Abstract

We present a comprehensive study of the one-dimensional modulation instability of partially spatially incoherent light in noninstantaneous self-focusing media. For this instability to occur, the nonlinearity has to exceed a specific threshold that depends on the coherence properties of the beam. Above this threshold a uniform-intensity partially spatially coherent wave front becomes unstable and breaks up into periodic trains of one-dimensional stripes.

© 2002 Optical Society of America

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  1. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).
  2. P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, “Generation of a periodic array of dark spatial solitons in the regime of effective amplification,” J. Opt. Soc. Am. B 11, 1254–1260 (1994).
    [Crossref]
  3. 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]
  4. 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]
  5. E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, “Generation of a train of fundamental solitons at a high repetition rate in optical fibers,” Opt. Lett. 14, 1008–1010 (1989).
    [Crossref] [PubMed]
  6. V. I. Karpman, “Self-modulation of nonlinear plane waves in dispersive media,” JETP Lett. 6, 277–280 (1967).
  7. V. I. Karpman and E. M. Kreushkal, “Modulated waves in non-linear dispersive media,” Sov. Phys. JETP 28, 277–281 (1969).
  8. A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulation instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
    [Crossref]
  9. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
    [Crossref] [PubMed]
  10. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [Crossref] [PubMed]
  11. S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
    [Crossref] [PubMed]
  12. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).
  13. For a recent review on optical spatial solitons, see G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
    [Crossref] [PubMed]
  14. A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. 9, 288–290 (1984).
    [Crossref] [PubMed]
  15. M. Artiglia, E. Ciaramella, and P. Gallina, “Demonstration of cw soliton trains at 10, 40 and 160 GHz by means of induced modulation instability,” in System Technologies, C. Menyuk and A. Willner, eds., Vol. 12 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 305–308.
  16. M. Soljačić, 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]
  17. D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
    [Crossref] [PubMed]
  18. J. Klinger, H. Martin, and Z. Chen, “Experiments on induced modulational instability of an incoherent optical beam,” Opt. Lett. 26, 271–273 (2000).
    [Crossref]
  19. We note the distinction between the MI threshold and the existence of a threshold for soliton formation, the former being unique to partially incoherent systems, whereas the latter can be found in many coherent systems. For example, there is a clear threshold for envelope soliton formation in weakly dissipative systems; see A. N. Slavin, “Thresholds of envelope soliton formation in a weakly dissipative medium,” Phys. Rev. Lett. 77, 4644–4647 (1996).
    [Crossref] [PubMed]
  20. We also note that, in principle, effects that are closely related to the MI of partially incoherent wave fronts could also be observed with diffusive nonlinearities, that is, nonlinearities for which the nonlinear index change is a function of the optical intensity averaged over the characteristic diffusion length. Such diffusive effects exist in several material systems, e.g., in semiconductor lasers with free-carrier nonlinearities, where diffusion of charge carriers washes out all structures more delicate than the diffusion length. However, to our knowledge, MI in such systems has not been studied.
  21. A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, “Charge density wave in two-dimensional electron liquid in weak magnetic field,” Phys. Rev. Lett. 76, 499–502 (1996).
    [Crossref] [PubMed]
  22. A. H. MacDonald and M. P. A. Fisher, “Quantum theory of quantum Hall smectics,” Phys. Rev. B 61, 5724–5733 (2000).
    [Crossref]
  23. M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for an anisotropic state of two-dimensional electrons in high Landau levels,” Phys. Rev. Lett. 82, 394–397 (1999).
    [Crossref]
  24. M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Anisotropic states of two-dimensional electron systems in high Landau levels: effect of an in-plane magnetic field,” Phys. Rev. Lett. 83, 824–827 (1999).
    [Crossref]
  25. W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and nu=5/2 under a tilted magnetic field,” Phys. Rev. Lett. 83, 820–823 (1999).
    [Crossref]
  26. Th. Busch and J. R. Anglin, “Motion of dark solitons in trapped Bose–Einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
    [Crossref] [PubMed]
  27. F. G. Bridges, A. Hatzes, and D. N. C. Lin, “Structure, stability and evolution of Saturn’s rings,” Nature 309, 333–335 (1984).
    [Crossref]
  28. K. Bekki, “Group-cluster merging and the formation of starburst galaxies,” Astrophys. J. 510, 15–19 (1999).
    [Crossref]
  29. V. J. Emery, S. A. Kivelson, and J. M. Tranquada, “Stripe phases in high-temperature superconductors,” Proc. Natl. Acad. Sci. U.S.A. 96, 8814–8817 (1999).
    [Crossref] [PubMed]
  30. S. R. White and D. J. Scalapino, “Competition between stripes and pairing in a t-t′-J model,” Phys. Rev. B 60, R753–R756 (1999).
    [Crossref]
  31. S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, “Pattern formation via symmetry breaking in nonlinear weakly correlated systems,” to be published in Phys. Rev. E.
  32. M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
    [Crossref] [PubMed]
  33. M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
    [Crossref]
  34. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
    [Crossref]
  35. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Multimode incoherent spatial solitons in logarithmically saturable nonlinear media,” Phys. Rev. Lett. 80, 2310–2313 (1998).
    [Crossref]
  36. M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
    [Crossref]
  37. A. W. Snyder and D. J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422–1424 (1998). This ray optics model of incoherent solitons is similar to the model of random-phase solitons in plasma, suggested in Refs. 38 and 39.
    [Crossref]
  38. A. Hasegawa, “Dynamics of an ensemble of plane waves in nonlinear dispersive media,” Phys. Fluids 18, 77–78 (1975).
    [Crossref]
  39. A. Hasegawa, “Envelope soliton of random phase waves,” Phys. Fluids 20, 2155–2156 (1977). This model assumes zero degree of coherence, that is, every single point on the beam acts as an independent point source. This view is problematic for several reasons. First, zero-correlated beams can be described neither by a paraxial equation nor by a Helmholtz equation: They require full vectorial formulation through Maxwell equations. Second, this model implies that the shape of incoherent solitons is completely arbitrary and their correlation statistical properties are delocalized, which is just an artifact of using a transport equation—see Refs. 33-35. But worse than all other artifacts of the total incoherence assumption is that such transport (or ray optics) models are modulationally stable, i.e., there is no MI in these models. The reason is obvious: The MI threshold is a function that is monotonically increasing with decreasing correlation distance. For fully spatially incoherent beams the transverse correlation distance is zero, which implies that the threshold for MI is infinity. This is obviously unphysical and is a direct artifact of the full incoherence assumption.
    [Crossref]
  40. G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).
  41. V. V. Shkunov and D. Z. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683–2686 (1998).
    [Crossref]
  42. Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
    [Crossref] [PubMed]
  43. D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
    [Crossref]
  44. N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially coherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
    [Crossref]
  45. O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5483 (1999).
    [Crossref]
  46. C. Anastassiou, M. Soljačić, M. Segev, D. Kip, E. Eugenieva, D. N. Christodoulides, and Z. Musslimani, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
    [Crossref] [PubMed]
  47. M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
    [Crossref]
  48. T. Coskun, D. N. Christodoulides, Y. Kim, Z. Chen, M. Soljačić, and M. Segev, “Bright spatial solitons on a partially incoherent background,” Phys. Rev. Lett. 84, 2374–2377 (2000).
    [Crossref] [PubMed]
  49. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
    [Crossref] [PubMed]
  50. 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]
  51. 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]
  52. M. Peccianti and G. Assanto, “Incoherent spatial solitary waves in nematic liquid crystal,” Opt. Lett. 26, 1791–1973 (2001).
    [Crossref]
  53. N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zel’dovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1–140 (1986).
    [Crossref]
  54. A Lorentzian power spectrum shape is not physical in 2D, since it would imply a logarithmically divergent intensity. Furthermore, one has reasons to doubt that it is a physically valid solution for all kx even in the 1D case. Nevertheless, it is a good approximation to the true shape (apart for the tails in large kx). Since this model can be solved in a closed formed analytically, we find it to be useful for studying and understanding MI. One should compare this situation with the approximation of parabolic waveguides in optical fibers, which serve as a good approximation of true graded-index waveguides, although of course true parabolic waveguides do not exist in reality.
  55. The rotating diffuser also broadens the linewidth of the laser light, because the rotation causes a Doppler shift. In our experiments, however, the speckles introduce a new phase every microsecond, and therefore the Doppler shift is of the order of megahertz, which is negligible compared with the ∼1 GHz laser linewidth that we use.

2001 (1)

2000 (7)

T. Coskun, D. N. Christodoulides, Y. Kim, Z. Chen, M. Soljačić, and M. Segev, “Bright spatial solitons on a partially incoherent background,” Phys. Rev. Lett. 84, 2374–2377 (2000).
[Crossref] [PubMed]

C. Anastassiou, M. Soljačić, M. Segev, D. Kip, E. Eugenieva, D. N. Christodoulides, and Z. Musslimani, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
[Crossref] [PubMed]

M. Soljačić, 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]

D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[Crossref] [PubMed]

J. Klinger, H. Martin, and Z. Chen, “Experiments on induced modulational instability of an incoherent optical beam,” Opt. Lett. 26, 271–273 (2000).
[Crossref]

A. H. MacDonald and M. P. A. Fisher, “Quantum theory of quantum Hall smectics,” Phys. Rev. B 61, 5724–5733 (2000).
[Crossref]

Th. Busch and J. R. Anglin, “Motion of dark solitons in trapped Bose–Einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[Crossref] [PubMed]

1999 (8)

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for an anisotropic state of two-dimensional electrons in high Landau levels,” Phys. Rev. Lett. 82, 394–397 (1999).
[Crossref]

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Anisotropic states of two-dimensional electron systems in high Landau levels: effect of an in-plane magnetic field,” Phys. Rev. Lett. 83, 824–827 (1999).
[Crossref]

W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and nu=5/2 under a tilted magnetic field,” Phys. Rev. Lett. 83, 820–823 (1999).
[Crossref]

K. Bekki, “Group-cluster merging and the formation of starburst galaxies,” Astrophys. J. 510, 15–19 (1999).
[Crossref]

V. J. Emery, S. A. Kivelson, and J. M. Tranquada, “Stripe phases in high-temperature superconductors,” Proc. Natl. Acad. Sci. U.S.A. 96, 8814–8817 (1999).
[Crossref] [PubMed]

S. R. White and D. J. Scalapino, “Competition between stripes and pairing in a t-t′-J model,” Phys. Rev. B 60, R753–R756 (1999).
[Crossref]

For a recent review on optical spatial solitons, see G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5483 (1999).
[Crossref]

1998 (7)

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
[Crossref]

V. V. Shkunov and D. Z. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683–2686 (1998).
[Crossref]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[Crossref] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[Crossref]

N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially coherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Multimode incoherent spatial solitons in logarithmically saturable nonlinear media,” Phys. Rev. Lett. 80, 2310–2313 (1998).
[Crossref]

A. W. Snyder and D. J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422–1424 (1998). This ray optics model of incoherent solitons is similar to the model of random-phase solitons in plasma, suggested in Refs. 38 and 39.
[Crossref]

1997 (3)

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[Crossref]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[Crossref]

1996 (5)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[Crossref] [PubMed]

We note the distinction between the MI threshold and the existence of a threshold for soliton formation, the former being unique to partially incoherent systems, whereas the latter can be found in many coherent systems. For example, there is a clear threshold for envelope soliton formation in weakly dissipative systems; see A. N. Slavin, “Thresholds of envelope soliton formation in a weakly dissipative medium,” Phys. Rev. Lett. 77, 4644–4647 (1996).
[Crossref] [PubMed]

A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, “Charge density wave in two-dimensional electron liquid in weak magnetic field,” Phys. Rev. Lett. 76, 499–502 (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]

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]

1995 (2)

1994 (2)

P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, “Generation of a periodic array of dark spatial solitons in the regime of effective amplification,” J. Opt. Soc. Am. B 11, 1254–1260 (1994).
[Crossref]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

1989 (1)

1988 (1)

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[Crossref] [PubMed]

1987 (1)

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[Crossref] [PubMed]

1986 (2)

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[Crossref] [PubMed]

N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zel’dovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1–140 (1986).
[Crossref]

1984 (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]

F. G. Bridges, A. Hatzes, and D. N. C. Lin, “Structure, stability and evolution of Saturn’s rings,” Nature 309, 333–335 (1984).
[Crossref]

1980 (1)

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulation instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

1977 (1)

A. Hasegawa, “Envelope soliton of random phase waves,” Phys. Fluids 20, 2155–2156 (1977). This model assumes zero degree of coherence, that is, every single point on the beam acts as an independent point source. This view is problematic for several reasons. First, zero-correlated beams can be described neither by a paraxial equation nor by a Helmholtz equation: They require full vectorial formulation through Maxwell equations. Second, this model implies that the shape of incoherent solitons is completely arbitrary and their correlation statistical properties are delocalized, which is just an artifact of using a transport equation—see Refs. 33-35. But worse than all other artifacts of the total incoherence assumption is that such transport (or ray optics) models are modulationally stable, i.e., there is no MI in these models. The reason is obvious: The MI threshold is a function that is monotonically increasing with decreasing correlation distance. For fully spatially incoherent beams the transverse correlation distance is zero, which implies that the threshold for MI is infinity. This is obviously unphysical and is a direct artifact of the full incoherence assumption.
[Crossref]

1975 (1)

A. Hasegawa, “Dynamics of an ensemble of plane waves in nonlinear dispersive media,” Phys. Fluids 18, 77–78 (1975).
[Crossref]

1974 (1)

G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).

1969 (1)

V. I. Karpman and E. M. Kreushkal, “Modulated waves in non-linear dispersive media,” Sov. Phys. JETP 28, 277–281 (1969).

1967 (1)

V. I. Karpman, “Self-modulation of nonlinear plane waves in dispersive media,” JETP Lett. 6, 277–280 (1967).

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Agrawal, G. P.

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[Crossref] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).

Akhmediev, N.

N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially coherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
[Crossref]

Anastassiou, C.

C. Anastassiou, M. Soljačić, M. Segev, D. Kip, E. Eugenieva, D. N. Christodoulides, and Z. Musslimani, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
[Crossref] [PubMed]

Anderson, D. Z.

V. V. Shkunov and D. Z. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683–2686 (1998).
[Crossref]

Anglin, J. R.

Th. Busch and J. R. Anglin, “Motion of dark solitons in trapped Bose–Einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[Crossref] [PubMed]

Artiglia, M.

M. Artiglia, E. Ciaramella, and P. Gallina, “Demonstration of cw soliton trains at 10, 40 and 160 GHz by means of induced modulation instability,” in System Technologies, C. Menyuk and A. Willner, eds., Vol. 12 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 305–308.

Assanto, G.

Baldwin, K. W.

W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and nu=5/2 under a tilted magnetic field,” Phys. Rev. Lett. 83, 820–823 (1999).
[Crossref]

Bang, O.

O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5483 (1999).
[Crossref]

Bekki, K.

K. Bekki, “Group-cluster merging and the formation of starburst galaxies,” Astrophys. J. 510, 15–19 (1999).
[Crossref]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Bosshard, C.

Bridges, F. G.

F. G. Bridges, A. Hatzes, and D. N. C. Lin, “Structure, stability and evolution of Saturn’s rings,” Nature 309, 333–335 (1984).
[Crossref]

Brinkman, W. F.

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulation instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

Busch, Th.

Th. Busch and J. R. Anglin, “Motion of dark solitons in trapped Bose–Einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[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.

Chen, Z.

J. Klinger, H. Martin, and Z. Chen, “Experiments on induced modulational instability of an incoherent optical beam,” Opt. Lett. 26, 271–273 (2000).
[Crossref]

T. Coskun, D. N. Christodoulides, Y. Kim, Z. Chen, M. Soljačić, and M. Segev, “Bright spatial solitons on a partially incoherent background,” Phys. Rev. Lett. 84, 2374–2377 (2000).
[Crossref] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[Crossref]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[Crossref] [PubMed]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[Crossref] [PubMed]

Chernikov, S. V.

Christodoulides, D. N.

C. Anastassiou, M. Soljačić, M. Segev, D. Kip, E. Eugenieva, D. N. Christodoulides, and Z. Musslimani, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
[Crossref] [PubMed]

T. Coskun, D. N. Christodoulides, Y. Kim, Z. Chen, M. Soljačić, and M. Segev, “Bright spatial solitons on a partially incoherent background,” Phys. Rev. Lett. 84, 2374–2377 (2000).
[Crossref] [PubMed]

M. Soljačić, 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]

D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[Crossref] [PubMed]

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Multimode incoherent spatial solitons in logarithmically saturable nonlinear media,” Phys. Rev. Lett. 80, 2310–2313 (1998).
[Crossref]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[Crossref] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[Crossref]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[Crossref]

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]

S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, “Pattern formation via symmetry breaking in nonlinear weakly correlated systems,” to be published in Phys. Rev. E.

Ciaramella, E.

M. Artiglia, E. Ciaramella, and P. Gallina, “Demonstration of cw soliton trains at 10, 40 and 160 GHz by means of induced modulation instability,” in System Technologies, C. Menyuk and A. Willner, eds., Vol. 12 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 305–308.

Cooper, K. B.

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for an anisotropic state of two-dimensional electrons in high Landau levels,” Phys. Rev. Lett. 82, 394–397 (1999).
[Crossref]

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Anisotropic states of two-dimensional electron systems in high Landau levels: effect of an in-plane magnetic field,” Phys. Rev. Lett. 83, 824–827 (1999).
[Crossref]

Coskun, T.

M. Soljačić, 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]

T. Coskun, D. N. Christodoulides, Y. Kim, Z. Chen, M. Soljačić, and M. Segev, “Bright spatial solitons on a partially incoherent background,” Phys. Rev. Lett. 84, 2374–2377 (2000).
[Crossref] [PubMed]

Coskun, T. H.

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[Crossref] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Multimode incoherent spatial solitons in logarithmically saturable nonlinear media,” Phys. Rev. Lett. 80, 2310–2313 (1998).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[Crossref]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[Crossref]

Crosignani, B.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

Dianov, E. M.

DiPorto, P.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

Du, R. R.

W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and nu=5/2 under a tilted magnetic field,” Phys. Rev. Lett. 83, 820–823 (1999).
[Crossref]

Edmundson, D.

O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5483 (1999).
[Crossref]

Eisenstein, J. P.

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Anisotropic states of two-dimensional electron systems in high Landau levels: effect of an in-plane magnetic field,” Phys. Rev. Lett. 83, 824–827 (1999).
[Crossref]

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for an anisotropic state of two-dimensional electrons in high Landau levels,” Phys. Rev. Lett. 82, 394–397 (1999).
[Crossref]

Emery, V. J.

V. J. Emery, S. A. Kivelson, and J. M. Tranquada, “Stripe phases in high-temperature superconductors,” Proc. Natl. Acad. Sci. U.S.A. 96, 8814–8817 (1999).
[Crossref] [PubMed]

Eugenieva, E.

C. Anastassiou, M. Soljačić, M. Segev, D. Kip, E. Eugenieva, D. N. Christodoulides, and Z. Musslimani, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
[Crossref] [PubMed]

D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[Crossref] [PubMed]

Fisher, M. P. A.

A. H. MacDonald and M. P. A. Fisher, “Quantum theory of quantum Hall smectics,” Phys. Rev. B 61, 5724–5733 (2000).
[Crossref]

Fogler, M. M.

A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, “Charge density wave in two-dimensional electron liquid in weak magnetic field,” Phys. Rev. Lett. 76, 499–502 (1996).
[Crossref] [PubMed]

Gallina, P.

M. Artiglia, E. Ciaramella, and P. Gallina, “Demonstration of cw soliton trains at 10, 40 and 160 GHz by means of induced modulation instability,” in System Technologies, C. Menyuk and A. Willner, eds., Vol. 12 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 305–308.

Hasegawa, A.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[Crossref] [PubMed]

A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. 9, 288–290 (1984).
[Crossref] [PubMed]

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulation instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

A. Hasegawa, “Envelope soliton of random phase waves,” Phys. Fluids 20, 2155–2156 (1977). This model assumes zero degree of coherence, that is, every single point on the beam acts as an independent point source. This view is problematic for several reasons. First, zero-correlated beams can be described neither by a paraxial equation nor by a Helmholtz equation: They require full vectorial formulation through Maxwell equations. Second, this model implies that the shape of incoherent solitons is completely arbitrary and their correlation statistical properties are delocalized, which is just an artifact of using a transport equation—see Refs. 33-35. But worse than all other artifacts of the total incoherence assumption is that such transport (or ray optics) models are modulationally stable, i.e., there is no MI in these models. The reason is obvious: The MI threshold is a function that is monotonically increasing with decreasing correlation distance. For fully spatially incoherent beams the transverse correlation distance is zero, which implies that the threshold for MI is infinity. This is obviously unphysical and is a direct artifact of the full incoherence assumption.
[Crossref]

A. Hasegawa, “Dynamics of an ensemble of plane waves in nonlinear dispersive media,” Phys. Fluids 18, 77–78 (1975).
[Crossref]

Hatzes, A.

F. G. Bridges, A. Hatzes, and D. N. C. Lin, “Structure, stability and evolution of Saturn’s rings,” Nature 309, 333–335 (1984).
[Crossref]

Iturbe-Castillo, M. D.

Karpman, V. I.

V. I. Karpman and E. M. Kreushkal, “Modulated waves in non-linear dispersive media,” Sov. Phys. JETP 28, 277–281 (1969).

V. I. Karpman, “Self-modulation of nonlinear plane waves in dispersive media,” JETP Lett. 6, 277–280 (1967).

Kim, Y.

T. Coskun, D. N. Christodoulides, Y. Kim, Z. Chen, M. Soljačić, and M. Segev, “Bright spatial solitons on a partially incoherent background,” Phys. Rev. Lett. 84, 2374–2377 (2000).
[Crossref] [PubMed]

Kip, D.

C. Anastassiou, M. Soljačić, M. Segev, D. Kip, E. Eugenieva, D. N. Christodoulides, and Z. Musslimani, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
[Crossref] [PubMed]

D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[Crossref] [PubMed]

Kivelson, S. A.

V. J. Emery, S. A. Kivelson, and J. M. Tranquada, “Stripe phases in high-temperature superconductors,” Proc. Natl. Acad. Sci. U.S.A. 96, 8814–8817 (1999).
[Crossref] [PubMed]

Klinger, J.

Koulakov, A. A.

A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, “Charge density wave in two-dimensional electron liquid in weak magnetic field,” Phys. Rev. Lett. 76, 499–502 (1996).
[Crossref] [PubMed]

Kreushkal, E. M.

V. I. Karpman and E. M. Kreushkal, “Modulated waves in non-linear dispersive media,” Sov. Phys. JETP 28, 277–281 (1969).

Krolikowski, W.

O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5483 (1999).
[Crossref]

N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially coherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
[Crossref]

Lilly, M. P.

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for an anisotropic state of two-dimensional electrons in high Landau levels,” Phys. Rev. Lett. 82, 394–397 (1999).
[Crossref]

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Anisotropic states of two-dimensional electron systems in high Landau levels: effect of an in-plane magnetic field,” Phys. Rev. Lett. 83, 824–827 (1999).
[Crossref]

Lin, D. N. C.

F. G. Bridges, A. Hatzes, and D. N. C. Lin, “Structure, stability and evolution of Saturn’s rings,” Nature 309, 333–335 (1984).
[Crossref]

MacDonald, A. H.

A. H. MacDonald and M. P. A. Fisher, “Quantum theory of quantum Hall smectics,” Phys. Rev. B 61, 5724–5733 (2000).
[Crossref]

Mamyshev, P. V.

Martin, H.

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422–1424 (1998). This ray optics model of incoherent solitons is similar to the model of random-phase solitons in plasma, suggested in Refs. 38 and 39.
[Crossref]

Mitchell, M.

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[Crossref]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[Crossref] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Multimode incoherent spatial solitons in logarithmically saturable nonlinear media,” Phys. Rev. Lett. 80, 2310–2313 (1998).
[Crossref]

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[Crossref]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[Crossref]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[Crossref]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[Crossref] [PubMed]

Musslimani, Z.

C. Anastassiou, M. Soljačić, M. Segev, D. Kip, E. Eugenieva, D. N. Christodoulides, and Z. Musslimani, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
[Crossref] [PubMed]

Pan, W.

W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and nu=5/2 under a tilted magnetic field,” Phys. Rev. Lett. 83, 820–823 (1999).
[Crossref]

Pasmanik, G. A.

G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).

Peccianti, M.

Pfeiffer, L. N.

W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and nu=5/2 under a tilted magnetic field,” Phys. Rev. Lett. 83, 820–823 (1999).
[Crossref]

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Anisotropic states of two-dimensional electron systems in high Landau levels: effect of an in-plane magnetic field,” Phys. Rev. Lett. 83, 824–827 (1999).
[Crossref]

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for an anisotropic state of two-dimensional electrons in high Landau levels,” Phys. Rev. Lett. 82, 394–397 (1999).
[Crossref]

Prokhorov, A. M.

Sanchez-Mondragon, J. J.

Scalapino, D. J.

S. R. White and D. J. Scalapino, “Competition between stripes and pairing in a t-t′-J model,” Phys. Rev. B 60, R753–R756 (1999).
[Crossref]

Sears, S. M.

S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, “Pattern formation via symmetry breaking in nonlinear weakly correlated systems,” to be published in Phys. Rev. E.

Segev, M.

C. Anastassiou, M. Soljačić, M. Segev, D. Kip, E. Eugenieva, D. N. Christodoulides, and Z. Musslimani, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
[Crossref] [PubMed]

T. Coskun, D. N. Christodoulides, Y. Kim, Z. Chen, M. Soljačić, and M. Segev, “Bright spatial solitons on a partially incoherent background,” Phys. Rev. Lett. 84, 2374–2377 (2000).
[Crossref] [PubMed]

D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[Crossref] [PubMed]

M. Soljačić, 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]

For a recent review on optical spatial solitons, see G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Multimode incoherent spatial solitons in logarithmically saturable nonlinear media,” Phys. Rev. Lett. 80, 2310–2313 (1998).
[Crossref]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[Crossref] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[Crossref]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[Crossref]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (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]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, “Pattern formation via symmetry breaking in nonlinear weakly correlated systems,” to be published in Phys. Rev. E.

Shih, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (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]

Shklovskii, B. I.

A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, “Charge density wave in two-dimensional electron liquid in weak magnetic field,” Phys. Rev. Lett. 76, 499–502 (1996).
[Crossref] [PubMed]

Shkunov, V. V.

V. V. Shkunov and D. Z. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683–2686 (1998).
[Crossref]

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]

Slavin, A. N.

We note the distinction between the MI threshold and the existence of a threshold for soliton formation, the former being unique to partially incoherent systems, whereas the latter can be found in many coherent systems. For example, there is a clear threshold for envelope soliton formation in weakly dissipative systems; see A. N. Slavin, “Thresholds of envelope soliton formation in a weakly dissipative medium,” Phys. Rev. Lett. 77, 4644–4647 (1996).
[Crossref] [PubMed]

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422–1424 (1998). This ray optics model of incoherent solitons is similar to the model of random-phase solitons in plasma, suggested in Refs. 38 and 39.
[Crossref]

N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially coherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
[Crossref]

Soljacic, M.

C. Anastassiou, M. Soljačić, M. Segev, D. Kip, E. Eugenieva, D. N. Christodoulides, and Z. Musslimani, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
[Crossref] [PubMed]

T. Coskun, D. N. Christodoulides, Y. Kim, Z. Chen, M. Soljačić, and M. Segev, “Bright spatial solitons on a partially incoherent background,” Phys. Rev. Lett. 84, 2374–2377 (2000).
[Crossref] [PubMed]

D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[Crossref] [PubMed]

M. Soljačić, 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]

S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, “Pattern formation via symmetry breaking in nonlinear weakly correlated systems,” to be published in Phys. Rev. E.

Stegeman, G. I.

For a recent review on optical spatial solitons, see G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

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[Crossref]

Stepanov, S. I.

Stormer, H. L.

W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and nu=5/2 under a tilted magnetic field,” Phys. Rev. Lett. 83, 820–823 (1999).
[Crossref]

Sukhov, A. V.

N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zel’dovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1–140 (1986).
[Crossref]

Tabiryan, N. V.

N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zel’dovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1–140 (1986).
[Crossref]

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K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[Crossref] [PubMed]

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K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[Crossref] [PubMed]

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Torres-Cisneros, M.

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V. J. Emery, S. A. Kivelson, and J. M. Tranquada, “Stripe phases in high-temperature superconductors,” Proc. Natl. Acad. Sci. U.S.A. 96, 8814–8817 (1999).
[Crossref] [PubMed]

Tsui, D. C.

W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and nu=5/2 under a tilted magnetic field,” Phys. Rev. Lett. 83, 820–823 (1999).
[Crossref]

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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 media with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

Vishwanath, A.

M. Soljačić, 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]

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[Crossref] [PubMed]

West, K. W.

W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and nu=5/2 under a tilted magnetic field,” Phys. Rev. Lett. 83, 820–823 (1999).
[Crossref]

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for an anisotropic state of two-dimensional electrons in high Landau levels,” Phys. Rev. Lett. 82, 394–397 (1999).
[Crossref]

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Anisotropic states of two-dimensional electron systems in high Landau levels: effect of an in-plane magnetic field,” Phys. Rev. Lett. 83, 824–827 (1999).
[Crossref]

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S. R. White and D. J. Scalapino, “Competition between stripes and pairing in a t-t′-J model,” Phys. Rev. B 60, R753–R756 (1999).
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Yariv, A.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

Zel’dovich, B. Ya.

N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zel’dovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1–140 (1986).
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V. I. Karpman, “Self-modulation of nonlinear plane waves in dispersive media,” JETP Lett. 6, 277–280 (1967).

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Mol. Cryst. Liq. Cryst. (1)

N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zel’dovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1–140 (1986).
[Crossref]

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M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
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[Crossref]

A. Hasegawa, “Envelope soliton of random phase waves,” Phys. Fluids 20, 2155–2156 (1977). This model assumes zero degree of coherence, that is, every single point on the beam acts as an independent point source. This view is problematic for several reasons. First, zero-correlated beams can be described neither by a paraxial equation nor by a Helmholtz equation: They require full vectorial formulation through Maxwell equations. Second, this model implies that the shape of incoherent solitons is completely arbitrary and their correlation statistical properties are delocalized, which is just an artifact of using a transport equation—see Refs. 33-35. But worse than all other artifacts of the total incoherence assumption is that such transport (or ray optics) models are modulationally stable, i.e., there is no MI in these models. The reason is obvious: The MI threshold is a function that is monotonically increasing with decreasing correlation distance. For fully spatially incoherent beams the transverse correlation distance is zero, which implies that the threshold for MI is infinity. This is obviously unphysical and is a direct artifact of the full incoherence assumption.
[Crossref]

Phys. Rev. A (1)

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[Crossref] [PubMed]

Phys. Rev. B (2)

A. H. MacDonald and M. P. A. Fisher, “Quantum theory of quantum Hall smectics,” Phys. Rev. B 61, 5724–5733 (2000).
[Crossref]

S. R. White and D. J. Scalapino, “Competition between stripes and pairing in a t-t′-J model,” Phys. Rev. B 60, R753–R756 (1999).
[Crossref]

Phys. Rev. Lett. (22)

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Multimode incoherent spatial solitons in logarithmically saturable nonlinear media,” Phys. Rev. Lett. 80, 2310–2313 (1998).
[Crossref]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[Crossref]

A. W. Snyder and D. J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422–1424 (1998). This ray optics model of incoherent solitons is similar to the model of random-phase solitons in plasma, suggested in Refs. 38 and 39.
[Crossref]

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for an anisotropic state of two-dimensional electrons in high Landau levels,” Phys. Rev. Lett. 82, 394–397 (1999).
[Crossref]

M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Anisotropic states of two-dimensional electron systems in high Landau levels: effect of an in-plane magnetic field,” Phys. Rev. Lett. 83, 824–827 (1999).
[Crossref]

W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and nu=5/2 under a tilted magnetic field,” Phys. Rev. Lett. 83, 820–823 (1999).
[Crossref]

Th. Busch and J. R. Anglin, “Motion of dark solitons in trapped Bose–Einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[Crossref] [PubMed]

We note the distinction between the MI threshold and the existence of a threshold for soliton formation, the former being unique to partially incoherent systems, whereas the latter can be found in many coherent systems. For example, there is a clear threshold for envelope soliton formation in weakly dissipative systems; see A. N. Slavin, “Thresholds of envelope soliton formation in a weakly dissipative medium,” Phys. Rev. Lett. 77, 4644–4647 (1996).
[Crossref] [PubMed]

A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, “Charge density wave in two-dimensional electron liquid in weak magnetic field,” Phys. Rev. Lett. 76, 499–502 (1996).
[Crossref] [PubMed]

M. Soljačić, 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]

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[Crossref] [PubMed]

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[Crossref] [PubMed]

V. V. Shkunov and D. Z. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683–2686 (1998).
[Crossref]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[Crossref]

N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially coherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
[Crossref]

O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5483 (1999).
[Crossref]

C. Anastassiou, M. Soljačić, M. Segev, D. Kip, E. Eugenieva, D. N. Christodoulides, and Z. Musslimani, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
[Crossref] [PubMed]

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
[Crossref]

T. Coskun, D. N. Christodoulides, Y. Kim, Z. Chen, M. Soljačić, and M. Segev, “Bright spatial solitons on a partially incoherent background,” Phys. Rev. Lett. 84, 2374–2377 (2000).
[Crossref] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

V. J. Emery, S. A. Kivelson, and J. M. Tranquada, “Stripe phases in high-temperature superconductors,” Proc. Natl. Acad. Sci. U.S.A. 96, 8814–8817 (1999).
[Crossref] [PubMed]

Science (3)

D. Kip, M. Soljačić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[Crossref] [PubMed]

For a recent review on optical spatial solitons, see G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[Crossref] [PubMed]

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Other (6)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).

M. Artiglia, E. Ciaramella, and P. Gallina, “Demonstration of cw soliton trains at 10, 40 and 160 GHz by means of induced modulation instability,” in System Technologies, C. Menyuk and A. Willner, eds., Vol. 12 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 305–308.

S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, “Pattern formation via symmetry breaking in nonlinear weakly correlated systems,” to be published in Phys. Rev. E.

We also note that, in principle, effects that are closely related to the MI of partially incoherent wave fronts could also be observed with diffusive nonlinearities, that is, nonlinearities for which the nonlinear index change is a function of the optical intensity averaged over the characteristic diffusion length. Such diffusive effects exist in several material systems, e.g., in semiconductor lasers with free-carrier nonlinearities, where diffusion of charge carriers washes out all structures more delicate than the diffusion length. However, to our knowledge, MI in such systems has not been studied.

A Lorentzian power spectrum shape is not physical in 2D, since it would imply a logarithmically divergent intensity. Furthermore, one has reasons to doubt that it is a physically valid solution for all kx even in the 1D case. Nevertheless, it is a good approximation to the true shape (apart for the tails in large kx). Since this model can be solved in a closed formed analytically, we find it to be useful for studying and understanding MI. One should compare this situation with the approximation of parabolic waveguides in optical fibers, which serve as a good approximation of true graded-index waveguides, although of course true parabolic waveguides do not exist in reality.

The rotating diffuser also broadens the linewidth of the laser light, because the rotation causes a Doppler shift. In our experiments, however, the speckles introduce a new phase every microsecond, and therefore the Doppler shift is of the order of megahertz, which is negligible compared with the ∼1 GHz laser linewidth that we use.

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Figures (7)

Fig. 1
Fig. 1

Intensity structure of a partially spatially incoherent beam at the output plane of the nonlinear crystal. The sample is illuminated homogeneously with partially spatially incoherent light with a correlation distance lc=13 µm. The displayed area is 500 µm×500 µm. The size of the nonlinear refractive-index change of the crystal is successively increased from the linear case with, A, Δn0=0 to, B, 3.5×10-4; C, 5×10-4; D, 9×10-4. Plots B and C show the cases at threshold (with only partial features) and above threshold (features throughout) for 1D incoherent MI that leads to 1D filaments. Above this first threshold and at significantly higher values of the nonlinearity, the 1D filaments become unstable and, D, start to form a regular two-dimensional pattern.

Fig. 2
Fig. 2

Threshold dependence of incoherent MI. Shown is the modulation m=(Imax-Imin)/(Imax+Imin) of the light pattern as a function of the nonlinearity Δn0 and an intensity ratio I0/Isat=1. The values of m are measured for correlation distances lc=6, 8, 10, 17.5 µm and for fully coherent light (lc). The dotted curves are guides for the eye.

Fig. 3
Fig. 3

Intensity cross sections of the stripes for lc=17.5 µm and nonlinear refractive-index changes Δn0 of, i, 2.75×10-4; ii, 4×10-4; iii, 5×10-4; iv, 8×10-4. The dotted lines indicate the base line of the respective profile. The stripes emerge as sinusoidal stripes (for nonlinearity just above threshold), turn into square-wave stripes at a higher nonlinearity, and eventually break up into filaments at a large enough nonlinearity.

Fig. 4
Fig. 4

Dominating spatial frequency fmax (stripes per unit length) as a function of correlation distance lc; A, experiment; B, theory. All measured spatial frequencies are for an experimental parameter range where the 1D filaments are stable and 2D instability is not yet visible. The theoretical curves are deduced from Eq. (10) and λ=514.5 nm, ne=2.3, and r33=260 pm/V. The dotted curve in A is a guide for the eye.

Fig. 5
Fig. 5

Dominating spatial frequency fmax as a function of nonlinear refractive-index change Δn0; A, experiment; B, theory. The dotted curves in A are guides for the eye; the theoretical curves in B are calculated with Eq. (10) and λ=514.5 nm, ne=2.3, and r33=260 pm/V.

Fig. 6
Fig. 6

Spatial frequency fmax, modulation m, and gain coefficient g as a function of intensity ratio I0/Isat of signal and background beam; A, C, experiment; B, D, theory. All curves are for a spatial correlation distance lc=8 µm. The dotted curves in A and C are merely guides for the eye. The theoretical curves are deduced from Eq. (10) and λ=514.5 nm, nc=2.3, and r33=260 pm/V.

Fig. 7
Fig. 7

Suppression of incoherent MI due to saturation of the nonlinearity. A, Intensity structure of a finite signal beam (Gaussian beam with a FWHM of 1 mm) at the output plane of the crystal. The intensity ratio (peak of beam to background intensity) is I0/Isat=3. Without nonlinearity (Δn0=0), the output beam shows no features. The photograph is taken for Δn0=6×10-4. The saturable nature of the nonlinearity clearly suppresses MI in the beam center, whereas strong modulation and filaments of random orientation occur in the margins of the beam. B, Intensity profile obtained with numerical simulations; this illustrates the MI inhibition caused by the saturation of the nonlinearity. Parameters match those used in the experimental configuration, with a beam intensity ratio of I0/Isat=3, a crystal nonlinearity of the saturable form, and Δn0=6×10-4. Ripples due to MI appear at the edges of the beam where the intensity tapers off.

Equations (13)

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Bz-ik2Brρ=in0kωc2{δn(r1, z)-δn(r2, z)}B,
B1z-ik2B1rρ=in0kωc2κB1r+ρ2, ρ=0, z-B1r-ρ2, ρ=0, zB0(ρ).
B1(r, ρ, z)=exp(gz)exp[i(αr+ϕ)]L(ρ)+exp(g*z)exp[-i(αr+ϕ)]L*(-ρ),
gM(ρ)+αkdM(ρ)dρ+2ωκcsinρα2B0(ρ)=0.
Mˆ(kx)g-iαkkx=iωκcBˆ0kx+α2-Bˆ0kx-α2,
Fˆ(kx)=12π-dρF(ρ)exp(ikxρ)
1=-ωκc-dkxBˆ0kx+α2-Bˆ0kx-α21ig+αkx/k,
gk=-(kx0/k)(|α|/k)+(|α|/k)κI0n0-α2k21/2.
1=-κk2n0-dkxkxdBˆ0dkx.
1=-k2κn0-dkxkxddαBˆ0kx+α2-Bˆ0kx-α2αmax.
δn=Δn01+I0IsatI(r)I(r)+Isat,
αmaxk=2A-θ022-(2Aθ02+θ04/4)1/21/2,
δn=Δn0I(r)I(r)+Isat.

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