(1 + 1)-Dimensional modulation instability of spatially incoherent light

Journal of the Optical Society of America B: Optical Physics

Volumn 19, Issue 3, 2002, Pages 502-512

  Kip, Detlef ^a,b   Soljačić, Marin ^c   Segev, Mordechai ^a,d   Sears, Suzanne M ^e,f   Christodoulides, Demetrios N ^f  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b UNIVERSITY OF OSNABRÜCK   (Germany)

^c MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^d Princeton University   (United States)

^e PRINCETON UNIVERSITY   (United States)

^f Lehigh University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COHERENT LIGHT; ELECTROMAGNETIC DISPERSION; PHASE MODULATION; WAVEFRONTS;

SPATIALLY INCOHERENT LIGHT;

NONLINEAR OPTICS;

EID: 0001182462     PISSN: 07403224     EISSN: None     Source Type: Journal    
DOI: 10.1364/JOSAB.19.000502     Document Type: Conference Paper

References (55)

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 soli tons in the regime of effective amplification," J. Opt. Soc. Am. B 11, 1254-1260 (1994).
3. 20 crystal," Opt. Lett. 20, 1853-1855 (1995).
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).
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).
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).
9. K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
10. G. P. Agrawal, "Modulation instability induced by cross-phase modulation," Phys. Rev. Lett. 59, 880-883 (1987).
11. S. Wabnitz, "Modulational polarization instability of light in a nonlinear birefringent dispersive medium," Phys. Rev. A 38, 2018-2021 (1988).
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).
14. A. Hasegawa, "Generation of a train of soliton pulses by induced modulational instability in optical fibers," Opt. Lett. 9, 288-290 (1984).
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).
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).
18. J. Klinger, H. Martin, and Z. Chen, "Experiments on induced modulational instability of an incoherent optical beam," Opt. Lett. 26, 271-273 (2000).
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).
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).
22. A. H. MacDonald and M. P. A. Fisher, "Quantum theory of quantum Hall smectics," Phys. Rev. B 61, 5724-5733 (2000).
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).
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).
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 v = 9/2 and nu = 5/2 under a tilted magnetic field," Phys. Rev. Lett. 83, 820-823 (1999).
26. Th. Busch and J. R. Anglin, "Motion of dark solitons in trapped Bose-Einstein condensates," Phys. Rev. Lett. 84, 2298-2301 (2000).
27. F. G. Bridges, A. Hatzes, and D. N. C. Lin, "Structure, stability and evolution of Saturn's rings," Nature 309, 333-335 (1984).
28. K. Bekki, "Group-cluster merging and the formation of starburst galaxies," Astrophys. J. 810, 15-19 (1999).
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).
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).
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).
33. M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature 387, 880-883 (1997).
34. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of incoherent self-focusing in biased photo-refractive media, " Phys. Rev. Lett. 78, 646-649 (1997).
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).
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).
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.
38. A. Hasegawa, "Dynamics of an ensemble of plane waves in nonlinear dispersive media," Phys. Fluids 18, 77-78 (1975).
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.
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).
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).
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).
44. N. Akhmediev, W. Krolikowski, and A. W. Snyder, "Partially coherent solitons of variable shape," Phys. Rev. Lett. 81, 4632-4635 (1998).
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).
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).
47. M. Mitchell, M. Segev, and D. N. Christodoulides, "Observation of multihump multimode solitons," Phys. Rev. Lett. 80, 4657-4660 (1998).
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).
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).
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).
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).
52. M. Peccianti and G. Assanto, "Incoherent spatial solitary waves in nematic liquid crystal," Opt. Lett. 26, 1791-1973 (2001).
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).
54. x). 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.