Abstract

Various physical structures exhibit a fundamentally probabilistic nature over diverse scales in space and time, to the point that the demarcation line between quantum and classic laws gets blurred. Here, we characterize the probability of intermittency in the laminar-turbulence transition of a partially mode-locked fiber laser system, whose degree of coherence is deteriorated by multiple mode mixing. Two competing processes, namely the proliferation and the decay of an optical turbulent puff, determine a critical behavior for the onset of turbulence in such a nonlinear dissipative system. A new kind of polarization rogue waves is introduced at the point of transition to polarization turbulence. The probabilistic description of the puff-mediated laminar-turbulence polarization transition provides an additional degree of freedom for our understanding of the complex physics of lasers.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  3. K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, “The onset of turbulence in pipe flow,” Science 333(6039), 192–196 (2011).
    [Crossref] [PubMed]
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    [Crossref]
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  7. D. Nagy, G. Szirmai, and P. Domokos, “Self-organization of a Bose-Einstein condensate in an optical cavity,” Eur. Phys. J. D 48(1), 127–137 (2008).
    [Crossref]
  8. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85(1), 299–366 (2013).
    [Crossref]
  9. M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
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    [Crossref] [PubMed]
  16. A. Picozzi, J. Garnier, T. Hansson, P. Suret, G. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
    [Crossref]
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    [Crossref] [PubMed]
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  22. S. V. Sergeyev, C. Mou, E. G. Turitsyna, A. Rozhin, S. K. Turitsyn, and K. Blow, “Spiral attractor created by vector solitons,” Light Sci. Appl. 3(1), e131 (2014).
    [Crossref]
  23. R. Nissim, A. Pejkic, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Ultrafast optical control by few photons in engineered fiber,” Science 345(6195), 417–419 (2014).
    [Crossref] [PubMed]
  24. K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
    [Crossref]
  25. M. Sano and K. Tamai, “A universal transition to turbulence in channel flow,” Nat. Phys. 12(3), 249–253 (2016).
    [Crossref]
  26. G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
    [Crossref]
  27. Z. Liu, S. Zhang, and F. W. Wise, “Rogue waves in a normal-dispersion fiber laser,” Opt. Lett. 40(7), 1366–1369 (2015).
    [Crossref] [PubMed]

2017 (1)

K. Krupa, K. Nithyanandan, and P. Grelu, “Vector dynamics of incoherent dissipative optical solitons,” Opt. Acta (Lond.) 4, 1239–1244 (2017).

2016 (4)

L. Gao, T. Zhu, S. Wabnitz, M. Liu, and W. Huang, “Coherence loss of partially mode-locked fibre laser,” Sci. Rep. 6(1), 24995 (2016).
[Crossref] [PubMed]

V. Kalashnikov, S. V. Sergeyev, G. Jacobsen, S. Popov, and S. K. Turitsyn, “Multi-scale polarisation phenomena,” Light Sci. Appl. 5(1), e16011 (2016).
[Crossref]

M. Sano and K. Tamai, “A universal transition to turbulence in channel flow,” Nat. Phys. 12(3), 249–253 (2016).
[Crossref]

G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
[Crossref]

2015 (5)

Z. Liu, S. Zhang, and F. W. Wise, “Rogue waves in a normal-dispersion fiber laser,” Opt. Lett. 40(7), 1366–1369 (2015).
[Crossref] [PubMed]

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

P. Walczak, S. Randoux, and P. Suret, “Optical rogue waves in integrable turbulence,” Phys. Rev. Lett. 114(14), 143903 (2015).
[Crossref] [PubMed]

D. V. Churkin, S. Sugavanam, N. Tarasov, S. Khorev, S. V. Smirnov, S. M. Kobtsev, and S. K. Turitsyn, “Stochasticity, periodicity and localized light structures in partially mode-locked fiber lasers,” Nat. Commun. 65, 700 (2015).

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

2014 (4)

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

A. Picozzi, J. Garnier, T. Hansson, P. Suret, G. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref]

S. V. Sergeyev, C. Mou, E. G. Turitsyna, A. Rozhin, S. K. Turitsyn, and K. Blow, “Spiral attractor created by vector solitons,” Light Sci. Appl. 3(1), e131 (2014).
[Crossref]

R. Nissim, A. Pejkic, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Ultrafast optical control by few photons in engineered fiber,” Science 345(6195), 417–419 (2014).
[Crossref] [PubMed]

2013 (4)

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
[Crossref]

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85(1), 299–366 (2013).
[Crossref]

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

2012 (2)

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(23), 233901 (2012).
[Crossref] [PubMed]

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nat. Photonics 6(7), 463–468 (2012).
[Crossref]

2011 (1)

K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, “The onset of turbulence in pipe flow,” Science 333(6039), 192–196 (2011).
[Crossref] [PubMed]

2010 (1)

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374(34), 3585–3589 (2010).
[Crossref]

2008 (1)

D. Nagy, G. Szirmai, and P. Domokos, “Self-organization of a Bose-Einstein condensate in an optical cavity,” Eur. Phys. J. D 48(1), 127–137 (2008).
[Crossref]

2007 (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

2000 (1)

S. Grossmann, “The onset of shear flow turbulence,” Rev. Mod. Phys. 72(2), 603–618 (2000).
[Crossref]

1994 (1)

1981 (1)

J. P. Eckmann, “Roads to turbulence in dissipative dynamical systems,” Rev. Mod. Phys. 53(4), 643–654 (1981).
[Crossref]

Akhmediev, N.

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(23), 233901 (2012).
[Crossref] [PubMed]

Alic, N.

R. Nissim, A. Pejkic, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Ultrafast optical control by few photons in engineered fiber,” Science 345(6195), 417–419 (2014).
[Crossref] [PubMed]

Arecchi, F. T.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

Avila, K.

K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, “The onset of turbulence in pipe flow,” Science 333(6039), 192–196 (2011).
[Crossref] [PubMed]

Avila, M.

K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, “The onset of turbulence in pipe flow,” Science 333(6039), 192–196 (2011).
[Crossref] [PubMed]

Babin, S. A.

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

Barkley, D.

K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, “The onset of turbulence in pipe flow,” Science 333(6039), 192–196 (2011).
[Crossref] [PubMed]

Blow, K.

S. V. Sergeyev, C. Mou, E. G. Turitsyna, A. Rozhin, S. K. Turitsyn, and K. Blow, “Spiral attractor created by vector solitons,” Light Sci. Appl. 3(1), e131 (2014).
[Crossref]

Bortolozzo, U.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

Carusotto, I.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85(1), 299–366 (2013).
[Crossref]

Caspani, L.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Christodoulides, D. N.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, G. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref]

Chu, S. T.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Churkin, D. V.

D. V. Churkin, S. Sugavanam, N. Tarasov, S. Khorev, S. V. Smirnov, S. M. Kobtsev, and S. K. Turitsyn, “Stochasticity, periodicity and localized light structures in partially mode-locked fiber lasers,” Nat. Commun. 65, 700 (2015).

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

Ciuti, C.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85(1), 299–366 (2013).
[Crossref]

Clerici, M.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

de Lozar, A.

K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, “The onset of turbulence in pipe flow,” Science 333(6039), 192–196 (2011).
[Crossref] [PubMed]

Dias, F.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

Domokos, P.

D. Nagy, G. Szirmai, and P. Domokos, “Self-organization of a Bose-Einstein condensate in an optical cavity,” Eur. Phys. J. D 48(1), 127–137 (2008).
[Crossref]

Dudley, J. M.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

Eckmann, J. P.

J. P. Eckmann, “Roads to turbulence in dissipative dynamical systems,” Rev. Mod. Phys. 53(4), 643–654 (1981).
[Crossref]

Erkintalo, M.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

Falkovich, G.

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

Ferrera, M.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Finot, C.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374(34), 3585–3589 (2010).
[Crossref]

Gao, L.

L. Gao, T. Zhu, S. Wabnitz, M. Liu, and W. Huang, “Coherence loss of partially mode-locked fibre laser,” Sci. Rep. 6(1), 24995 (2016).
[Crossref] [PubMed]

Garnier, J.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, G. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref]

Genty, G.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

Goda, K.

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
[Crossref]

Grelu, P.

K. Krupa, K. Nithyanandan, and P. Grelu, “Vector dynamics of incoherent dissipative optical solitons,” Opt. Acta (Lond.) 4, 1239–1244 (2017).

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(23), 233901 (2012).
[Crossref] [PubMed]

Grossmann, S.

S. Grossmann, “The onset of shear flow turbulence,” Rev. Mod. Phys. 72(2), 603–618 (2000).
[Crossref]

Haelterman, M.

Hammani, K.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374(34), 3585–3589 (2010).
[Crossref]

Hansson, T.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, G. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref]

Herink, G.

G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
[Crossref]

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nat. Photonics 6(7), 463–468 (2012).
[Crossref]

Hof, B.

K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, “The onset of turbulence in pipe flow,” Science 333(6039), 192–196 (2011).
[Crossref] [PubMed]

Huang, W.

L. Gao, T. Zhu, S. Wabnitz, M. Liu, and W. Huang, “Coherence loss of partially mode-locked fibre laser,” Sci. Rep. 6(1), 24995 (2016).
[Crossref] [PubMed]

Jacobsen, G.

V. Kalashnikov, S. V. Sergeyev, G. Jacobsen, S. Popov, and S. K. Turitsyn, “Multi-scale polarisation phenomena,” Light Sci. Appl. 5(1), e16011 (2016).
[Crossref]

Jalali, B.

G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
[Crossref]

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
[Crossref]

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nat. Photonics 6(7), 463–468 (2012).
[Crossref]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Jestin, Y.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Kalashnikov, V.

V. Kalashnikov, S. V. Sergeyev, G. Jacobsen, S. Popov, and S. K. Turitsyn, “Multi-scale polarisation phenomena,” Light Sci. Appl. 5(1), e16011 (2016).
[Crossref]

Khorev, S.

D. V. Churkin, S. Sugavanam, N. Tarasov, S. Khorev, S. V. Smirnov, S. M. Kobtsev, and S. K. Turitsyn, “Stochasticity, periodicity and localized light structures in partially mode-locked fiber lasers,” Nat. Commun. 65, 700 (2015).

Kibler, B.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374(34), 3585–3589 (2010).
[Crossref]

Kobtsev, S. M.

D. V. Churkin, S. Sugavanam, N. Tarasov, S. Khorev, S. V. Smirnov, S. M. Kobtsev, and S. K. Turitsyn, “Stochasticity, periodicity and localized light structures in partially mode-locked fiber lasers,” Nat. Commun. 65, 700 (2015).

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Krupa, K.

K. Krupa, K. Nithyanandan, and P. Grelu, “Vector dynamics of incoherent dissipative optical solitons,” Opt. Acta (Lond.) 4, 1239–1244 (2017).

Kues, M.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Kuo, B. P.

R. Nissim, A. Pejkic, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Ultrafast optical control by few photons in engineered fiber,” Science 345(6195), 417–419 (2014).
[Crossref] [PubMed]

Lecaplain, C.

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(23), 233901 (2012).
[Crossref] [PubMed]

Little, B. E.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Liu, M.

L. Gao, T. Zhu, S. Wabnitz, M. Liu, and W. Huang, “Coherence loss of partially mode-locked fibre laser,” Sci. Rep. 6(1), 24995 (2016).
[Crossref] [PubMed]

Liu, Z.

Millot, G.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, G. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref]

Montina, A.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

Morandotti, R.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Moss, D. J.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Mou, C.

S. V. Sergeyev, C. Mou, E. G. Turitsyna, A. Rozhin, S. K. Turitsyn, and K. Blow, “Spiral attractor created by vector solitons,” Light Sci. Appl. 3(1), e131 (2014).
[Crossref]

Moxey, D.

K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, “The onset of turbulence in pipe flow,” Science 333(6039), 192–196 (2011).
[Crossref] [PubMed]

Myslivets, E.

R. Nissim, A. Pejkic, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Ultrafast optical control by few photons in engineered fiber,” Science 345(6195), 417–419 (2014).
[Crossref] [PubMed]

Nagy, D.

D. Nagy, G. Szirmai, and P. Domokos, “Self-organization of a Bose-Einstein condensate in an optical cavity,” Eur. Phys. J. D 48(1), 127–137 (2008).
[Crossref]

Nissim, R.

R. Nissim, A. Pejkic, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Ultrafast optical control by few photons in engineered fiber,” Science 345(6195), 417–419 (2014).
[Crossref] [PubMed]

Nithyanandan, K.

K. Krupa, K. Nithyanandan, and P. Grelu, “Vector dynamics of incoherent dissipative optical solitons,” Opt. Acta (Lond.) 4, 1239–1244 (2017).

Onorato, M.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

Pasquazi, A.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Peccianti, M.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Pejkic, A.

R. Nissim, A. Pejkic, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Ultrafast optical control by few photons in engineered fiber,” Science 345(6195), 417–419 (2014).
[Crossref] [PubMed]

Picozzi, A.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, G. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref]

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374(34), 3585–3589 (2010).
[Crossref]

Podivilov, E. V.

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

Popov, S.

V. Kalashnikov, S. V. Sergeyev, G. Jacobsen, S. Popov, and S. K. Turitsyn, “Multi-scale polarisation phenomena,” Light Sci. Appl. 5(1), e16011 (2016).
[Crossref]

Radic, S.

R. Nissim, A. Pejkic, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Ultrafast optical control by few photons in engineered fiber,” Science 345(6195), 417–419 (2014).
[Crossref] [PubMed]

Randoux, G.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, G. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref]

Randoux, S.

P. Walczak, S. Randoux, and P. Suret, “Optical rogue waves in integrable turbulence,” Phys. Rev. Lett. 114(14), 143903 (2015).
[Crossref] [PubMed]

Reimer, C.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Residori, S.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

Ropers, C.

G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
[Crossref]

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nat. Photonics 6(7), 463–468 (2012).
[Crossref]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Rozhin, A.

S. V. Sergeyev, C. Mou, E. G. Turitsyna, A. Rozhin, S. K. Turitsyn, and K. Blow, “Spiral attractor created by vector solitons,” Light Sci. Appl. 3(1), e131 (2014).
[Crossref]

Roztocki, P.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Sano, M.

M. Sano and K. Tamai, “A universal transition to turbulence in channel flow,” Nat. Phys. 12(3), 249–253 (2016).
[Crossref]

Sciamanna, M.

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

Sergeyev, S. V.

V. Kalashnikov, S. V. Sergeyev, G. Jacobsen, S. Popov, and S. K. Turitsyn, “Multi-scale polarisation phenomena,” Light Sci. Appl. 5(1), e16011 (2016).
[Crossref]

S. V. Sergeyev, C. Mou, E. G. Turitsyna, A. Rozhin, S. K. Turitsyn, and K. Blow, “Spiral attractor created by vector solitons,” Light Sci. Appl. 3(1), e131 (2014).
[Crossref]

Shore, K. A.

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

Shu, X.

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

Smirnov, S. V.

D. V. Churkin, S. Sugavanam, N. Tarasov, S. Khorev, S. V. Smirnov, S. M. Kobtsev, and S. K. Turitsyn, “Stochasticity, periodicity and localized light structures in partially mode-locked fiber lasers,” Nat. Commun. 65, 700 (2015).

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

Solli, D. R.

G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
[Crossref]

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nat. Photonics 6(7), 463–468 (2012).
[Crossref]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Soto-Crespo, J. M.

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(23), 233901 (2012).
[Crossref] [PubMed]

Sugavanam, S.

D. V. Churkin, S. Sugavanam, N. Tarasov, S. Khorev, S. V. Smirnov, S. M. Kobtsev, and S. K. Turitsyn, “Stochasticity, periodicity and localized light structures in partially mode-locked fiber lasers,” Nat. Commun. 65, 700 (2015).

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

Suret, P.

P. Walczak, S. Randoux, and P. Suret, “Optical rogue waves in integrable turbulence,” Phys. Rev. Lett. 114(14), 143903 (2015).
[Crossref] [PubMed]

A. Picozzi, J. Garnier, T. Hansson, P. Suret, G. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref]

Szirmai, G.

D. Nagy, G. Szirmai, and P. Domokos, “Self-organization of a Bose-Einstein condensate in an optical cavity,” Eur. Phys. J. D 48(1), 127–137 (2008).
[Crossref]

Tamai, K.

M. Sano and K. Tamai, “A universal transition to turbulence in channel flow,” Nat. Phys. 12(3), 249–253 (2016).
[Crossref]

Tarasov, N.

D. V. Churkin, S. Sugavanam, N. Tarasov, S. Khorev, S. V. Smirnov, S. M. Kobtsev, and S. K. Turitsyn, “Stochasticity, periodicity and localized light structures in partially mode-locked fiber lasers,” Nat. Commun. 65, 700 (2015).

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

Trillo, S.

Turitsyn, S. K.

V. Kalashnikov, S. V. Sergeyev, G. Jacobsen, S. Popov, and S. K. Turitsyn, “Multi-scale polarisation phenomena,” Light Sci. Appl. 5(1), e16011 (2016).
[Crossref]

D. V. Churkin, S. Sugavanam, N. Tarasov, S. Khorev, S. V. Smirnov, S. M. Kobtsev, and S. K. Turitsyn, “Stochasticity, periodicity and localized light structures in partially mode-locked fiber lasers,” Nat. Commun. 65, 700 (2015).

S. V. Sergeyev, C. Mou, E. G. Turitsyna, A. Rozhin, S. K. Turitsyn, and K. Blow, “Spiral attractor created by vector solitons,” Light Sci. Appl. 3(1), e131 (2014).
[Crossref]

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

Turitsyna, E. G.

S. V. Sergeyev, C. Mou, E. G. Turitsyna, A. Rozhin, S. K. Turitsyn, and K. Blow, “Spiral attractor created by vector solitons,” Light Sci. Appl. 3(1), e131 (2014).
[Crossref]

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

Wabnitz, S.

L. Gao, T. Zhu, S. Wabnitz, M. Liu, and W. Huang, “Coherence loss of partially mode-locked fibre laser,” Sci. Rep. 6(1), 24995 (2016).
[Crossref] [PubMed]

M. Haelterman, S. Trillo, and S. Wabnitz, “Polarization multistability and instability in a nonlinear dispersive ring cavity,” J. Opt. Soc. Am. B 11(3), 446–456 (1994).
[Crossref]

Walczak, P.

P. Walczak, S. Randoux, and P. Suret, “Optical rogue waves in integrable turbulence,” Phys. Rev. Lett. 114(14), 143903 (2015).
[Crossref] [PubMed]

Wetzel, B.

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

Wise, F. W.

Zhang, S.

Zhu, T.

L. Gao, T. Zhu, S. Wabnitz, M. Liu, and W. Huang, “Coherence loss of partially mode-locked fibre laser,” Sci. Rep. 6(1), 24995 (2016).
[Crossref] [PubMed]

Eur. Phys. J. D (1)

D. Nagy, G. Szirmai, and P. Domokos, “Self-organization of a Bose-Einstein condensate in an optical cavity,” Eur. Phys. J. D 48(1), 127–137 (2008).
[Crossref]

J. Opt. Soc. Am. B (1)

Light Sci. Appl. (2)

V. Kalashnikov, S. V. Sergeyev, G. Jacobsen, S. Popov, and S. K. Turitsyn, “Multi-scale polarisation phenomena,” Light Sci. Appl. 5(1), e16011 (2016).
[Crossref]

S. V. Sergeyev, C. Mou, E. G. Turitsyna, A. Rozhin, S. K. Turitsyn, and K. Blow, “Spiral attractor created by vector solitons,” Light Sci. Appl. 3(1), e131 (2014).
[Crossref]

Nat. Commun. (2)

C. Reimer, M. Kues, L. Caspani, B. Wetzel, P. Roztocki, M. Clerici, Y. Jestin, M. Ferrera, M. Peccianti, A. Pasquazi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip,” Nat. Commun. 6(1), 8236 (2015).
[Crossref] [PubMed]

D. V. Churkin, S. Sugavanam, N. Tarasov, S. Khorev, S. V. Smirnov, S. M. Kobtsev, and S. K. Turitsyn, “Stochasticity, periodicity and localized light structures in partially mode-locked fiber lasers,” Nat. Commun. 65, 700 (2015).

Nat. Photonics (6)

E. G. Turitsyna, S. V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. A. Babin, E. V. Podivilov, D. V. Churkin, G. Falkovich, and S. K. Turitsyn, “The laminar-turbulent transition in a fiber laser,” Nat. Photonics 7(10), 783–786 (2013).
[Crossref]

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

D. R. Solli, G. Herink, B. Jalali, and C. Ropers, “Fluctuations and correlations in modulation instability,” Nat. Photonics 6(7), 463–468 (2012).
[Crossref]

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016).
[Crossref]

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
[Crossref]

Nat. Phys. (1)

M. Sano and K. Tamai, “A universal transition to turbulence in channel flow,” Nat. Phys. 12(3), 249–253 (2016).
[Crossref]

Nature (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Opt. Acta (Lond.) (1)

K. Krupa, K. Nithyanandan, and P. Grelu, “Vector dynamics of incoherent dissipative optical solitons,” Opt. Acta (Lond.) 4, 1239–1244 (2017).

Opt. Lett. (1)

Phys. Lett. A (1)

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374(34), 3585–3589 (2010).
[Crossref]

Phys. Rep. (2)

A. Picozzi, J. Garnier, T. Hansson, P. Suret, G. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref]

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

Phys. Rev. Lett. (2)

P. Walczak, S. Randoux, and P. Suret, “Optical rogue waves in integrable turbulence,” Phys. Rev. Lett. 114(14), 143903 (2015).
[Crossref] [PubMed]

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(23), 233901 (2012).
[Crossref] [PubMed]

Rev. Mod. Phys. (3)

J. P. Eckmann, “Roads to turbulence in dissipative dynamical systems,” Rev. Mod. Phys. 53(4), 643–654 (1981).
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S. Grossmann, “The onset of shear flow turbulence,” Rev. Mod. Phys. 72(2), 603–618 (2000).
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[Crossref]

Sci. Rep. (1)

L. Gao, T. Zhu, S. Wabnitz, M. Liu, and W. Huang, “Coherence loss of partially mode-locked fibre laser,” Sci. Rep. 6(1), 24995 (2016).
[Crossref] [PubMed]

Science (2)

R. Nissim, A. Pejkic, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Ultrafast optical control by few photons in engineered fiber,” Science 345(6195), 417–419 (2014).
[Crossref] [PubMed]

K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, “The onset of turbulence in pipe flow,” Science 333(6039), 192–196 (2011).
[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 Schematic of our fiber laser cavity and measurement methods. (a) EDF, erbium-doped fiber (LiekkiER 80-8/125); WDM, wavelength division multiplexer; ISO, polarization independent optical isolator; PC, polarization controller; DCF, dispersion compensation fiber; OC, optical coupler; OSA, optical spectrum analyzer; PD, photo-detector; EDFA, erbium-doped fiber amplifier; TF, tunable filter; PSA, polarization state analyzer.
Fig. 2
Fig. 2 Typical outputs for PML. (a) Oscilloscope trace. (b) Autocorrelation trace (inset is the coherent peak in a larger scale). (c) Example of single-shot spectrum together with the averaged spectrum. The labels of (i), (ii), (iii), (iv), (v), (vi) are wavelength at 1547.6 nm, 1556.1 nm, 1561.2 nm, 1565 nm, 1574.1 nm, 1582.7 nm, respectively.
Fig. 3
Fig. 3 The build-up of PML as the formation of optical turbulent puff. (a) Normalized intracavity power in forming an optical puff, which demarcates two regimes, namely the proliferation phase (as fitted exponentially by means of a solid red line) and the decay phase. About 850 round trips are required for the optical puff to proliferate from pure noise to a maximum intensity value. Next, the puff decays within a time frame, which is much smaller than the proliferation time. The time t0 for the turbulence formation as shown by single-shot spectra in panel (e) is about 10 trt. Whenever the total energy is insufficient to sustain mode-locking, pulse relaminarizes. (b)-(d) single-shot spectra for various pump powers. The corresponding single-shot spectrum appears as stochastically varying at each round trip. (e) Single-shot spectra with almost identical puff proliferation and decay times, corresponding to the critical point for transition to turbulence, for P = 650 mW. (f) temporal outputs for various pump powers.
Fig. 4
Fig. 4 Statistical properties of temporal laser emission. (a) Measured probability of optical puffs starting to decay up to time t. All data are fitted with the expression exp[−(tt0)/ τ(P)], as illustrated by the straight lines. The formation time of an optical puff, t0, is about 10trt as deduced from experimental data. (b) Variation of characteristic time τ with pump power P (inset in linear scale). This was extracted by exponentially fitting data in (a). Error bars represent absolute errors based on experimental uncertainties. The error is relatively high for P larger than 600 mW, since the corresponding τ is smaller. This error may have a contribution from the oscilloscope trigger error, and the difficulty to distinguish two adjacent optical puffs when P is high. (c) Histograms of the temporal trace (inset in log scale), showing that the signal intensity distribution exhibits a transition from laminar to turbulent state. Optical rogue waves can be well distinguished from stochastic events.
Fig. 5
Fig. 5 Scattering plots for filtered PML at various wavelengths. The wavelength are 1547.6 nm, 1556.1 nm, 1561.2 nm, 1565 nm, 1574.1 nm, 1582.7 nm, respectively, which are indicated by (i), (ii), (iii), (iv), (v), (vi) in Fig. 3(c). The labels are the same in Fig. 2(c).
Fig. 6
Fig. 6 Optical polarization rogue waves in the polarization laminar-turbulent transition. (a)-(c), Normalized Stokes parameters s2, s3 for filtered wavelength at different pump powers. Irregular points with distance far away from others are observed along with unpredictable probabilities and positions, indicating the emergence optical polarization rogue waves. (d)-(f), Logarithm of the PDF of the distance between each polarization point corresponding to cases in (a)-(c). The PDF evolves into a nearly lognormal distribution when more optical turbulent puffs are formed, where the SOP is fully randomized.

Equations (2)

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P proliferating ( P , t ) = exp [ ( t t 0 ) / τ ( P ) ]
r = m , n = 1 m n N | S m S n | = m , n = 1 m n N ( s m 1 s n 1 ) 2 + ( s m 2 s n 2 ) 2 + ( s m 3 s n 3 ) 2

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