Abstract

One of the extraordinary aspects of nonlinear wave evolution which has been observed as the spontaneous occurrence of astonishing and statistically extraordinary amplitude wave is called rogue wave. We show that the eigenvalues of the associated equation of nonlinear Schrödinger equation are almost constant in the vicinity of rogue wave and we validate that optical rogue waves are formed by the collision between quasi-solitons in anomalous dispersion fiber exhibiting weak third order dispersion.

© 2015 Optical Society of America

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References

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  1. Paul C. Lin, “A chronology of freaque wave encounters,” Geofizika 24(1), 57–70 (2007).
  2. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
    [Crossref] [PubMed]
  3. G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phy. Lett. 374, 989–996 (2010).
    [Crossref]
  4. M. Erkintalo, G. Genty, and J. M. Dudley, “Giant dispersive wave generation by soliton collision,” Opt. Lett. 35(5), 658–660 (2010).
    [Crossref] [PubMed]
  5. Bertrand Kibler, Kamal Hammani, Claire Michel, Christophe Finot, and Antonio Picozzi, “Rogue waves, rational solitons and wave turbulence theory,” Phy. Lett. 375, 3149–3155 (2011).
    [Crossref]
  6. Kamal Hammani, Bertrand Kibler, Christophe Finot, and Antonio Picozzi, “Emergence of rogue waves from optical turbulence,” Phy. Lett. 374, 3585–3589 (2010).
    [Crossref]
  7. N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 06020 (2013).
    [Crossref]
  8. V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, Heidelberg, New York, 1992).
    [Crossref]
  9. C. Kharif and E. Pelinovsky, “Physical mechanisms of the rogue wave phenomenon,” Eur. J. Mech. 22(6), 603–634 (2003).
    [Crossref]
  10. V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress freak wave formation?” J. Fluids Mech. 604, 263–296 (2008).
    [Crossref]
  11. Govind P. Agrawal, Nonlinear Fiber Optics (Oxford, 2013).
  12. T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water. Part 1. Theory,” J. Fluid Mech. 27(3), 417–430 (1967).
    [Crossref]
  13. D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nature Photon. 6, 463–468 (2012).
    [Crossref]
  14. D. R. Solli, C. Ropers, and B. Jalali, “Measuring single-shot modulation instability and supercontinuum spectra at megahertz rates,” Nonlinearity 26(3), R85–R92 (2013).
    [Crossref]
  15. A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17(19), 17010–17015 (2009).
    [Crossref] [PubMed]
  16. J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17(24), 21497–21508 (2009).
    [Crossref] [PubMed]
  17. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford, 1995).
  18. 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(1), 118–134 (1972).
  19. http://www.opc.ncep.noaa.gov/perfectstorm/mpc_ps_rogue.shtml .

2013 (2)

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 06020 (2013).
[Crossref]

D. R. Solli, C. Ropers, and B. Jalali, “Measuring single-shot modulation instability and supercontinuum spectra at megahertz rates,” Nonlinearity 26(3), R85–R92 (2013).
[Crossref]

2012 (1)

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nature Photon. 6, 463–468 (2012).
[Crossref]

2011 (1)

Bertrand Kibler, Kamal Hammani, Claire Michel, Christophe Finot, and Antonio Picozzi, “Rogue waves, rational solitons and wave turbulence theory,” Phy. Lett. 375, 3149–3155 (2011).
[Crossref]

2010 (3)

Kamal Hammani, Bertrand Kibler, Christophe Finot, and Antonio Picozzi, “Emergence of rogue waves from optical turbulence,” Phy. Lett. 374, 3585–3589 (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,” Phy. Lett. 374, 989–996 (2010).
[Crossref]

M. Erkintalo, G. Genty, and J. M. Dudley, “Giant dispersive wave generation by soliton collision,” Opt. Lett. 35(5), 658–660 (2010).
[Crossref] [PubMed]

2009 (2)

2008 (1)

V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress freak wave formation?” J. Fluids Mech. 604, 263–296 (2008).
[Crossref]

2007 (2)

Paul C. Lin, “A chronology of freaque wave encounters,” Geofizika 24(1), 57–70 (2007).

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[Crossref] [PubMed]

2003 (1)

C. Kharif and E. Pelinovsky, “Physical mechanisms of the rogue wave phenomenon,” Eur. J. Mech. 22(6), 603–634 (2003).
[Crossref]

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(1), 118–134 (1972).

1967 (1)

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water. Part 1. Theory,” J. Fluid Mech. 27(3), 417–430 (1967).
[Crossref]

Agrawal, Govind P.

Govind P. Agrawal, Nonlinear Fiber Optics (Oxford, 2013).

Akhmediev, N.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 06020 (2013).
[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,” Phy. Lett. 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(24), 21497–21508 (2009).
[Crossref] [PubMed]

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,” Phy. Lett. 374, 989–996 (2010).
[Crossref]

Benjamin, T. B.

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water. Part 1. Theory,” J. Fluid Mech. 27(3), 417–430 (1967).
[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,” Phy. Lett. 374, 989–996 (2010).
[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,” Phy. Lett. 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(24), 21497–21508 (2009).
[Crossref] [PubMed]

Douay, M.

Dudley, J. M.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 06020 (2013).
[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,” Phy. Lett. 374, 989–996 (2010).
[Crossref]

M. Erkintalo, G. Genty, and J. M. Dudley, “Giant dispersive wave generation by soliton collision,” Opt. Lett. 35(5), 658–660 (2010).
[Crossref] [PubMed]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17(24), 21497–21508 (2009).
[Crossref] [PubMed]

Erkintalo, M.

Falkovich, G.

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, Heidelberg, New York, 1992).
[Crossref]

Feir, J. E.

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water. Part 1. Theory,” J. Fluid Mech. 27(3), 417–430 (1967).
[Crossref]

Finot, Christophe

Bertrand Kibler, Kamal Hammani, Claire Michel, Christophe Finot, and Antonio Picozzi, “Rogue waves, rational solitons and wave turbulence theory,” Phy. Lett. 375, 3149–3155 (2011).
[Crossref]

Kamal Hammani, Bertrand Kibler, Christophe Finot, and Antonio Picozzi, “Emergence of rogue waves from optical turbulence,” Phy. Lett. 374, 3585–3589 (2010).
[Crossref]

Genty, G.

Hammani, Kamal

Bertrand Kibler, Kamal Hammani, Claire Michel, Christophe Finot, and Antonio Picozzi, “Rogue waves, rational solitons and wave turbulence theory,” Phy. Lett. 375, 3149–3155 (2011).
[Crossref]

Kamal Hammani, Bertrand Kibler, Christophe Finot, and Antonio Picozzi, “Emergence of rogue waves from optical turbulence,” Phy. Lett. 374, 3585–3589 (2010).
[Crossref]

Hasegawa, A.

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford, 1995).

Herink, G.

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nature Photon. 6, 463–468 (2012).
[Crossref]

Jalali, B.

D. R. Solli, C. Ropers, and B. Jalali, “Measuring single-shot modulation instability and supercontinuum spectra at megahertz rates,” Nonlinearity 26(3), R85–R92 (2013).
[Crossref]

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nature Photon. 6, 463–468 (2012).
[Crossref]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[Crossref] [PubMed]

Kharif, C.

C. Kharif and E. Pelinovsky, “Physical mechanisms of the rogue wave phenomenon,” Eur. J. Mech. 22(6), 603–634 (2003).
[Crossref]

Kibler, B.

Kibler, Bertrand

Bertrand Kibler, Kamal Hammani, Claire Michel, Christophe Finot, and Antonio Picozzi, “Rogue waves, rational solitons and wave turbulence theory,” Phy. Lett. 375, 3149–3155 (2011).
[Crossref]

Kamal Hammani, Bertrand Kibler, Christophe Finot, and Antonio Picozzi, “Emergence of rogue waves from optical turbulence,” Phy. Lett. 374, 3585–3589 (2010).
[Crossref]

Kodama, Y.

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford, 1995).

Kolobov, M.

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[Crossref] [PubMed]

Kudlinski, A.

L’vov, V. S.

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, Heidelberg, New York, 1992).
[Crossref]

Lin, Paul C.

Paul C. Lin, “A chronology of freaque wave encounters,” Geofizika 24(1), 57–70 (2007).

Louvergneaux, E.

Michel, Claire

Bertrand Kibler, Kamal Hammani, Claire Michel, Christophe Finot, and Antonio Picozzi, “Rogue waves, rational solitons and wave turbulence theory,” Phy. Lett. 375, 3149–3155 (2011).
[Crossref]

Mussot, A.

Pelinovsky, E.

C. Kharif and E. Pelinovsky, “Physical mechanisms of the rogue wave phenomenon,” Eur. J. Mech. 22(6), 603–634 (2003).
[Crossref]

Picozzi, Antonio

Bertrand Kibler, Kamal Hammani, Claire Michel, Christophe Finot, and Antonio Picozzi, “Rogue waves, rational solitons and wave turbulence theory,” Phy. Lett. 375, 3149–3155 (2011).
[Crossref]

Kamal Hammani, Bertrand Kibler, Christophe Finot, and Antonio Picozzi, “Emergence of rogue waves from optical turbulence,” Phy. Lett. 374, 3585–3589 (2010).
[Crossref]

Ropers, C.

D. R. Solli, C. Ropers, and B. Jalali, “Measuring single-shot modulation instability and supercontinuum spectra at megahertz rates,” Nonlinearity 26(3), R85–R92 (2013).
[Crossref]

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nature Photon. 6, 463–468 (2012).
[Crossref]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[Crossref] [PubMed]

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(1), 118–134 (1972).

Shrira, V. I.

V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress freak wave formation?” J. Fluids Mech. 604, 263–296 (2008).
[Crossref]

Solli, D. R.

D. R. Solli, C. Ropers, and B. Jalali, “Measuring single-shot modulation instability and supercontinuum spectra at megahertz rates,” Nonlinearity 26(3), R85–R92 (2013).
[Crossref]

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 06020 (2013).
[Crossref]

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nature Photon. 6, 463–468 (2012).
[Crossref]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[Crossref] [PubMed]

Taki, M.

Thomas, G.

V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress freak wave formation?” J. Fluids Mech. 604, 263–296 (2008).
[Crossref]

Turitsyn, S. K.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 06020 (2013).
[Crossref]

Voronovich, V. V.

V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress freak wave formation?” J. Fluids Mech. 604, 263–296 (2008).
[Crossref]

Zakharov, V. E.

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(1), 118–134 (1972).

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, Heidelberg, New York, 1992).
[Crossref]

Eur. J. Mech. (1)

C. Kharif and E. Pelinovsky, “Physical mechanisms of the rogue wave phenomenon,” Eur. J. Mech. 22(6), 603–634 (2003).
[Crossref]

Geofizika (1)

Paul C. Lin, “A chronology of freaque wave encounters,” Geofizika 24(1), 57–70 (2007).

J. Fluid Mech. (1)

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water. Part 1. Theory,” J. Fluid Mech. 27(3), 417–430 (1967).
[Crossref]

J. Fluids Mech. (1)

V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress freak wave formation?” J. Fluids Mech. 604, 263–296 (2008).
[Crossref]

J. Opt. (1)

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 06020 (2013).
[Crossref]

Nature (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[Crossref] [PubMed]

Nature Photon. (1)

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nature Photon. 6, 463–468 (2012).
[Crossref]

Nonlinearity (1)

D. R. Solli, C. Ropers, and B. Jalali, “Measuring single-shot modulation instability and supercontinuum spectra at megahertz rates,” Nonlinearity 26(3), R85–R92 (2013).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phy. Lett. (3)

Bertrand Kibler, Kamal Hammani, Claire Michel, Christophe Finot, and Antonio Picozzi, “Rogue waves, rational solitons and wave turbulence theory,” Phy. Lett. 375, 3149–3155 (2011).
[Crossref]

Kamal Hammani, Bertrand Kibler, Christophe Finot, and Antonio Picozzi, “Emergence of rogue waves from optical turbulence,” Phy. Lett. 374, 3585–3589 (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,” Phy. Lett. 374, 989–996 (2010).
[Crossref]

Sov. Phys. JETP (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(1), 118–134 (1972).

Other (4)

http://www.opc.ncep.noaa.gov/perfectstorm/mpc_ps_rogue.shtml .

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford, 1995).

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, Heidelberg, New York, 1992).
[Crossref]

Govind P. Agrawal, Nonlinear Fiber Optics (Oxford, 2013).

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Figures (10)

Fig. 1
Fig. 1 Basic concept of optical rogue wave generation.
Fig. 2
Fig. 2 Modulational instability process.
Fig. 3
Fig. 3 (a) Collision of two solitons, and (b) variations of imaginary part of eigenvalues in the vicinity of collision.
Fig. 4
Fig. 4 Initial waveform.
Fig. 5
Fig. 5 (a) Observed optical rogue wave profile at Z = 827.47, (b) Contour plot, and (c) Variations of imaginary part of eigenvalues in the vicinity of Z = 827.47 for σ = 0.
Fig. 6
Fig. 6 Maximum achieved peak power vs. TOD coefficient.
Fig. 7
Fig. 7 (a) Observed localized wave profile at Z = 1706.37, (b) Contour plot, and (c) Variations of imaginary part of eigenvalues in the vicinity of Z = 1706.37 for σ = 0.015.
Fig. 8
Fig. 8 (a) Observed localized wave profile at Z = 1885.75, (b) Contour plot, and (c) Variations of imaginary part of eigenvalues in the vicinity of Z = 1885.75 for σ = 0.02.
Fig. 9
Fig. 9 Variations of imaginary part of two largest eigenvalues against TOD.
Fig. 10
Fig. 10 Stability of quasi-solitons for different TOD coefficients.

Equations (9)

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i u Z + 1 2 2 u T 2 + | u | 2 u = i σ 3 u T 3 ,
{ i ψ 1 T + u ψ 2 = ζ ψ 1 , i ψ 2 T u * ψ 1 = ζ ψ 2 ,
f ˜ ( Ω ) = 1 2 π f ( T ) e i Ω T d T ,
{ Ω ψ ˜ 1 ( Z , Ω ) + 1 2 π u ˜ ( Z , Ω Ω ) ψ ˜ 2 ( Z , Ω ) d Ω = ζ ψ ˜ 1 ( Z , Ω ) , Ω ψ ˜ 2 ( Z , Ω ) 1 2 π u ˜ * ( Z , Ω Ω ) ψ ˜ 1 ( Z , Ω ) d Ω = ζ ψ ˜ 2 ( Z , Ω ) ,
{ Ω n ψ ˜ 1 ( Ω n ) + Δ Ω 2 π m = 1 N u ˜ ( Ω n Ω m ) ψ ˜ 2 ( Ω m ) = ζ ψ ˜ 1 ( Ω n ) , Ω n ψ ˜ 2 ( Ω n ) Δ Ω 2 π m = 1 N u ˜ * ( Ω m Ω n ) ψ ˜ 1 ( Ω m ) = ζ ψ ˜ 2 ( Ω n ) ,
[ A B B * A ] [ Ψ ˜ 2 Ψ ˜ 2 ] = ζ [ Ψ ˜ 1 Ψ ˜ 2 ]
a j k = { Ω j ( j = k ) , 0 ( otherwise ) ,
b j k = { 1 2 π u ˜ ( Ω n j k ) Δ Ω ( 1 n j k N ) , 0 ( otherwise ) ,
u ( Z = 0 , T ) = A sech [ A ( T + Δ T / 2 ) ] exp ( i B T ) + A sech [ A ( T Δ T / 2 ) ] exp ( i B T ) .

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