OSA's Digital Library

Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Vol. 18, Iss. 5 — May. 1, 2001
  • pp: 628–638

Transverse modulational instability in counterpropagating two-wave mixing with frequency-detuned pump beams

M. Schwab, C. Denz, and M. Saffman  »View Author Affiliations

JOSA B, Vol. 18, Issue 5, pp. 628-638 (2001)

View Full Text Article

Acrobat PDF (252 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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 γ l=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

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(190.5330) Nonlinear optics : Photorefractive optics

M. Schwab, C. Denz, and M. Saffman, "Transverse modulational instability in counterpropagating two-wave mixing with frequency-detuned pump beams," J. Opt. Soc. Am. B 18, 628-638 (2001)

Sort:  Author  |  Year  |  Journal  |  Reset


  1. D. Walgraef, Spatio-Temporal Pattern Formation (Springer, New York, 1995).
  2. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1113 (1993).
  3. M. A. Vorontsov and W. B. Miller, eds., Self-Organization in Optical Systems and Applications in Information Technology (Springer, Berlin, 1995).
  4. 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).
  5. J. Pender and L. Hesselink, “Degenerate conical emission in atomic sodium vapor,” J. Opt. Soc. Am. B 7, 1361–1373 (1990).
  6. 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).
  7. 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)].
  8. W. J. Firth, A. Fitzgerald, and C. Pare, “Transverse instabilities due to counterpropagation in Kerr media,” J. Opt. Soc. Am. B 7, 1087–1097 (1990).
  9. R. Macdonald and H. J. Eichler, “Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror,” Opt. Commun. 89, 289–295 (1992).
  10. 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).
  11. B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in liquid crystal light valve feedback system,” Opt. Commun. 102, 111–115 (1993).
  12. 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).
  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).
  14. T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598–600 (1993).
  15. T. Honda, “Flow and controlled rotation of the spontaneous optical hexagon in KNbO3,” Opt. Lett. 20, 851–853 (1995).
  16. 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).
  17. A. V. Mamaev and M. Saffman, “Hexagonal optical patterns in anisotropic nonlinear media,” Europhys. Lett. 34, 669–674 (1996).
  18. 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).
  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).
  20. M. Schwab, C. Denz, and M. Saffman, “Multiple-pattern stability in a photorefractive feedback system,” Appl. Phys. B 69, 429–433 (1999).
  21. 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).
  22. P. M. Lushnikov and A. V. Mamaev, “Spontaneous hexagon formation in photorefractive crystal with a single pump wave,” Opt. Lett. 24, 1511–1513 (1999).
  23. 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).
  24. T. Honda and P. P. Banerjee, “Threshold for spontaneous pattern formation in reflection-grating-dominated photore-fractive media with mirror feedback,” Opt. Lett. 21, 779–781 (1996).
  25. 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, 1754–1760 (1997).
  26. O. Sandfuchs, J. Leonardy, F. Kaiser, and M. R. Belić, “Transverse instabilities in photorefractive counterpropagating two-wave mixing,” Opt. Lett. 22, 498–500 (1997).
  27. 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, 2070–2078 (1998).
  28. 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)].
  29. 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).
  30. L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, Oxford, 1996).
  31. 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.
  32. 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).
  33. 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).
  34. 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.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited