Abstract

We report picosecond pulsed experiments and numerical simulations of spatially induced modulational instability, which we used to form stable periodic arrays of bright soliton beams in a planar Kerr-like CS2 waveguide. We have found that the generation stage of these arrays is accurately described by the usual nonlinear Schrödinger wave equation, whereas the temporal dynamics of the nonlinearity is beneficial for subsequent stable propagation of the soliton arrays. In the picosecond regime the finite molecular relaxation time of the reorientational nonlinear index inhibits the Fermi–Pasta–Ulam recurrence predicted for an instantaneous Kerr nonlinearity. Moreover, the inhibition is associated with a novel spatiotemporal dynamics confirmed by numeric and streak-camera recordings.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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  4. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  5. A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
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  7. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
    [CrossRef] [PubMed]
  8. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).
  9. V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494–500 (1974).
  10. A. Barthélémy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
    [CrossRef]
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  14. C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85, 4888–4891 (2000).
    [CrossRef] [PubMed]
  15. K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability,” Appl. Phys. Lett. 49, 236–238 (1986).
    [CrossRef]
  16. G. Millot, E. Seve, S. Wabnitz, and M. Haelterman, “Observation of induced modulation polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber,” J. Opt. Soc. Am. B 15, 1266–1277 (1998).
    [CrossRef]
  17. E. Seve, G. Millot, S. Wabnitz, T. Sylvestre, and H. Maillotte, “Generation of vector dark-soliton trains by induced modulation instability in a highly birefringent fiber,” J. Opt. Soc. Am. B 16, 1642–1650 (1999).
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  18. M. Nakazawa, A. Suzuki, and H. A. Haus, “The modulational instability laser. I. Experiment,” IEEE J. Quantum Electron. 25, 2036–2044 (1989).
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  19. M. Nakazawa, A. Suzuki, H. Kubota, and H. A. Haus, “The modulational instability laser. II. Theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
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  20. S. Coen and M. Haelterman, “Impedance-matched modulational instability laser for background-free pulse train generation in the THz range,” Opt. Commun. 146, 339–346 (1998).
    [CrossRef]
  21. S. Coen and M. Haelterman, “Continuous-wave ultrahigh-repetition-rate pulse train generation through modulational instability in a passive fiber cavity,” Opt. Lett. 26, 39–41 (2001).
    [CrossRef]
  22. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
    [CrossRef]
  23. A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
    [CrossRef] [PubMed]
  24. R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
    [CrossRef]
  25. H. Fang, R. Malendevich, R. Schiek, and G. I. Stegeman, “Spatial modulational instability in one-dimensional lithium niobate slab waveguides,” Opt. Lett. 25, 1786–1788 (2000).
    [CrossRef]
  26. N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in optical fiber: exact solutions,” Sov. Phys. JETP 62, 894–899 (1985).
  27. N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, London, 1997), pp. 27–76.
  28. P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, “Restoration of dual frequency signals with nonlinear propagation in fibers with positive group velocity dispersion,” Appl. Phys. Lett. 64, 3374–3376 (1994).
    [CrossRef]
  29. D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave-mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
    [CrossRef]
  30. 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]
  31. P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
    [CrossRef]
  32. 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]
  33. G. Van Simaeys, P. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi–Pasta–Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
    [CrossRef]
  34. R. A. Sammut, C. Pask, and Q. Y. Li, “Theoretical study of spatial solitons in planar waveguides,” J. Opt. Soc. Am. B 10, 485–491 (1993).
    [CrossRef]
  35. S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulation instability in optical fibers,” Opt. Lett. 16, 986–988 (1991).
    [CrossRef] [PubMed]
  36. S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency-pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
    [CrossRef] [PubMed]
  37. S. Maneuf and F. Reynaud, “Quasi-steady state self-trapping of first, second and third order sub-nanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
    [CrossRef]
  38. In fact, in Ref. 33 the recurrence was experimentally demonstrated by induction of MI with square-shaped pulses in a fiber. In this way, MI was negligibly influenced by either the rear or the front side of the pulse.
  39. N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP 67, 89–95 (1988).
  40. Note that, rigorously, frequency detuning should be in-creased when I0 increases, in accordance with the change of variables of Eqs. (5), to keep the same dimensionless form of Eq. (4). Nevertheless, this is not strictly necessary when one is attempting to determine whether the MI-generated solitonlike arrays are recurrent.
  41. J.-M. Halbout and C. L. Tang, “Femtosecond interferometry for nonlinear optics,” Appl. Phys. Lett. 40, 765–767 (1982).
    [CrossRef]
  42. D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron. 24, 443–454 (1988).
    [CrossRef]
  43. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 159–190.
  44. E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
    [CrossRef]
  45. S. Trillo, S. Wabnitz, G. I. Stegeman, and E. M. Wright, “Parametric amplification and modulational instabilities in dispersive nonlinear directional couplers with relaxing nonlinearity,” J. Opt. Soc. Am. B 6, 889–900 (1989).
    [CrossRef]
  46. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995), pp. 50–55.

2001 (2)

S. Coen and M. Haelterman, “Continuous-wave ultrahigh-repetition-rate pulse train generation through modulational instability in a passive fiber cavity,” Opt. Lett. 26, 39–41 (2001).
[CrossRef]

G. Van Simaeys, P. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi–Pasta–Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef]

2000 (2)

H. Fang, R. Malendevich, R. Schiek, and G. I. Stegeman, “Spatial modulational instability in one-dimensional lithium niobate slab waveguides,” Opt. Lett. 25, 1786–1788 (2000).
[CrossRef]

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

1999 (2)

1998 (5)

N. N. Akhmediev, “Spatial solitons in Kerr and Kerr-like media,” Opt. Quantum Electron. 30, 535–569 (1998).
[CrossRef]

M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

G. Millot, E. Seve, S. Wabnitz, and M. Haelterman, “Observation of induced modulation polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber,” J. Opt. Soc. Am. B 15, 1266–1277 (1998).
[CrossRef]

S. Coen and M. Haelterman, “Impedance-matched modulational instability laser for background-free pulse train generation in the THz range,” Opt. Commun. 146, 339–346 (1998).
[CrossRef]

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave-mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

1997 (1)

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

1996 (2)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[CrossRef]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

1994 (4)

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, “Restoration of dual frequency signals with nonlinear propagation in fibers with positive group velocity dispersion,” Appl. Phys. Lett. 64, 3374–3376 (1994).
[CrossRef]

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]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency-pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
[CrossRef] [PubMed]

H. Maillotte, J. Monneret, A. Barthélémy, and C. Froehly, “Laser beam self-splitting into solitons by optical Kerr nonlinearity,” Opt. Commun. 109, 265–271 (1994).
[CrossRef]

1993 (1)

1991 (2)

S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulation instability in optical fibers,” Opt. Lett. 16, 986–988 (1991).
[CrossRef] [PubMed]

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
[CrossRef]

1989 (4)

1988 (3)

D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron. 24, 443–454 (1988).
[CrossRef]

S. Maneuf and F. Reynaud, “Quasi-steady state self-trapping of first, second and third order sub-nanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[CrossRef]

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP 67, 89–95 (1988).

1986 (3)

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability,” Appl. Phys. Lett. 49, 236–238 (1986).
[CrossRef]

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

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
[CrossRef]

1985 (2)

A. Barthélémy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[CrossRef]

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in optical fiber: exact solutions,” Sov. Phys. JETP 62, 894–899 (1985).

1984 (1)

1982 (1)

J.-M. Halbout and C. L. Tang, “Femtosecond interferometry for nonlinear optics,” Appl. Phys. Lett. 40, 765–767 (1982).
[CrossRef]

1980 (1)

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

1974 (1)

V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494–500 (1974).

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1966 (2)

E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. 19, 347–349 (1966).
[CrossRef]

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

Akhmediev, N. N.

N. N. Akhmediev, “Spatial solitons in Kerr and Kerr-like media,” Opt. Quantum Electron. 30, 535–569 (1998).
[CrossRef]

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP 67, 89–95 (1988).

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in optical fiber: exact solutions,” Sov. Phys. JETP 62, 894–899 (1985).

Anastassiou, C.

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

Anderson, D. Z.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

Baboiu, D.-M.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Barthélémy, A.

H. Maillotte, J. Monneret, A. Barthélémy, and C. Froehly, “Laser beam self-splitting into solitons by optical Kerr nonlinearity,” Opt. Commun. 109, 265–271 (1994).
[CrossRef]

A. Barthélémy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[CrossRef]

Beletic, J. W.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave-mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[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.

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, “Restoration of dual frequency signals with nonlinear propagation in fibers with positive group velocity dispersion,” Appl. Phys. Lett. 64, 3374–3376 (1994).
[CrossRef]

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]

Brinkman, W. F.

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

Chernikov, S. V.

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
[CrossRef]

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]

Chiao, R. Y.

E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. 19, 347–349 (1966).
[CrossRef]

Christodoulides, D. N.

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

Coen, S.

S. Coen and M. Haelterman, “Continuous-wave ultrahigh-repetition-rate pulse train generation through modulational instability in a passive fiber cavity,” Opt. Lett. 26, 39–41 (2001).
[CrossRef]

S. Coen and M. Haelterman, “Impedance-matched modulational instability laser for background-free pulse train generation in the THz range,” Opt. Commun. 146, 339–346 (1998).
[CrossRef]

Dianov, E. M.

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
[CrossRef]

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]

Eleonskii, V. M.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in optical fiber: exact solutions,” Sov. Phys. JETP 62, 894–899 (1985).

Emplit, P.

G. Van Simaeys, P. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi–Pasta–Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef]

Eugenieva, E. D.

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

Fang, H.

Froehly, C.

H. Maillotte, J. Monneret, A. Barthélémy, and C. Froehly, “Laser beam self-splitting into solitons by optical Kerr nonlinearity,” Opt. Commun. 109, 265–271 (1994).
[CrossRef]

A. Barthélémy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[CrossRef]

Fuerst, R. A.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Garmire, E.

E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. 19, 347–349 (1966).
[CrossRef]

Haelterman, M.

S. Coen and M. Haelterman, “Continuous-wave ultrahigh-repetition-rate pulse train generation through modulational instability in a passive fiber cavity,” Opt. Lett. 26, 39–41 (2001).
[CrossRef]

G. Van Simaeys, P. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi–Pasta–Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef]

G. Millot, E. Seve, S. Wabnitz, and M. Haelterman, “Observation of induced modulation polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber,” J. Opt. Soc. Am. B 15, 1266–1277 (1998).
[CrossRef]

S. Coen and M. Haelterman, “Impedance-matched modulational instability laser for background-free pulse train generation in the THz range,” Opt. Commun. 146, 339–346 (1998).
[CrossRef]

Halbout, J.-M.

J.-M. Halbout and C. L. Tang, “Femtosecond interferometry for nonlinear optics,” Appl. Phys. Lett. 40, 765–767 (1982).
[CrossRef]

Hart, D. L.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave-mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

Hasegawa, A.

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability,” Appl. Phys. Lett. 49, 236–238 (1986).
[CrossRef]

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 modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[CrossRef]

Haus, H. A.

M. Nakazawa, A. Suzuki, and H. A. Haus, “The modulational instability laser. I. Experiment,” IEEE J. Quantum Electron. 25, 2036–2044 (1989).
[CrossRef]

M. Nakazawa, A. Suzuki, H. Kubota, and H. A. Haus, “The modulational instability laser. II. Theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

Jewell, J. L.

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability,” Appl. Phys. Lett. 49, 236–238 (1986).
[CrossRef]

Judy, A. F.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave-mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

Kennedy, T. A. B.

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency-pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
[CrossRef] [PubMed]

Kenney-Wallace, G. A.

D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron. 24, 443–454 (1988).
[CrossRef]

Kip, D.

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

Korneev, V. I.

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP 67, 89–95 (1988).

Kubota, H.

M. Nakazawa, A. Suzuki, H. Kubota, and H. A. Haus, “The modulational instability laser. II. Theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

Kulagin, N. E.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in optical fiber: exact solutions,” Sov. Phys. JETP 62, 894–899 (1985).

Kuznetsov, E. A.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
[CrossRef]

Lawrence, B.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Li, Q. Y.

Lotshaw, W. T.

D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron. 24, 443–454 (1988).
[CrossRef]

Maillotte, H.

E. Seve, G. Millot, S. Wabnitz, T. Sylvestre, and H. Maillotte, “Generation of vector dark-soliton trains by induced modulation instability in a highly birefringent fiber,” J. Opt. Soc. Am. B 16, 1642–1650 (1999).
[CrossRef]

H. Maillotte, J. Monneret, A. Barthélémy, and C. Froehly, “Laser beam self-splitting into solitons by optical Kerr nonlinearity,” Opt. Commun. 109, 265–271 (1994).
[CrossRef]

Malendevich, R.

Mamaev, A. V.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[CrossRef]

Mamyshev, P. V.

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, “Restoration of dual frequency signals with nonlinear propagation in fibers with positive group velocity dispersion,” Appl. Phys. Lett. 64, 3374–3376 (1994).
[CrossRef]

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]

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
[CrossRef]

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]

Maneuf, S.

S. Maneuf and F. Reynaud, “Quasi-steady state self-trapping of first, second and third order sub-nanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[CrossRef]

A. Barthélémy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[CrossRef]

McMorrow, D.

D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron. 24, 443–454 (1988).
[CrossRef]

Millot, G.

Mitskevich, N. V.

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP 67, 89–95 (1988).

Monneret, J.

H. Maillotte, J. Monneret, A. Barthélémy, and C. Froehly, “Laser beam self-splitting into solitons by optical Kerr nonlinearity,” Opt. Commun. 109, 265–271 (1994).
[CrossRef]

Musslimani, Z. H.

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

Nakazawa, M.

M. Nakazawa, A. Suzuki, and H. A. Haus, “The modulational instability laser. I. Experiment,” IEEE J. Quantum Electron. 25, 2036–2044 (1989).
[CrossRef]

M. Nakazawa, A. Suzuki, H. Kubota, and H. A. Haus, “The modulational instability laser. II. Theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

Pask, C.

Prokhorov, A. M.

Reynaud, F.

S. Maneuf and F. Reynaud, “Quasi-steady state self-trapping of first, second and third order sub-nanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[CrossRef]

Roy, R.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave-mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

Rubenchik, A. M.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
[CrossRef]

V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494–500 (1974).

Saffman, M.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[CrossRef]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

Sammut, R. A.

Schiek, R.

Sears, S.

M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

Segev, M.

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

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[CrossRef] [PubMed]

M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

Seve, E.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Soljacic, M.

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

M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

Stegeman, G. I.

H. Fang, R. Malendevich, R. Schiek, and G. I. Stegeman, “Spatial modulational instability in one-dimensional lithium niobate slab waveguides,” Opt. Lett. 25, 1786–1788 (2000).
[CrossRef]

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[CrossRef] [PubMed]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

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]

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, “Restoration of dual frequency signals with nonlinear propagation in fibers with positive group velocity dispersion,” Appl. Phys. Lett. 64, 3374–3376 (1994).
[CrossRef]

S. Trillo, S. Wabnitz, G. I. Stegeman, and E. M. Wright, “Parametric amplification and modulational instabilities in dispersive nonlinear directional couplers with relaxing nonlinearity,” J. Opt. Soc. Am. B 6, 889–900 (1989).
[CrossRef]

Suzuki, A.

M. Nakazawa, A. Suzuki, and H. A. Haus, “The modulational instability laser. I. Experiment,” IEEE J. Quantum Electron. 25, 2036–2044 (1989).
[CrossRef]

M. Nakazawa, A. Suzuki, H. Kubota, and H. A. Haus, “The modulational instability laser. II. Theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

Sylvestre, T.

Tai, K.

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

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability,” Appl. Phys. Lett. 49, 236–238 (1986).
[CrossRef]

Talanov, V. I.

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

Tang, C. L.

J.-M. Halbout and C. L. Tang, “Femtosecond interferometry for nonlinear optics,” Appl. Phys. Lett. 40, 765–767 (1982).
[CrossRef]

Tomita, A.

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

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability,” Appl. Phys. Lett. 49, 236–238 (1986).
[CrossRef]

Torres, J. P.

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

Torruellas, W. E.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Townes, C. H.

E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. 19, 347–349 (1966).
[CrossRef]

Trillo, S.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency-pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
[CrossRef] [PubMed]

S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulation instability in optical fibers,” Opt. Lett. 16, 986–988 (1991).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, G. I. Stegeman, and E. M. Wright, “Parametric amplification and modulational instabilities in dispersive nonlinear directional couplers with relaxing nonlinearity,” J. Opt. Soc. Am. B 6, 889–900 (1989).
[CrossRef]

Van Simaeys, G.

G. Van Simaeys, P. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi–Pasta–Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef]

Wabnitz, S.

Wigley, P. G.

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, “Restoration of dual frequency signals with nonlinear propagation in fibers with positive group velocity dispersion,” Appl. Phys. Lett. 64, 3374–3376 (1994).
[CrossRef]

Wilson, J.

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, “Restoration of dual frequency signals with nonlinear propagation in fibers with positive group velocity dispersion,” Appl. Phys. Lett. 64, 3374–3376 (1994).
[CrossRef]

Wright, E. M.

Zakharov, V. E.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
[CrossRef]

V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494–500 (1974).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Zozulya, A. A.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[CrossRef]

Appl. Phys. Lett. (3)

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability,” Appl. Phys. Lett. 49, 236–238 (1986).
[CrossRef]

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, “Restoration of dual frequency signals with nonlinear propagation in fibers with positive group velocity dispersion,” Appl. Phys. Lett. 64, 3374–3376 (1994).
[CrossRef]

J.-M. Halbout and C. L. Tang, “Femtosecond interferometry for nonlinear optics,” Appl. Phys. Lett. 40, 765–767 (1982).
[CrossRef]

Europhys. Lett. (1)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[CrossRef]

IEEE J. Quantum Electron. (5)

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

M. Nakazawa, A. Suzuki, and H. A. Haus, “The modulational instability laser. I. Experiment,” IEEE J. Quantum Electron. 25, 2036–2044 (1989).
[CrossRef]

M. Nakazawa, A. Suzuki, H. Kubota, and H. A. Haus, “The modulational instability laser. II. Theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, “Femtosecond optical Kerr studies on the origin of the nonlinear responses in simple liquids,” IEEE J. Quantum Electron. 24, 443–454 (1988).
[CrossRef]

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
[CrossRef]

J. Opt. Soc. Am. B (5)

JETP Lett. (1)

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

Opt. Commun. (4)

A. Barthélémy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[CrossRef]

H. Maillotte, J. Monneret, A. Barthélémy, and C. Froehly, “Laser beam self-splitting into solitons by optical Kerr nonlinearity,” Opt. Commun. 109, 265–271 (1994).
[CrossRef]

S. Coen and M. Haelterman, “Impedance-matched modulational instability laser for background-free pulse train generation in the THz range,” Opt. Commun. 146, 339–346 (1998).
[CrossRef]

S. Maneuf and F. Reynaud, “Quasi-steady state self-trapping of first, second and third order sub-nanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[CrossRef]

Opt. Lett. (5)

Opt. Quantum Electron. (1)

N. N. Akhmediev, “Spatial solitons in Kerr and Kerr-like media,” Opt. Quantum Electron. 30, 535–569 (1998).
[CrossRef]

Phys. Rep. (1)

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
[CrossRef]

Phys. Rev. A (2)

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency-pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
[CrossRef] [PubMed]

Phys. Rev. E (1)

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave-mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

Phys. Rev. Lett. (6)

E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. 19, 347–349 (1966).
[CrossRef]

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

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

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

G. Van Simaeys, P. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi–Pasta–Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef]

Science (1)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[CrossRef] [PubMed]

Sov. Phys. JETP (4)

V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494–500 (1974).

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in optical fiber: exact solutions,” Sov. Phys. JETP 62, 894–899 (1985).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP 67, 89–95 (1988).

Other (6)

Note that, rigorously, frequency detuning should be in-creased when I0 increases, in accordance with the change of variables of Eqs. (5), to keep the same dimensionless form of Eq. (4). Nevertheless, this is not strictly necessary when one is attempting to determine whether the MI-generated solitonlike arrays are recurrent.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 159–190.

In fact, in Ref. 33 the recurrence was experimentally demonstrated by induction of MI with square-shaped pulses in a fiber. In this way, MI was negligibly influenced by either the rear or the front side of the pulse.

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, London, 1997), pp. 27–76.

H. Maillotte and R. Grasser, “Generation and propagation of stable periodic arrays of soliton stripes in a bulk Kerr liquid,” in Nonlinear Guided Waves and Their Applications, Vol. 5 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 167–169.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995), pp. 50–55.

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Figures (10)

Fig. 1
Fig. 1

Analytical results. (a) Unstable evolution [from Eq. (4)] of a (1+1)D plane wave modulated with period p=125 µm, with C0=20% initial contrast and where I0=130 MW/cm2. (b) Evolution of the spatial Fourier spectrum, showing periodic energy exchange between the pump and the ±nΩ harmonics (three are displayed here). (c) Intensity versus phase, showing the periodic backevolution to the initially modulated wave (FPU recurrence). Filled arrows, the evolution of a minimum of the initial sine modulation; open arrows, that of a maximum. For a non-plane-wave initial condition (here C0=20%), a cumulative phase shift is introduced after each recurrence period in z (two are displayed here).

Fig. 2
Fig. 2

Simplified experimental schematic of picosecond spatially induced MI.

Fig. 3
Fig. 3

Experimental observation of the formation of a solitonlike periodic array after 7 cm of propagation with a 125-µm modulation period and 20% initial contrast: (a) linear regime with a sinusoidal output pattern, (b) nonlinear regime for I0135 MW/cm2, (c) nonlinear output profile exhibiting the solitonlike array, (d) corresponding output linear and nonlinear intensity spatial spectra obtained in the Fourier domain. (e)–(g) Comparison of a cw one-dimensional SSF numerical simulation with the above experimental spatial parameters and I0=130 MW/cm2. (e) Top view of the propagation in the waveguide with (f) corresponding input (dotted curve) and output (solid curve) profiles and (g) evolution of the Fourier spectrum.

Fig. 4
Fig. 4

Pulsed SSF simulation of spatially induced MI with only the +Ω sideband at the beginning of the process. The spatial parameters are the same as in Fig. 3, and the initial Gaussian temporal envelope lasts for 38 ps (FWHM). (a)–(c) Time-integrated patterns showing (a) the top view of the propagation, (b) input sinusoidal modulation (dotted curve) and output profile at the maximum amplification stage (solid curve) at 6.2 cm of propagation and (c) the corresponding Fourier intensity spatial spectra. Bending of the whole array and asymmetric background intensity in the output profiles can be seen.

Fig. 5
Fig. 5

SSF simulation of splitting of the 38-ps soliton array temporal envelope owing to spatially induced MI (spatial parameters are p=110 µm and C0=15%). (a) Space–time pattern after propagation for L=7 cm with I0=400 MW/cm2 (twice that necessary for spontaneous generation at period p). (b) Pulse envelope profiles: solid curve, a maximum of the initial spatial sine modulation, dashed curve, a minimum.

Fig. 6
Fig. 6

Computed output contrast of the spatial MI-induced periodic array (p=125 µm,C0=20%) versus the input mean intensity I0 with a fixed propagation length (L=7 cm.) Comparison of the steady-state spatially sine-modulated cw plane wave (filled stars) and the pulsed wave (the same spatial distribution supported by a 38-ps Gaussian pulse) with (open circles) and without (open stars) relaxation of the nonlinearity. In the pulsed cases, we calculate modulation contrast by accounting for time integration by the CCD camera.

Fig. 7
Fig. 7

Experimental output images with 110-µm period and 15% initial contrast: (a) generation of a periodic array at I0184 MW/cm2, (b) spatial-frequency doubling with increasing intensity (I0400 MW/cm2), (c) corresponding profiles showing the array formation threshold (dashed curve) and spatial-frequency doubling at higher intensity (solid curve).

Fig. 8
Fig. 8

Pulsed SSF simulations of the generation and propagation of a solitonlike array with p=110 µm, C0=15%, a 38-ps Gaussian pulse, and I0=400 MW/cm2. The profiles are given at the z coordinate that corresponds to the second maximum amplification distance at ∼6.5 cm. Time-integrated transverse profiles and top views for (a) instantaneous nonlinearity, for which the expected periodic recurrent behavior is partially hidden because of the pulsed excitation, (b) with a 2-ps relaxation time of the nonlinearity, for which there are inhibition of the recurrence and spatial-frequency doubling.

Fig. 9
Fig. 9

Intensity versus phase that correspond to the simulations in Fig. 8: (a) without and (b) with a 2-ps relaxation time, showing the z evolution at time t that corresponds to the peak of the Gaussian pulse. Solid curves, a maximum of the initial spatial sine modulation; dashed curves, a minimum. With the relaxation, damping oscillation of the maxima and an almost asymptotic decrease to zero in the minima can be seen.

Fig. 10
Fig. 10

Evolution of the temporal envelope of the array at the output of the waveguide, corresponding to Figs. 7(b) and 8(b). (a) Streak-camera experimental image, showing the spatiotemporal dynamics of the soliton array, (b) simulated space–time envelope with 2-ps relaxation.

Equations (11)

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n(x, y)=n0(y)+n2I(x, y),
iψz+12β2ψx2+12c0n0γ|ψ|2ψ=0,
δiK=ImΩΩ24β2-n2I0n01/2
φ(X, Z)
=(1-4a)cosh(ΔZ/2)+2a cos(ΓX)+iΔ sinh(ΔZ/2)2[cosh(ΔZ/2)-2a cos(ΓX)]×exp(iZ/2).
Z=(q2/λ0)z,X=q(β/λ0)1/2x,
φ=1q12c0n0λ0γ1/2ψ,
iφt+uφz+122φx2+Pφ=0,AˆP=|φ|2.
AˆP=τPt+P=|φ|2,
ψx, z+Δz2, t+=ψx, z+Δz2, t- exp(iΔβΔz),
Δβ=12γ-t[I(x, z, t)+I(x, z+Δz, t)]exp[-(t-t)/τ]dt0+exp[-t/τ]dt=γ2τ-t[I(x, z, t)+I(x, z+Δz, t)]exp[-(t-t)/τ]dt.

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