2+|B2|.
F(z)=F¯0(z){1+f+ exp[i(k·r-δt)]+f- exp[-i(k·r-δt)]},
B(z)=B¯0(z){1+b+ exp[i(k·r-δt)]+b- exp[-i(k·r-δt)]},
F¯0(z)=F0(z)exp[i(k0z-ω0t)],
B¯0(z)=B0(z)exp[i(-k0z-ω0t)]exp(-iΩt),
τ t+1g=r0 exp(iΩt)+r0r-×exp[i(k·r+(Ω-δ)t)]+r0r+ exp{i[-k·r+(Ω+δ)t]},
r0=iγ F0B0*|F0|2+|B0|2,
r-=1|F0|2+|B0|2[|F0|2(b-*-f-*)+|B0|2(f+-b+)],
r+=1|F0|2+|B0|2[|F0|2(b+*-f+*)+|B0|2(f--b-)].
g(t)=g0 exp(iΩt)+g1 exp{i[k·r+(Ω-δ)t]}+g2 exp{i[-k·r+(Ω+δ)t]}.
g=χ0r0+χ-r0r- exp{i[k·r+(Ω-δ)t]}+χ+r0r+ exp{i[-k·r+(Ω+δ)t]},
χ0=11+iΩτ,
χ±=11+i(Ω±δ)τ.
z+ikdf+=-iγA{χ-(f-*-b-*)+[χ0+q(χ0-χ-)](f+-b+)},
z-ikdf-*=iγ*A{χ+*(f+-b+)+[χ0*+q(χ0*-χ+*)](f-*-b-*)},
z+ikdb+=-iγ*A{χ+*(f-*-b-*)+[χ0*+q-1(χ0*-χ+*)](f+-b+)},
z-ikdb-*=iγA{χ-(f+-b+)+[χ0+q-1(χ0-χ-)](f-*-b-*)},
A=|F0|2|B0|2(|F0|2+|B0|2)2,
kd=k22k0n0,
q=|B0|2|F0|2.
z+ikdf+=-iγ/4[χ-(f-*-b-*)+(2χ0-χ-)(f+-b+)],
z-ikdf-*=iγ*/4[χ+*(f+-b+)+(2χ0*-χ+*)(f-*-b-*)],
z-ikdb+=-iγ*/4[χ+*(f-*-b-*)+(2χ0*-χ+*)(f+-b+)],
z+ikdb-*=iγ/4[χ-(f+-b+)+(2χ0-χ-)(f*-b-*)].
u=f+f-*b+b-*
ddz u=D(γ, δ, Ω, kd)·u.
f+(0)=f-*(0)=0,
b+(l)=b-*(l)=0.
B·b+(0)b-*(0)=0,
ρ(γI, γR, Ω, δ)=2γR[(χ0+χ0*)-(χ-+χ+*)]+2iγI[(χ0-χ0*)-(χ--χ+*)],
ζ(γI, γR, Ω, δ)=2iγR[(χ0-χ0*)-(χ--χ+*)]-2γI[(χ0+χ0*)-(χ-+χ+*)],
cosh12χ0γIl+18ζl
+cos(w1l)cos(w2l)+η2 sin(w1l)sin(w2l)w1w2=0,
w1=kd2+kdχ0γR-14χ02γI2,
w2=kd2+14kdρ-164ζ2,
η=-18χ0γIζ+kd2+(χ0γR+kd)kd+14ρ.
γext.appl.field=γ0 1+iE01+iE0/(Ed+Eq)=γ0 1+f1(E0, Ed, Eq)1+f2(E0, Ed, Eq),
cosh12γIl(1+Δ)+cos(w10l)cos(w20l)
+η02 sin(w0l)sin(w20l)w10w20=0,
w10=kd2+kdγR-14γI2,
w20=kd2-kdγRΔ-14γI2Δ2,
η0=2kd2+γRkd-(γI2/2+γR2+kdγR)Δ,
Δ=iδτ1-iδτ.
d=|F0(L)|2/|F0(0)|2=(1+M)/[M+exp(γl)],
τ dISdt=2(γ-γth)γthIs.
IS(t)=IS(0)exp2 γ-γthγth tτ.
τ dISdt=2(γ-γth)γthIS-βIS2,
I¯S=-2β+2γβγth.

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