Abstract

Optical “rogue” waves are rare and very high intensity pulses of light that occur in optical devices such as communication fibers. They appear suddenly and can cause transmission errors and damage in optical communication systems. Indeed, the physics governing their dynamics is very similar to “monster” or “freak” waves on the Earth’s oceans, which are known to harm shipping. It is therefore important to characterize rogue wave generation, dynamics and, if possible, predictability. Here we demonstrate a simple cascade mechanism that drives the formation and emergence of rogue waves in the generalized non-linear Schrödinger equation with third-order dispersion. This generation mechanism is based on inelastic collisions of quasi-solitons and is well described by a resonant-like scattering behaviour for the energy transfer in pair-wise quasi-soliton collisions. Our theoretical and numerical results demonstrate a threshold for rogue wave emergence and the existence of a period of reduced amplitudes — a “calm before the storm” — preceding the arrival of a rogue wave event. Comparing with ultra-long time window simulations of 3.865 × 106ps we observe the statistics of rogue waves in optical fibres with an unprecedented level of detail and accuracy, unambiguously establishing the long-ranged character of the rogue wave power-distribution function over seven orders of magnitude.

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2016 (5)

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

C. Brée, G. Steinmeyer, I. Babushkin, U. Morgner, and A. Demircan, “Controlling formation and suppression of fiber-optical rogue waves,” Opt. Lett. 41, 3515 (2016).
[Crossref] [PubMed]

H. Degueldre, J. M. Jakob, T. Geisel, and R. Fleischmann, “Random focusing of tsunami waves,” Nat. Phys. 12, 259–262 (2016).

A. Slunyaev and E. Pelinovsky, “Role of Multiple Soliton Interactions in the Generation of Rogue Waves: The Modified Korteweg-de Vries Framework,” Phys. Rev. Lett. 117, 214501 (2016).
[Crossref] [PubMed]

Y.-H. Sun, “Soliton synchronization in the focusing nonlinear Schrödinger equation,” Phys. Rev. E 93, 052222 (2016).
[Crossref]

2015 (6)

A. Armaroli, C. Conti, and F. Biancalana, “Rogue solitons in optical fibers: a dynamical process in a complex energy landscape?” Optica 2, 497 (2015).
[Crossref]

G. Weerasekara, A. Tokunaga, H. Terauchi, M. Eberhard, and A. Maruta, “Soliton’s eigenvalue based analysis on the generation mechanism of rogue wave phenomenon in optical fibers exhibiting weak third order dispersion,” Opt. Express 23, 143 (2015).
[Crossref] [PubMed]

M. Erkintalo, “Rogue waves: Predicting the unpredictable?” Nat. Photonics 9, 560–562 (2015).
[Crossref]

T. A. A. Adcock, P. H. Taylor, and S. Draper, “Nonlinear dynamics of wave-groups in random seas: unexpected walls of water in the open ocean,” Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 471, 20150660 (2015).
[Crossref]

P. Walczak, S. Randoux, and P. Suret, “Optical rogue waves in integrable turbulence,” Phys. Rev. Lett. 114, 143903 (2015).
[Crossref] [PubMed]

S. Birkholz, C. Brée, A. Demircan, and G. Steinmeyer, “Predictability of Rogue Events,” Phys. Rev. Lett. 114, 213901 (2015).
[Crossref] [PubMed]

2014 (4)

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nature 8, 755–764 (2014).

S. Randoux, P. Walczak, M. Onorato, and P. Suret, “Intermittency in integrable turbulence,” Phys. Rev. Lett. 113, 113902 (2014).
[Crossref] [PubMed]

A. Demircan, S. Amiranashvili, C. Brée, C. Mahnke, F. Mitschke, and G. Steinmeyer, “Rogue wave formation by accelerated solitons at an optical event horizon,” Appl. Phys. B 115, 343–354 (2014).
[Crossref]

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. Christodoulides, “Optical wave turbulence,” Phys. Rep. 542, 1–132 (2014).
[Crossref]

2013 (2)

A. Chabchoub, N. Hoffmann, M. Onorato, G. Genty, J. M. Dudley, and N. Akhmediev, “Hydrodynamic supercontinuum,.” Phys. Rev. Lett. 111, 054104 (2013).
[Crossref] [PubMed]

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

2012 (3)

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref] [PubMed]

S. Randoux and P. Suret, “Experimental evidence of extreme value statistics in Raman fiber lasers,” Opt. Lett. 37, 500–502 (2012).
[Crossref] [PubMed]

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. M. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[Crossref] [PubMed]

2011 (2)

C. Bonatto, M. Feyereisen, S. Barland, M. Giudici, C. Masoller, J. R. R. Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
[Crossref] [PubMed]

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

2010 (8)

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

Z. Chen, A. J. Taylor, and A. Efimov, “Soliton dynamics in non-uniform fiber tapers: analytical description through an improved moment method,” J. Opt. Soc. Am. B 27, 1022 (2010).
[Crossref]

N. Akhmediev and E. Pelinovsky, “Editorial – Introductory remarks on “Discussion &Rogue waves–towards a unifying concept: discussion and debates,” Eur. Phys. J. Spec. Top. 185, 1–4 (2010).
[Crossref]

V. Ruban, Y. Kodama, M. Ruderman, J. Dudley, R. Grimshaw, P. V. E. McClintock, M. Onorato, C. Kharif, E. Pelinovsky, T. Soomere, G. Lindgren, N. Akhmediev, A. Slunyaev, D. Solli, C. Ropers, B. Jalali, F. Dias, and A. Osborne, “Rogue waves – towards a unifying concept?: Discussions and debates,” Eur. Phys. J. Spec. Top. 185, 5–15 (2010).
[Crossref]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[Crossref]

R. Höhmann, U. Kuhl, H.-J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett. 104, 093901 (2010).
[Crossref] [PubMed]

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

2009 (7)

2008 (3)

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

D. R. Solli, C. Ropers, and B. Jalali, “Active control of rogue waves for stimulated supercontinuum generation,” Phys. Rev. Lett. 101, 233902 (2008).
[Crossref] [PubMed]

K. Hammani, C. Finot, J. M. Dudley, and G. Millot, “Optical rogue-wave-like extreme value fluctuations in fiber Raman amplifiers,” Opt. Express 16, 16467 (2008).
[Crossref] [PubMed]

2007 (1)

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

2006 (2)

2005 (1)

M. Onorato, A. R. Osborne, M. Serio, and L. Cavaleri, “Modulational instability and non-Gaussian statistics in experimental random water-wave trains,” Phys. Flu. 17, 078101 (2005).
[Crossref]

2004 (2)

M. Hopkin, “Sea snapshots will map frequency of freak waves,” Nature 430, 492 (2004).
[Crossref] [PubMed]

M. Onorato, A. R. Osborne, M. Serio, L. Cavaleri, C. Brandini, and C. T. Stansberg, “Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments,” Phys. Rev. E 70, 067302 (2004).
[Crossref]

2003 (2)

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

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
[Crossref]

1995 (1)

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

1994 (1)

A. V. Buryak and N. N. Akhmediev, “Internal friction between solitons in near-integrable systems,” Phys. Rev. E 50, 3126–3133 (1994).
[Crossref]

1988 (1)

V. Zakharov, A. Pushkarev, V. Shvets, and V. Yan’kov, “Soliton turbulence,” Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 48, 79–82 (1988).

1986 (1)

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

1984 (1)

1981 (1)

V. Karpman and V. Solov’ev, “A perturbational approach to the two-soliton systems,” Phys. D: Nonlin. Phenom. 3, 487–502 (1981).
[Crossref]

1974 (1)

J. Mallory, “Abnormal waves in the south-east coast of South Africa,” Int. Hydrog. Rev. 51, 89–129 (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).

1971 (1)

L. Draper, “Severe wave conditions at sea,” J. Inst. Navig. 24, 273–277 (1971).
[Crossref]

1964 (1)

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J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
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Akhmediev, N.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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A. Chabchoub, N. Hoffmann, M. Onorato, G. Genty, J. M. Dudley, and N. Akhmediev, “Hydrodynamic supercontinuum,.” Phys. Rev. Lett. 111, 054104 (2013).
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C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
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B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. M. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[Crossref] [PubMed]

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106, 204502 (2011).
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B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
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N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Could rogue waves be used as efficient weapons against enemy ships?” Eur. Phys. J. Spec. Top. 185, 259–266 (2010).
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G. Genty, C. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
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N. Akhmediev and E. Pelinovsky, “Editorial – Introductory remarks on “Discussion &Rogue waves–towards a unifying concept: discussion and debates,” Eur. Phys. J. Spec. Top. 185, 1–4 (2010).
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V. Ruban, Y. Kodama, M. Ruderman, J. Dudley, R. Grimshaw, P. V. E. McClintock, M. Onorato, C. Kharif, E. Pelinovsky, T. Soomere, G. Lindgren, N. Akhmediev, A. Slunyaev, D. Solli, C. Ropers, B. Jalali, F. Dias, and A. Osborne, “Rogue waves – towards a unifying concept?: Discussions and debates,” Eur. Phys. J. Spec. Top. 185, 5–15 (2010).
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N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
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Akhmediev, N. N.

A. V. Buryak and N. N. Akhmediev, “Internal friction between solitons in near-integrable systems,” Phys. Rev. E 50, 3126–3133 (1994).
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Amiranashvili, S.

A. Demircan, S. Amiranashvili, C. Brée, C. Mahnke, F. Mitschke, and G. Steinmeyer, “Rogue wave formation by accelerated solitons at an optical event horizon,” Appl. Phys. B 115, 343–354 (2014).
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Amiraranashvili, S.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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Ankiewicz, A.

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Could rogue waves be used as efficient weapons against enemy ships?” Eur. Phys. J. Spec. Top. 185, 259–266 (2010).
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Arecchi, F.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
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Arecchi, F. T.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian statistics and extreme waves in a nonlinear optical cavity,” Phys. Rev. Lett. 103, 173901 (2009).
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Armaroli, A.

Babushkin, I.

Bang, O.

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

C. Bonatto, M. Feyereisen, S. Barland, M. Giudici, C. Masoller, J. R. R. Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
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Baronio, F.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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Béjot, P.

Belic, M.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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Bendahmane, A.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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Birkholz, S.

S. Birkholz, C. Brée, A. Demircan, and G. Steinmeyer, “Predictability of Rogue Events,” Phys. Rev. Lett. 114, 213901 (2015).
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C. Bonatto, M. Feyereisen, S. Barland, M. Giudici, C. Masoller, J. R. R. Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
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Bortolozzo, U.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
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A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian statistics and extreme waves in a nonlinear optical cavity,” Phys. Rev. Lett. 103, 173901 (2009).
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U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, “Optical wave turbulence and the condensation of light,” J. Opt. Soc. Am. B 26, 2280 (2009).
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Brandini, C.

M. Onorato, A. R. Osborne, M. Serio, L. Cavaleri, C. Brandini, and C. T. Stansberg, “Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments,” Phys. Rev. E 70, 067302 (2004).
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Bree, C.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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Brée, C.

C. Brée, G. Steinmeyer, I. Babushkin, U. Morgner, and A. Demircan, “Controlling formation and suppression of fiber-optical rogue waves,” Opt. Lett. 41, 3515 (2016).
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S. Birkholz, C. Brée, A. Demircan, and G. Steinmeyer, “Predictability of Rogue Events,” Phys. Rev. Lett. 114, 213901 (2015).
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A. Demircan, S. Amiranashvili, C. Brée, C. Mahnke, F. Mitschke, and G. Steinmeyer, “Rogue wave formation by accelerated solitons at an optical event horizon,” Appl. Phys. B 115, 343–354 (2014).
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Broderick, N. G. R.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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Buryak, A. V.

A. V. Buryak and N. N. Akhmediev, “Internal friction between solitons in near-integrable systems,” Phys. Rev. E 50, 3126–3133 (1994).
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Cavaleri, L.

M. Onorato, A. R. Osborne, M. Serio, and L. Cavaleri, “Modulational instability and non-Gaussian statistics in experimental random water-wave trains,” Phys. Flu. 17, 078101 (2005).
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M. Onorato, A. R. Osborne, M. Serio, L. Cavaleri, C. Brandini, and C. T. Stansberg, “Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments,” Phys. Rev. E 70, 067302 (2004).
[Crossref]

Chabchoub, A.

A. Chabchoub, N. Hoffmann, M. Onorato, G. Genty, J. M. Dudley, and N. Akhmediev, “Hydrodynamic supercontinuum,.” Phys. Rev. Lett. 111, 054104 (2013).
[Crossref] [PubMed]

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

Chang, W.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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Christodoulides, D.

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Coulibaly, S.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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de Sterke, C.

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

H. Degueldre, J. M. Jakob, T. Geisel, and R. Fleischmann, “Random focusing of tsunami waves,” Nat. Phys. 12, 259–262 (2016).

Demircan, A.

C. Brée, G. Steinmeyer, I. Babushkin, U. Morgner, and A. Demircan, “Controlling formation and suppression of fiber-optical rogue waves,” Opt. Lett. 41, 3515 (2016).
[Crossref] [PubMed]

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

S. Birkholz, C. Brée, A. Demircan, and G. Steinmeyer, “Predictability of Rogue Events,” Phys. Rev. Lett. 114, 213901 (2015).
[Crossref] [PubMed]

A. Demircan, S. Amiranashvili, C. Brée, C. Mahnke, F. Mitschke, and G. Steinmeyer, “Rogue wave formation by accelerated solitons at an optical event horizon,” Appl. Phys. B 115, 343–354 (2014).
[Crossref]

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J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nature 8, 755–764 (2014).

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. M. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[Crossref] [PubMed]

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

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[Crossref]

V. Ruban, Y. Kodama, M. Ruderman, J. Dudley, R. Grimshaw, P. V. E. McClintock, M. Onorato, C. Kharif, E. Pelinovsky, T. Soomere, G. Lindgren, N. Akhmediev, A. Slunyaev, D. Solli, C. Ropers, B. Jalali, F. Dias, and A. Osborne, “Rogue waves – towards a unifying concept?: Discussions and debates,” Eur. Phys. J. Spec. Top. 185, 5–15 (2010).
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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).
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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, 17010–17015 (2009).
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Draper, S.

T. A. A. Adcock, P. H. Taylor, and S. Draper, “Nonlinear dynamics of wave-groups in random seas: unexpected walls of water in the open ocean,” Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 471, 20150660 (2015).
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Dudley, J.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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V. Ruban, Y. Kodama, M. Ruderman, J. Dudley, R. Grimshaw, P. V. E. McClintock, M. Onorato, C. Kharif, E. Pelinovsky, T. Soomere, G. Lindgren, N. Akhmediev, A. Slunyaev, D. Solli, C. Ropers, B. Jalali, F. Dias, and A. Osborne, “Rogue waves – towards a unifying concept?: Discussions and debates,” Eur. Phys. J. Spec. Top. 185, 5–15 (2010).
[Crossref]

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

Dudley, J. M.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nature 8, 755–764 (2014).

A. Chabchoub, N. Hoffmann, M. Onorato, G. Genty, J. M. Dudley, and N. Akhmediev, “Hydrodynamic supercontinuum,.” Phys. Rev. Lett. 111, 054104 (2013).
[Crossref] [PubMed]

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. M. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[Crossref] [PubMed]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
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M. Erkintalo, G. Genty, and J. M. Dudley, “Rogue-wave-like characteristics in femtosecond supercontinuum generation,” Opt. Lett. 34, 2468 (2009).
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Figures (5)

Fig. 1
Fig. 1 (a) PDFs of the intensity |u|2 from the gNLSE (1) at β3 = 2.64 × 10−42s3m−1 using a large time window of Δt = 3.865 × 106ps. The PDFs have been computed at distances z = 100m, 200m, 500m, 1000m and 1500m. The left vertical axis denotes the values of the normalized PDF while the right vertical axis gives the event count per bin. The PDFs have been fitted using the Weibull function (full black lines). The brown points represnt the PDF calculate for a small time window of 200ps at 1500m. The inset shows results for β3 = 0. (b) PDFs of the intensity |u|2 from the cascade model for the same distances as in (a), using the same symbol and axes conventions. The PDFs have been fitted using the Weibull function (full black lines). The inset shows a comparison between the results from the gNLSE (colored lines) and the cascade model (black lines and symbol outlines) for z = 500m, 1000m and 1500m. Only every 50th symbol is shown.

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Fig. 2
Fig. 2 (a) Intensity |u(z, t)|2 for β3 = 2.64 × 10−42s3m−1 of the gNLSE Eq. (1) as function of the time t and distance z in a selected time frame of Δt = 15ps and distance range Δz = 1.5km. (b) |u|2 with β3 = 0 for a zoomed-in distance and time region, (c) |u|2 with β3 value as in (a) for a region of (a) with Δt and Δz chosen identical to (b). (d) Intensities |u|2 as computed from the effective cascade model using the same shading/color scale as in (a). Note that we start the effective model at z0 = 100m to mimic the effects of the modulation instability in (a).

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Fig. 3
Fig. 3 (a+b) Intensities |u(z, t)|2 of the scattering between two quasi-solitons. The phase difference ϕ was chosen to correspond to (a) the minimum and (b) the maximum of ΔE1/E2(ϕ). (c) ΔE1/E2(ϕ) for various choices of initial speeds. The data points represents results of the gNLSE (1) while the lines denote the fit (5). (d) ∊ values have been obtained comparing the PDF from the gNLSE and the cascade model at short distance using a Kolmogorov-Smirnov-like (KS) test and minimizing the variance (cf. appendix B). A KS significance measure is also shown [60].

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Fig. 4
Fig. 4 Normalized averaged powers |u(Δt)|2/〈|u(Δt)|2〉 for times Δt in the vicinity of a RW event at Δt = 0. Panel (a), (b) corresponds to 200 and 500m, respectively. Solid lines in both panels indicate averaged results for two gNLSE runs (with parameters as in Fig. 2), while dashed lines show the corresponding results for the cascade model. In both panels, we identify RWs as corresponding to powers equal to or larger than 150W. The colours are chosen to indicate distances compatible with a full set of results z = 150, . . ., 1500m given in the appendix. Note that |u(0)|2/〈|u(0)|2〉 > 10 in both panels.

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Fig. 5
Fig. 5 (a) Normalized peak power distribution PDF(P) for β3 = 0 at 1.5 km. The data points denotes (blue squares) denote the data while the solid (red) line shows the fit with Eq. (7). The dashed black line is at the fitted value P0 = 31.4 W (b)Relative variance r (filled squares) and largest difference D (open circles) calculated for different values of ∊eff at β3 = 2.64 × 10−42s3m−1. Parabolic fits to the data are shown as lines. The vertical dotted line denotes the estimated ∊eff = (1.23 ± 0.05)fs/m at which D is minimal, the grey region indicates the error of that estimate. The vertical dashed-dotted line denotes the estimate ∊eff = (1.32 ± 0.05)fs/m from r.

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Tables (1)

Tables Icon

Table 1 Values of the coefficients a, b and F0 needed to fit the various PDF(|u|2) distributions from Fig. 1 at three representative distances z = 500 m, 1000 m and 1500 m, using (W) the Weibull function of Eq. (14), (F) the stretched exponential of Eq. (15) and (P) the Pareto function as in Eq. (16). The reduced χ2 value is also displayed.

View Table

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

∂ z u + i β 2 2 ∂ t 2 u − β 3 6 ∂ t 3 u − i γ | u | 2 u = 0
u q ( z , t ) = P q sech [ ( t − t q ) + z / v q T q ] exp [ i ϕ q ]
T q = | β 2 + β 3 Ω q | γ P q
v q − 1 = β 2 Ω q + β 3 2 Ω q 2 + β 3 6 T q 2
Δ E 1 E 2 = ∊ eff | v 1 − 1 − v 2 − 1 | sin 2 ( ϕ 1 − ϕ 2 2 ) ,
β 3 ≥ 3 | β 2 | π | β 2 | γ P ≈ 0.9 × ( 2.64 × 10 − 42 ) s 3 m − 1 ,
ρ ( P q ) = b P 0 ( P q P 0 ) b − 1 exp [ − ( P q P 0 ) b ] ,
u ( z , t ) = P sech [ t − q ( z ) T ] × exp { − i Ω [ t − q ( z ) ] − i C [ t − q ( z ) ] 2 2 T 2 } ,
d T d z = ( β 2 + β 3 Ω ) C T , d C d z = ( 4 π 2 + C 2 ) ( β 2 + β 3 Ω ) T 2 + 4 γ P π 2 , d q d z = β 2 Ω + β 3 2 Ω 2 + β 3 6 T 2 ( 1 + π 2 4 C 2 ) , d Ω d z = 0 .
Δ E 1 = 1 | v 1 − 1 − v 2 − 1 | ∫ 𝒢 ( P 1 , Ω 1 , P 2 , Ω 2 ; s , w , ϕ ) d s d w .
Δ E 1 E 2 = 1 | v 1 − 1 − v 2 − 1 | ∑ n = 0 ∞ ∊ P 1 , Ω 1 , P 2 , Ω 2 ( n ) cos [ n ( ϕ − ϕ 0 ) ] ,
Δ E 1 E 2 ≃ ∊ P 1 , Ω 1 , P 2 , Ω 2 | v 1 − 1 − v 2 − 1 | sin 2 ( ϕ − ϕ o 2 ) ,
r ( ∊ eff ) = ∑ i [ log   PDF gNLSE ( | u i | 2 ) − log   PDF CM ( | u i | 2 , ∊ eff ) ] 2 ∑ i [ log   PDF gNLSE ( | u i | 2 ) ] 2
W ( | u | 2 ) = b a − b ( | u | 2 ) b − 1 exp [ − ( | u | 2 a ) b ]
F ( | u | 2 ) = F 0 exp [ − ( | u | 2 a ) b ] ,
Q ( | u | 2 ) = a b a | u | 2 ( a + 1 ) .

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