OSA's Digital Library

Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Vol. 21, Iss. 12 — Dec. 1, 2004
  • pp: 2197–2205

Pattern formation in a ring cavity with temporally incoherent feedback

Tal Schwartz, Jason W. Fleischer, Oren Cohen, Hrvoje Buljan, Mordechai Segev, and Tal Carmon  »View Author Affiliations

JOSA B, Vol. 21, Issue 12, pp. 2197-2205 (2004)

View Full Text Article

Acrobat PDF (219 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We present a theoretical and experimental study of modulation instability and pattern formation in a passive nonlinear optical cavity that is longer than the coherence length of the light circulating in it. Pattern formation in this cavity exhibits various features of a second-order phase transition, closely resembling laser action.

© 2004 Optical Society of America

OCIS Codes
(190.3100) Nonlinear optics : Instabilities and chaos
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

Tal Schwartz, Jason W. Fleischer, Oren Cohen, Hrvoje Buljan, Mordechai Segev, and Tal Carmon, "Pattern formation in a ring cavity with temporally incoherent feedback," J. Opt. Soc. Am. B 21, 2197-2205 (2004)

Sort:  Author  |  Year  |  Journal  |  Reset


  1. For an extensive review on pattern formation in various nonlinear systems, see M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
  2. A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. 9, 288–290 (1984).
  3. F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
  4. P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B 6, R60–R75 (2004).
  5. C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A 38, 5960–5963 (1988).
  6. K. Staliunas, G. Slekys, and C. O. Weiss, “Nonlinear pattern formation in active optical systems—shocks, domains of tilted waves, and cross-roll patterns,” Phys. Rev. Lett. 79, 2658–2661 (1997).
  7. G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
  8. M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
  9. J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
  10. S. R. Liu and G. Indebetouw, “Periodic and chaotic spatiotemporal states in a phase-conjugate resonator using a photorefractive BaTiO3 phase-conjugate mirror,” J. Opt. Soc. Am. B 9, 1507–1520 (1992).
  11. K. Staliunas, M. F. H. Tarroja, G. Slekys, and C. O. Weiss, “Analogy between photorefractive oscillators and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
  12. W. J. Firth and A. J. Scroggie, “Optical bullet holes—robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
  13. K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence—chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
  14. M. Haelterman, S. Wabnitz, and S. Trillo, “Additive-modulation-instability ring laser in the normal dispersion regime of a fiber,” Opt. Lett. 17, 745–747 (1992).
  15. S. Coen and M. Haelterman, “Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber,” Phys. Rev. Lett. 79, 4139–4142 (1997).
  16. S. J. Bentley, R. W. Boyd, W. E. Butler, and A. C. Melissinos, “Spatial patterns induced in a laser beam by thermal nonlinearities,” Opt. Lett. 26, 1084–1086 (2001).
  17. S. J. Jensen, M. Schwab, and C. Denz, “Manipulation, stabilization, and control of pattern formation using Fourier space filtering,” Phys. Rev. Lett. 81, 1614–1617 (1998).
  18. M. D. Iturbe-Castillo, M. Torres-Cisneros, J. J. Sanchez-Mondragon, S. Chavez-Cerda, S. I. Stepanov, V. A. Vysloukh, and G. E. Torres-Cisneros, “Experimental evidence of modulation instability in a photorefractive Bi12TiO20 crystal,” Opt. Lett. 20, 1853–1855 (1995).
  19. A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinearmedia: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
  20. M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Modulational instability of quasi-plane-wave optical beams biased in photorefractive crystals,” Opt. Commun. 126, 167–174 (1996).
  21. W. J. Firth and C. Paré, “Transverse modulational instabilities for counterpropagating beams in Kerr media,” Opt. Lett. 13, 1096–1098 (1988).
  22. T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598–600 (1993).
  23. L. A. Lugiato and C. Oldano, “Stationary spatial patterns in passive optical systems: two-level atoms,” Phys. Rev. A 37, 3896–3908 (1988).
  24. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
  25. T. Carmon, M. Soljacic, and M. Segev, “Pattern formation in a cavity longer than the coherence length of the light in it,” Phys. Rev. Lett. 89, 183902 (2002).
  26. H. Buljan, M. Soljacic, T. Carmon, and M. Segev, “Cavity pattern formation with incoherent light,” Phys. Rev. E 68, 016616 (2003).
  27. T. Carmon, H. Buljan, and M. Seger, “Spontaneous pattern formation in a cavity with incoherent light,” Opt. Express 12, 3481–3487 (2004), http://www.opticsexpress.org.
  28. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1988), p. 579.
  29. This interaction resembles cross-phase modulation [See M. Haelterman, S. Trillo, and S. Wabnitz, “Polarization multistability and instability in a nonlinear dispersive ring cavity,” J. Opt. Soc. Am. B 11, 446–456 (1994)], yet it leads to different phenomena. In the cross-phase modulation the two orthogonal polarizations interact through the sum of their intensities alone, but, in contrast to our case, the fields of each polarization are coherent. Therefore, in that case, each polarization has its own set of resonant frequencies, and the pattern formation process generally depends on these frequencies.
  30. In transforming to the dimensionless equation, we use x0=λ/[2π(2nΔn0)1/2] and z0=λ/2πΔn0 as the characteristic transverse and propagation scales, respectively, where λ is the wavelength in vacuum, n is the material refractive index, and Δn0 is the typical scale of the nonlinear change in the refractive index.
  31. M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
  32. We define the bandwidth Δq as the standard deviation of the spatial power density: Δq=[∫0(q− 〈q〉)2S(q)dq]1/2/∫0S(q)dq, where S(q) is the perturbations’ spatial power spectrum and 〈q〉= ∫0qS(q)dq/∫0S(q)dq is the mean transverse wave number.
  33. H. Haken, Synergetics: An Introduction, 3rd ed. (Springer-Verlag, Berlin, 1983), p. 229.
  34. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
  35. M. Segev, M. Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13, 706–718 (1996).
  36. D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628–1633 (1995).

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