Abstract

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

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  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).
    [CrossRef]
  2. A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. 9, 288–290 (1984).
    [CrossRef] [PubMed]
  3. F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
    [CrossRef]
  4. P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B 6, R60–R75 (2004).
    [CrossRef]
  5. C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A 38, 5960–5963 (1988).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef] [PubMed]
  8. M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
    [CrossRef]
  9. J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  21. W. J. Firth and C. Paré, “Transverse modulational instabilities for counterpropagating beams in Kerr media,” Opt. Lett. 13, 1096–1098 (1988).
    [CrossRef] [PubMed]
  22. T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598–600 (1993).
    [CrossRef]
  23. L. A. Lugiato and C. Oldano, “Stationary spatial patterns in passive optical systems: two-level atoms,” Phys. Rev. A 37, 3896–3908 (1988).
    [CrossRef] [PubMed]
  24. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  26. H. Buljan, M. Soljacic, T. Carmon, and M. Segev, “Cavity pattern formation with incoherent light,” Phys. Rev. E 68, 016616 (2003).
    [CrossRef]
  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.
    [CrossRef] [PubMed]
  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.
    [CrossRef]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef]

2004 (2)

2003 (1)

H. Buljan, M. Soljacic, T. Carmon, and M. Segev, “Cavity pattern formation with incoherent light,” Phys. Rev. E 68, 016616 (2003).
[CrossRef]

2002 (1)

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).
[CrossRef] [PubMed]

2001 (1)

2000 (1)

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).
[CrossRef] [PubMed]

1999 (3)

F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[CrossRef]

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
[CrossRef] [PubMed]

1998 (1)

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).
[CrossRef]

1997 (2)

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).
[CrossRef]

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).
[CrossRef]

1996 (4)

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

1995 (3)

1994 (3)

1993 (2)

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

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).
[CrossRef]

1992 (2)

1988 (3)

C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A 38, 5960–5963 (1988).
[CrossRef] [PubMed]

L. A. Lugiato and C. Oldano, “Stationary spatial patterns in passive optical systems: two-level atoms,” Phys. Rev. A 37, 3896–3908 (1988).
[CrossRef] [PubMed]

W. J. Firth and C. Paré, “Transverse modulational instabilities for counterpropagating beams in Kerr media,” Opt. Lett. 13, 1096–1098 (1988).
[CrossRef] [PubMed]

1987 (1)

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

1984 (1)

1980 (1)

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).
[CrossRef]

Akimoto, O.

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).
[CrossRef]

Anderson, D. Z.

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).
[CrossRef] [PubMed]

Arecchi, F. T.

F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Bentley, S. J.

Boccaletti, S.

F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Boyd, R. W.

Brambilla, M.

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).
[CrossRef] [PubMed]

Buljan, H.

Butler, W. E.

Carmon, T.

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.
[CrossRef] [PubMed]

H. Buljan, M. Soljacic, T. Carmon, and M. Segev, “Cavity pattern formation with incoherent light,” Phys. Rev. E 68, 016616 (2003).
[CrossRef]

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).
[CrossRef] [PubMed]

Carvalho, M. I.

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).
[CrossRef]

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).
[CrossRef]

Chavez-Cerda, S.

Christodoulides, D. N.

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef]

Coen, S.

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).
[CrossRef]

Coskun, T.

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).
[CrossRef] [PubMed]

Crosignani, B.

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).
[CrossRef] [PubMed]

Cross, M. C.

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).
[CrossRef]

Daido, H.

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).
[CrossRef]

Denz, C.

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).
[CrossRef]

DiPorto, P.

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).
[CrossRef] [PubMed]

Fabre, C.

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[CrossRef]

Firth, W. J.

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).
[CrossRef] [PubMed]

W. J. Firth and C. Paré, “Transverse modulational instabilities for counterpropagating beams in Kerr media,” Opt. Lett. 13, 1096–1098 (1988).
[CrossRef] [PubMed]

Haelterman, M.

Hasegawa, A.

Hohenberg, P. C.

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).
[CrossRef]

Honda, T.

Ikeda, K.

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).
[CrossRef]

Indebetouw, G.

Iturbe-Castillo, M. D.

Jensen, S. J.

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).
[CrossRef]

Lefever, R.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

Liu, S. R.

Lugiato, L. A.

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).
[CrossRef] [PubMed]

L. A. Lugiato and C. Oldano, “Stationary spatial patterns in passive optical systems: two-level atoms,” Phys. Rev. A 37, 3896–3908 (1988).
[CrossRef] [PubMed]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

Maitre, A.

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[CrossRef]

Mamaev, A. V.

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).
[CrossRef] [PubMed]

Mandel, P.

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B 6, R60–R75 (2004).
[CrossRef]

Melissinos, A. C.

Oldano, C.

L. A. Lugiato and C. Oldano, “Stationary spatial patterns in passive optical systems: two-level atoms,” Phys. Rev. A 37, 3896–3908 (1988).
[CrossRef] [PubMed]

Oppo, G. L.

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).
[CrossRef] [PubMed]

Orenstein, M.

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
[CrossRef] [PubMed]

Paré, C.

Ramazza, P. L.

F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Saffman, M.

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).
[CrossRef] [PubMed]

Sanchez-Mondragon, J. J.

Scheuer, J.

J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
[CrossRef] [PubMed]

Schwab, M.

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).
[CrossRef]

Scroggie, A. J.

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).
[CrossRef] [PubMed]

Seger, M.

Segev, M.

H. Buljan, M. Soljacic, T. Carmon, and M. Segev, “Cavity pattern formation with incoherent light,” Phys. Rev. E 68, 016616 (2003).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

Shih, M.

Singh, S. R.

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).
[CrossRef]

Slekys, G.

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).
[CrossRef]

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).
[CrossRef] [PubMed]

Soljacic, M.

H. Buljan, M. Soljacic, T. Carmon, and M. Segev, “Cavity pattern formation with incoherent light,” Phys. Rev. E 68, 016616 (2003).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Staliunas, K.

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).
[CrossRef]

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).
[CrossRef] [PubMed]

Stepanov, S. I.

Tamm, C.

C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A 38, 5960–5963 (1988).
[CrossRef] [PubMed]

Tarroja, M. F. H.

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).
[CrossRef] [PubMed]

Tlidi, M.

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B 6, R60–R75 (2004).
[CrossRef]

Torres-Cisneros, G. E.

Torres-Cisneros, M.

Trillo, S.

Valley, G. C.

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).
[CrossRef]

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).
[CrossRef] [PubMed]

Vaupel, M.

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[CrossRef]

Vishwanath, A.

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).
[CrossRef] [PubMed]

Vysloukh, V. A.

Wabnitz, S.

Weiss, C. O.

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[CrossRef]

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[CrossRef] [PubMed]

Yariv, A.

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[CrossRef] [PubMed]

Zozulya, A. A.

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[CrossRef] [PubMed]

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[CrossRef]

Opt. Express (1)

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Phys. Rep. (1)

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[CrossRef]

Phys. Rev. A (5)

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[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Phys. Rev. E (1)

H. Buljan, M. Soljacic, T. Carmon, and M. Segev, “Cavity pattern formation with incoherent light,” Phys. Rev. E 68, 016616 (2003).
[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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Other (4)

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1988), p. 579.

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.

H. Haken, Synergetics: An Introduction, 3rd ed. (Springer-Verlag, Berlin, 1983), p. 229.

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.

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Figures (8)

Fig. 1
Fig. 1

Schematical sketch of the nonlinear (NL) ring cavity system.

Fig. 2
Fig. 2

(a) Calculated exponential gain factor g(q) and (b) spatial spectral density of the perturbation at the output face of the nonlinear medium for different intensity feedback values below threshold, based on the analytic solution (calculated with ψ0=1 and L=1).

Fig. 3
Fig. 3

Amplitude of the dominant spatial frequency at the output of the nonlinear medium (solid curve) and the line narrowing of the spatial bandwidth Δq (dashed curve) as a function of the feedback in the cavity below the threshold, based on the analytic solution. The cavity threshold is marked by an arrow (calculated with ψ0=1 and L=1).

Fig. 4
Fig. 4

Typical results of the numerical simulation with Kerr nonlinearity, showing the intensity pattern at the output of the nonlinear medium (normalized by the mean intensity in the cavity) without feedback (thin curve), slightly below the cavity threshold (thick curve), and slightly above the threshold (dashed curve), demonstrating the sudden increase in the modulation depth.

Fig. 5
Fig. 5

Calculated visibility versus intensity feedback for the numerical simulations with the Kerr nonlinearity, with different strengths of initial noise, showing the transition from low-visibility perturbations (below threshold) to a highly modulated intensity pattern (above threshold). The inset shows the magnification of the region below the cavity threshold.

Fig. 6
Fig. 6

Bandwidth of the perturbations’ spatial spectrum versus intensity feedback calculated with the numerical simulation (circles) below and above the threshold, and with the analytic solution (solid curve) below the cavity threshold for Kerr nonlinearity, showing line narrowing as the feedback is increased.

Fig. 7
Fig. 7

Comparisons between the modulation depth calculated for the numerical simulation (circles) and experimental results (squares) with the screening nonlinearity. The cavity threshold is marked by an arrow.

Fig. 8
Fig. 8

Bandwidth of the perturbations’ spatial spectrum versus intensity feedback in the (a) numerical simulation and (b) experimental results taken from Ref. 25, marked by circles. The solid curves in both figures show the theoretical dependence of the bandwidth based on the analytic solution. The dashed curve above the threshold in (b) is a guide to the eye.

Equations (29)

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i Ψiζ+2Ψiξ2+Δn(I)Ψi=0,
I=i=0|ψi+ϕi|2=i[|ψi|2+ψi*ϕi+ψiϕi*+O(ϕi2)]|ψ0|21-ε2+i(ψi*ϕi+ψiϕi*),
-Γi(ψi+ϕi)+i ϕiζ+2ϕiξ2+|ψ0|21-ε2+j(ψj*ϕj+ψjϕj*)(ψi+ϕi)=0.
-Γiψi+|ψ0|21-ε2ψi=0,
i ϕiζ+2ϕiξ2+j(ψj*ϕj+ψjϕj*)ψi=0.
i ϕ¯iζ+2ϕ¯iξ2+j|ψj|2(ϕ¯j+ϕ¯j*)=0.
i ϕ¯iζ+2ϕ¯iξ2+ψ02jε2j(ϕ¯j+ϕ¯j*)=0.
ϕ¯i-ζ+2ϕ¯i+ξ2+2ψ02jε2jϕ¯j+=0,
ϕ¯i+ζ-2ϕ¯i-ξ2=0.
Θζ+2Φξ2+2ψ021-ε2Φ=0,
Φζ-2Θξ2=0.
I=ψ021-ε2+iψi2(ϕ¯i+ϕ¯i*)=ψ021-ε2+ψ02iε2iϕ¯i+=ψ021-ε2+Φ(ξ, ζ),
Θˆζ-q2Φˆ+2ψ021-ε2 Φˆ=0,
Φˆζ+q2Θˆ=0,
2Φˆζ2=q22ψ021-ε2-q2Φˆg2Φˆ,
Φˆ(q, ζ)=A exp(gζ)+B exp(-gζ),
Θˆ(q, ζ)=igq2[A exp(gζ)-B exp(-gζ)].
Φˆ(q, L)exp[g(q)L]1-ε2  exp[g(q)L],
g(q)=|q|2ψ021-ε2-q21/2.
2ϕˆi+ζ2+q4ϕˆi+=2q2[A exp(gζ)+B exp(-gζ)].
i ϕ¯iζ+2ϕ¯iξ2+(1-ε2) ψ021+ψ02 jε2j(ϕ¯j+ϕ¯j*)=0,
g(q)=|q|2ψ021+ψ02-q21/2.
εth2=exp-ψ021+ψ02L.
E(x)=-Vl η[1+I(x)],
η1l 0l  dx1+I-1.
Δn(x)=-Δn01l 0l  dx1+I-1  1(1+I),
Δn(x)=-Δn0  1+I01+I(x).
i dΨdζ+d2Ψdξ2-1+I01+|Ψ|2Ψ=0,
i dψ1dζ+d2ψ1dξ2+ψ021+ψ02(ψ1+ψ1*)=0.

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