Abstract

We study, numerically, the influence of third-order dispersion (TOD) on modulation instability (MI) in optical fibers described by the extended nonlinear Schrödinger equation. We consider two MI scenarios. One starts with a continuous wave (CW) with a small amount of white noise, while the second one starts with a CW with a small harmonic perturbation at the highest value of the growth rate. In each case, the MI spectra show an additional spectral feature that is caused by Cherenkov radiation. We give an analytic expression for its frequency. Taking a single frequency of modulation instead of a noisy CW leads to the Fermi–Pasta–Ulam (FPU) recurrence dynamics. In this case, the radiation spectral feature multiplies due to the four-wave mixing process. FPU recurrence dynamics is quite pronounced at small values of TOD, disappears at intermediate values, and is restored again at high TOD when the Cherenkov frequency enters the MI band. Our results may lead to a better understanding of the role of TOD in optical fibers.

© 2012 Optical Society of America

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  1. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibres,” Phys. Rev. 51, 2602–2607 (1995).
    [CrossRef]
  2. J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
    [CrossRef]
  3. M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Phys. Lett. A 374, 691–695 (2010).
    [CrossRef]
  4. G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
    [CrossRef]
  5. N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Could rogue waves be used as efficient weapons against enemy ships?,” Eur. J. Phys. Special Topics 185, 259–266 (2010).
    [CrossRef]
  6. N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of a continuous signal in an optical fibre taking into account third order dispersion,” Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika 33, 111–117, (1990).
  7. M. I. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
    [CrossRef]
  8. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, 1997).
  9. M. Droques, B. Barviau, A. Kudlinski, M. Taki, A. Boucon, T. Sylvestre, and A. Mussot, “Symmetry-breaking dynamics of the modulational instability spectrum,” Opt. Lett. 36, 1359–1361 (2011).
    [CrossRef]
  10. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310, (1966).
  11. T. B. Benjamin and J. E. Feir, “The disintegration of wavetrains on deep water. Part 1: theory,” J. Fluid Mech. 27, 417–430 (1967).
    [CrossRef]
  12. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [CrossRef]
  13. D. Kip, Marin Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
    [CrossRef]
  14. A. V. Gorbach, X. Zhao, and D. V. Skryabin, “Dispersion of nonlinearity and modulation instability in sub-wavelength semiconductor waveguides,” Opt. Express 19, 9345–9351 (2011).
    [CrossRef]
  15. K. Porsezian, K. Senthilnathan, and S. Devipriya, “Modulation instability in Fiber Bragg grating with non-Kerr nonlinearity,” IEEE J. Quantum Electron. 41, 789–796 (2005).
    [CrossRef]
  16. N. C. Panoiu, X. F. Chen, and R. M. Osgood, “Modulation instability in silicon photonic nanowiresa,” Opt. Lett. 31, 3609–3611 (2006).
    [CrossRef]
  17. K. Kasamatsu and M. Tsubota, “Modulation instability and solitary-wave formation in two-component Bose-Einstein condensates,” Phys. Rev. A 74, 013617 (2006).
    [CrossRef]
  18. N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, and J. M. Dudley, “Universal triangular spectra in parametrically-driven systems,” Phys. Lett. 375, 775–779 (2011).
    [CrossRef]
  19. 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]
  20. N. Akhmediev, “Deja vu in optics,” Nature 413, 267–268 (2001).
    [CrossRef]
  21. S. Wabnitz and N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
    [CrossRef]
  22. M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
    [CrossRef]
  23. M. J. Potasek, “Modulation instability in an extended nonlinear Schrödinger equation,” Opt. Lett. 12, 921–923 (1987).
    [CrossRef]
  24. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
    [CrossRef]
  25. S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
    [CrossRef]
  26. C. Mahnke and F. Mitschke, “Possibility of an Akhmediev breather decaying into solitons,” Phys. Rev. A 85, 033808 (2012).
    [CrossRef]
  27. M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029–2034 (2011).
    [CrossRef]
  28. N. Devine, A. Ankiewicz, G. Genty, J. M. Dudley, and N. Akhmediev, “Recurrence phase shift in Fermi-Pasta-Ulam nonlinear dynamics,” Phys. Lett. A 375, 4158–4161 (2011).
    [CrossRef]
  29. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

2012 (1)

C. Mahnke and F. Mitschke, “Possibility of an Akhmediev breather decaying into solitons,” Phys. Rev. A 85, 033808 (2012).
[CrossRef]

2011 (7)

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029–2034 (2011).
[CrossRef]

N. Devine, A. Ankiewicz, G. Genty, J. M. Dudley, and N. Akhmediev, “Recurrence phase shift in Fermi-Pasta-Ulam nonlinear dynamics,” Phys. Lett. A 375, 4158–4161 (2011).
[CrossRef]

M. Droques, B. Barviau, A. Kudlinski, M. Taki, A. Boucon, T. Sylvestre, and A. Mussot, “Symmetry-breaking dynamics of the modulational instability spectrum,” Opt. Lett. 36, 1359–1361 (2011).
[CrossRef]

A. V. Gorbach, X. Zhao, and D. V. Skryabin, “Dispersion of nonlinearity and modulation instability in sub-wavelength semiconductor waveguides,” Opt. Express 19, 9345–9351 (2011).
[CrossRef]

M. I. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[CrossRef]

N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, and J. M. Dudley, “Universal triangular spectra in parametrically-driven systems,” Phys. Lett. 375, 775–779 (2011).
[CrossRef]

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

2010 (4)

M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Phys. Lett. A 374, 691–695 (2010).
[CrossRef]

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Could rogue waves be used as efficient weapons against enemy ships?,” Eur. J. Phys. Special Topics 185, 259–266 (2010).
[CrossRef]

S. Wabnitz and N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
[CrossRef]

2009 (1)

2006 (2)

N. C. Panoiu, X. F. Chen, and R. M. Osgood, “Modulation instability in silicon photonic nanowiresa,” Opt. Lett. 31, 3609–3611 (2006).
[CrossRef]

K. Kasamatsu and M. Tsubota, “Modulation instability and solitary-wave formation in two-component Bose-Einstein condensates,” Phys. Rev. A 74, 013617 (2006).
[CrossRef]

2005 (1)

K. Porsezian, K. Senthilnathan, and S. Devipriya, “Modulation instability in Fiber Bragg grating with non-Kerr nonlinearity,” IEEE J. Quantum Electron. 41, 789–796 (2005).
[CrossRef]

2001 (2)

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]

N. Akhmediev, “Deja vu in optics,” Nature 413, 267–268 (2001).
[CrossRef]

2000 (1)

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

1995 (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibres,” Phys. Rev. 51, 2602–2607 (1995).
[CrossRef]

1991 (1)

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

1990 (1)

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of a continuous signal in an optical fibre taking into account third order dispersion,” Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika 33, 111–117, (1990).

1987 (2)

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

M. J. Potasek, “Modulation instability in an extended nonlinear Schrödinger equation,” Opt. Lett. 12, 921–923 (1987).
[CrossRef]

1986 (1)

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

1967 (1)

T. B. Benjamin and J. E. Feir, “The disintegration of wavetrains on deep water. Part 1: theory,” J. Fluid Mech. 27, 417–430 (1967).
[CrossRef]

1966 (1)

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

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

Agrawal, G. P.

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

Akhmediev, N.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, and J. M. Dudley, “Universal triangular spectra in parametrically-driven systems,” Phys. Lett. 375, 775–779 (2011).
[CrossRef]

N. Devine, A. Ankiewicz, G. Genty, J. M. Dudley, and N. Akhmediev, “Recurrence phase shift in Fermi-Pasta-Ulam nonlinear dynamics,” Phys. Lett. A 375, 4158–4161 (2011).
[CrossRef]

S. Wabnitz and N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
[CrossRef]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Could rogue waves be used as efficient weapons against enemy ships?,” Eur. J. Phys. Special Topics 185, 259–266 (2010).
[CrossRef]

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

N. Akhmediev, “Deja vu in optics,” Nature 413, 267–268 (2001).
[CrossRef]

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibres,” Phys. Rev. 51, 2602–2607 (1995).
[CrossRef]

N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, 1997).

Akhmediev, N. N.

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of a continuous signal in an optical fibre taking into account third order dispersion,” Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika 33, 111–117, (1990).

Ankiewicz, A.

N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, and J. M. Dudley, “Universal triangular spectra in parametrically-driven systems,” Phys. Lett. 375, 775–779 (2011).
[CrossRef]

N. Devine, A. Ankiewicz, G. Genty, J. M. Dudley, and N. Akhmediev, “Recurrence phase shift in Fermi-Pasta-Ulam nonlinear dynamics,” Phys. Lett. A 375, 4158–4161 (2011).
[CrossRef]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Could rogue waves be used as efficient weapons against enemy ships?,” Eur. J. Phys. Special Topics 185, 259–266 (2010).
[CrossRef]

N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, 1997).

Bang, O.

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

Barviau, B.

Benjamin, T. B.

T. B. Benjamin and J. E. Feir, “The disintegration of wavetrains on deep water. Part 1: theory,” J. Fluid Mech. 27, 417–430 (1967).
[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).

Boucon, A.

Cavalcanti, S. B.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

Chen, X. F.

Christodoulides, D. N.

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

Cressoni, J. C.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

da Cruz, H. R.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

de Sterke, C. M.

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

Devine, N.

N. Devine, A. Ankiewicz, G. Genty, J. M. Dudley, and N. Akhmediev, “Recurrence phase shift in Fermi-Pasta-Ulam nonlinear dynamics,” Phys. Lett. A 375, 4158–4161 (2011).
[CrossRef]

Devipriya, S.

K. Porsezian, K. Senthilnathan, and S. Devipriya, “Modulation instability in Fiber Bragg grating with non-Kerr nonlinearity,” IEEE J. Quantum Electron. 41, 789–796 (2005).
[CrossRef]

Dias, F.

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

Douay, M.

M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Phys. Lett. A 374, 691–695 (2010).
[CrossRef]

Droques, M.

Dudley, J. M.

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029–2034 (2011).
[CrossRef]

N. Devine, A. Ankiewicz, G. Genty, J. M. Dudley, and N. Akhmediev, “Recurrence phase shift in Fermi-Pasta-Ulam nonlinear dynamics,” Phys. Lett. A 375, 4158–4161 (2011).
[CrossRef]

N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, and J. M. Dudley, “Universal triangular spectra in parametrically-driven systems,” Phys. Lett. 375, 775–779 (2011).
[CrossRef]

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

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]

Erkintalo, M.

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029–2034 (2011).
[CrossRef]

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

Eugenieva, E.

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

Feir, J. E.

T. B. Benjamin and J. E. Feir, “The disintegration of wavetrains on deep water. Part 1: theory,” J. Fluid Mech. 27, 417–430 (1967).
[CrossRef]

Finot, C.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

Genty, G.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

N. Devine, A. Ankiewicz, G. Genty, J. M. Dudley, and N. Akhmediev, “Recurrence phase shift in Fermi-Pasta-Ulam nonlinear dynamics,” Phys. Lett. A 375, 4158–4161 (2011).
[CrossRef]

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029–2034 (2011).
[CrossRef]

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

Gorbach, A. V.

Gouveia-Neto, A.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

Haelterman, M.

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]

Hammani, K.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

Hasegawa, A.

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

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibres,” Phys. Rev. 51, 2602–2607 (1995).
[CrossRef]

Kasamatsu, K.

K. Kasamatsu and M. Tsubota, “Modulation instability and solitary-wave formation in two-component Bose-Einstein condensates,” Phys. Rev. A 74, 013617 (2006).
[CrossRef]

Kibler, B.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

Kip, D.

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

Kolobov, M.

M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Phys. Lett. A 374, 691–695 (2010).
[CrossRef]

Kolobov, M. I.

M. I. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[CrossRef]

Korneev, V. I.

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of a continuous signal in an optical fibre taking into account third order dispersion,” Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika 33, 111–117, (1990).

Kudlinski, A.

M. Droques, B. Barviau, A. Kudlinski, M. Taki, A. Boucon, T. Sylvestre, and A. Mussot, “Symmetry-breaking dynamics of the modulational instability spectrum,” Opt. Lett. 36, 1359–1361 (2011).
[CrossRef]

M. I. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[CrossRef]

M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Phys. Lett. A 374, 691–695 (2010).
[CrossRef]

Louvergneaux, E.

M. I. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[CrossRef]

M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Phys. Lett. A 374, 691–695 (2010).
[CrossRef]

Mahnke, C.

C. Mahnke and F. Mitschke, “Possibility of an Akhmediev breather decaying into solitons,” Phys. Rev. A 85, 033808 (2012).
[CrossRef]

Mitschke, F.

C. Mahnke and F. Mitschke, “Possibility of an Akhmediev breather decaying into solitons,” Phys. Rev. A 85, 033808 (2012).
[CrossRef]

Mitskevich, N. V.

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of a continuous signal in an optical fibre taking into account third order dispersion,” Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika 33, 111–117, (1990).

Mussot, A.

M. Droques, B. Barviau, A. Kudlinski, M. Taki, A. Boucon, T. Sylvestre, and A. Mussot, “Symmetry-breaking dynamics of the modulational instability spectrum,” Opt. Lett. 36, 1359–1361 (2011).
[CrossRef]

M. I. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[CrossRef]

M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Phys. Lett. A 374, 691–695 (2010).
[CrossRef]

Osgood, R. M.

Panoiu, N. C.

Porsezian, K.

K. Porsezian, K. Senthilnathan, and S. Devipriya, “Modulation instability in Fiber Bragg grating with non-Kerr nonlinearity,” IEEE J. Quantum Electron. 41, 789–796 (2005).
[CrossRef]

Potasek, M. J.

Segev, M.

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

Senthilnathan, K.

K. Porsezian, K. Senthilnathan, and S. Devipriya, “Modulation instability in Fiber Bragg grating with non-Kerr nonlinearity,” IEEE J. Quantum Electron. 41, 789–796 (2005).
[CrossRef]

Skryabin, D. V.

Soljacic, Marin

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

Soto-Crespo, J. M.

N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, and J. M. Dudley, “Universal triangular spectra in parametrically-driven systems,” Phys. Lett. 375, 775–779 (2011).
[CrossRef]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Could rogue waves be used as efficient weapons against enemy ships?,” Eur. J. Phys. Special Topics 185, 259–266 (2010).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

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]

Taki, M.

M. I. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[CrossRef]

M. Droques, B. Barviau, A. Kudlinski, M. Taki, A. Boucon, T. Sylvestre, and A. Mussot, “Symmetry-breaking dynamics of the modulational instability spectrum,” Opt. Lett. 36, 1359–1361 (2011).
[CrossRef]

M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Phys. Lett. A 374, 691–695 (2010).
[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).

Tomita, A.

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

Tsubota, M.

K. Kasamatsu and M. Tsubota, “Modulation instability and solitary-wave formation in two-component Bose-Einstein condensates,” Phys. Rev. A 74, 013617 (2006).
[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.

S. Wabnitz and N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
[CrossRef]

Wetzel, B.

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029–2034 (2011).
[CrossRef]

Zhao, X.

Eur. J. Phys. Special Topics (1)

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Could rogue waves be used as efficient weapons against enemy ships?,” Eur. J. Phys. Special Topics 185, 259–266 (2010).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Porsezian, K. Senthilnathan, and S. Devipriya, “Modulation instability in Fiber Bragg grating with non-Kerr nonlinearity,” IEEE J. Quantum Electron. 41, 789–796 (2005).
[CrossRef]

Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika (1)

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of a continuous signal in an optical fibre taking into account third order dispersion,” Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika 33, 111–117, (1990).

J. Fluid Mech. (1)

T. B. Benjamin and J. E. Feir, “The disintegration of wavetrains on deep water. Part 1: theory,” J. Fluid Mech. 27, 417–430 (1967).
[CrossRef]

JETP Lett. (1)

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

Nature (1)

N. Akhmediev, “Deja vu in optics,” Nature 413, 267–268 (2001).
[CrossRef]

Opt. Commun. (1)

S. Wabnitz and N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Lett. (1)

N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, and J. M. Dudley, “Universal triangular spectra in parametrically-driven systems,” Phys. Lett. 375, 775–779 (2011).
[CrossRef]

Phys. Lett. A (4)

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029–2034 (2011).
[CrossRef]

N. Devine, A. Ankiewicz, G. Genty, J. M. Dudley, and N. Akhmediev, “Recurrence phase shift in Fermi-Pasta-Ulam nonlinear dynamics,” Phys. Lett. A 375, 4158–4161 (2011).
[CrossRef]

M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Phys. Lett. A 374, 691–695 (2010).
[CrossRef]

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

Phys. Rev. (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibres,” Phys. Rev. 51, 2602–2607 (1995).
[CrossRef]

Phys. Rev. A (4)

K. Kasamatsu and M. Tsubota, “Modulation instability and solitary-wave formation in two-component Bose-Einstein condensates,” Phys. Rev. A 74, 013617 (2006).
[CrossRef]

M. I. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[CrossRef]

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

C. Mahnke and F. Mitschke, “Possibility of an Akhmediev breather decaying into solitons,” Phys. Rev. A 85, 033808 (2012).
[CrossRef]

Phys. Rev. Lett. (4)

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

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

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

Science (1)

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

Other (2)

N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, 1997).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

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Figures (10)

Fig. 1.
Fig. 1.

Spectral output intensity for a propagation distance z=20, β3=0.01 and for various values of β2. The two sidebands are related to MI, while the additional spectral band on the right represents the resonant radiation. The green dashed line, defined by Eq. (14), is in perfect agreement with the numerical simulations.

Fig. 2.
Fig. 2.

Spectral intensity output of the fiber for a propagation distance of z=20 and β2=1, as a function of β3. The white dashed line is given by Eq. (14).

Fig. 3.
Fig. 3.

Recurrence spectrum of the NLSE with small TOD, namely β3=0.01, while β2=1. The TOD resonant radiation can be seen on the right-hand-side of the spectrum. Its position is indicated by the dashed blue line. Its influence is negligible until the radiation grows to higher amplitudes and distorts the initially symmetric AB spectrum. However, this distortion occurs at very long propagation distances. The two red vertical lines show the limits of the MI region (±2). The initial condition contains only a single pair of sidebands at the maximum growth rate (ω=2). The periodicity is preserved on propagation by imposing periodic boundary conditions, such that the whole spectrum remains discrete. The discreteness cannot be seen on this plot and in the plots below, since the curves are drawn by joining the discrete spectrum values with straight lines.

Fig. 4.
Fig. 4.

The same as in Fig. 3, except that the TOD is higher: β3=0.02. According to Eq. (14), TOD resonant radiation is now closer to the pump. It hardly influences the first recurrence, but due to the FWM process through the pump, the resonant radiation appears almost symmetrically on the left-hand-side of the spectrum. Moreover, an additional “sideband” of the resonant radiation appears at an equal distance on the right-hand-side of the spectrum. The asymmetry between the left-hand-side and the right-hand-side of the spectrum is due to a delay in the transfer of the spectral energy to the left-hand-side of the spectrum. The influence of the radiation is stronger than in the previous case, as the radiation appears on top of stronger spectral components. The radiation accumulates faster and recurrence survives shorter distances. Four cycles of recurrence can be seen here.

Fig. 5.
Fig. 5.

The same as in Figs. 3 and 4, except that the TOD is further increased to β3=0.04. TOD resonant radiation (shown by the blue dashed line) is now much closer to the pump, but is still out of the instability band shown by the red dashed lines. The FWM process through the pump adds distortions on each side of the spectrum. The resonant radiation appears at higher values of the background spectrum, thus increasing its influence. It accumulates much faster, and FPU recurrence is lost right from the first broadening of the spectrum. No cycles of recurrence can be seen here.

Fig. 6.
Fig. 6.

Field evolution, starting with MI, for three small values of TOD. The solid red curve is for β3=0.02, the green dotted line is for β3=0.03, and the blue dashed curve for β3=0.04. Up to z=12, the three curves almost coincide. Multiple recurrence is clearly seen at the initial stages of evolution for the smallest values of β3. Increasing the value of the TOD parameter causes the periodic behavior to deteriorate.

Fig. 7.
Fig. 7.

FPU recurrence dynamics of MI with almost perfect triangular spectrum of the sidebands when β3=1.0. The resonant frequency here is within the MI band. It defines the fundamental frequency of the sidebands.

Fig. 8.
Fig. 8.

Optical field evolution, starting with MI for three larger values of TOD: the red solid line stands for β3=0.3, the green dotted line for β3=0.4, and the dotted blue line for β3=0.5. Recurrence, which was lost for small values of β3, starts to be seen again for larger values of TOD, namely for β3>0.3.

Fig. 9.
Fig. 9.

Evolution of the two nearest sidebands, ±ω1 and ±ω2, in the MI recurrence dynamics at β3=1.0. The plus (red curves) and minus (blue curves) signs denote the right-hand-side and left-hand-side sidebands, respectively. The first sidebands (b) are almost symmetric, while the second sidebands (a) have a significant amount of asymmetry.

Fig. 10.
Fig. 10.

Recurrent trajectories of MI dynamics for β3=0 (green curve), β3=0.02 (red curve), and β3=1 (blue curve).

Equations (17)

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iΨZβ(2)2ΨTT+n2ω0c|Ψ|2Ψ=iβ(3)6ΨTTT,
z=Z/λ,t=T/λ,ψ=n2/(2π)Ψ,
iψz+β22ψtt+|ψ|2ψ=iβ3ψttt,
β2=β(2),β3=β(3)6λ.
ψ=BeiB2z.
ψ(t,z)=[1Ω22cosh(δz)+iδsinh(δz)cosh(δz)δΩcos(Ωt)]exp(iz).
δ=Ω1Ω24.
ψ(t,z)=[1μδ(Ω+i2δΩ)eδzcos(Ωt)]eiz+iϕ,
A0(z)=1iδsinh(δz)+(Ω2/2)cosh(δz)cosh2(δz)δ2Ω2,
An(z)=iδsinhδz+(Ω2/2)cosh(δz)cosh2(δz)δ2Ω2×[Ωcosh(δz)cosh2(δz)δ2Ω2δ]|n|,
ψ=μei(kzωt),
kω2β2/2=β3ω3.
B2=1=ω2β2/2+β3ω3,
ω3β22β3ω21β3=0.
ω=χ2+β2χ+β226β3χ,
ψ(z=0,t)=1+a(t)+ib(t),
ψ(z=0,t)=1+μcos(ωt),

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