Abstract

We show, both experimentally and theoretically, that the loss of coherence of a long cavity optical coherence tomography (OCT) laser can be described as a transition from laminar to turbulent flows. We demonstrate that in this strongly dissipative system, the transition happens either via an absolute or a convective instability depending on the laser parameters. In the latter case, the transition occurs via formation of localised structures in the laminar regime, which trigger the formation of growing and drifting puffs of turbulence. Experimentally, we demonstrate that these turbulent bursts are seeded by appearance of Nozaki-Bekki holes, characterised by the zero field amplitude and π phase jumps. Our experimental results are supported with numerical simulations based on the delay differential equations model.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  24. P. Coullet, T. Frisch, and F. Plaza, “Sources and sinks of wave patterns,” Phys. D: Nonlinear Phenom. 62, 75–79 (1993).
    [Crossref]
  25. A. Couairon and J. M. Chomaz, “Global instability in fully nonlinear systems,” Phys. Rev. Lett. 77, 4015 (1996).
    [Crossref] [PubMed]
  26. R. J. Deissler, “Noise-sustained structure, intermittency, and the ginzburg-landau equation,” J. Stat. Phys. 40, 371–395 (1985).
    [Crossref]
  27. S. Slepneva, B. Kelleher, B. O’Shaughnessy, S. Hegarty, A. G. Vladimirov, and G. Huyet, “Dynamics of Fourier domain mode-locked lasers,” Opt. Express 21, 19240–19251 (2013).
    [Crossref] [PubMed]
  28. R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier domain mode locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14, 3225–3237 (2006).
    [Crossref] [PubMed]
  29. T. Pfeiffer, M. Petermann, W. Draxinger, C. Jirauschek, and R. Huber, “Ultra low noise fourier domain mode locked laser for high quality megahertz optical coherence tomography,” Biomed. Opt. Express 9, 4130–4148 (2018).
    [Crossref]
  30. C. Jirauschek and R. Huber, “Wavelength shifting of intra-cavity photons: Adiabatic wavelength tuning in rapidly wavelength-swept lasers,” Biomed. Opt. Express 6, 2448–2465 (2015).
    [Crossref] [PubMed]
  31. D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709 (2007).
    [Crossref]
  32. S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A time-encoded technique for fibre-based hyperspectral broadband stimulated raman microscopy,” Nat. Comm. 6, 6784 (2015).
    [Crossref] [PubMed]
  33. C. M. Eigenwillig, W. Wieser, S. Todor, B. R. Biedermann, T. Klein, C. Jirauschek, and R. Huber, “Picosecond pulses from wavelength-swept continuous-wave fourier domain mode-locked lasers,” Nat. Comm. 4, 1848 (2013).
    [Crossref]
  34. G. Giacomelli and A. Politi, “Relationship between delayed and spatially extended dynamical systems,” Phys. Rev. Lett. 76, 2686 (1996).
    [Crossref] [PubMed]
  35. D. Goulding, T. Butler, B. Kelleher, S. Slepneva, S. P. Hegarty, and G. Huyet, “Visualisation of the intensity and phase dynamics of semiconductor lasers via electric field reconstructions,” in Nonlinear Dynamics: Materials, Theory and Experiments (Springer, 2016), pp. 3–29.
  36. K. Nozaki and N. Bekki, “Exact solutions of the generalized ginzburg-landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
    [Crossref]
  37. N. Bekki and K. Nozaki, “Formations of spatial patterns and holes in the generalized ginzburg-landau equation,” Phys. Lett. A 110, 133–135 (1985).
    [Crossref]
  38. W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex ginzburg-landau equations,” Phys. D: Nonlinear Phenom. 56, 303–367 (1992).
    [Crossref]
  39. J. Burguete, H. Chaté, F. Daviaud, and N. Mukolobwiez, “Bekki-Nozaki amplitude holes in hydrothermal nonlinear waves,” Phys. Rev. Lett. 82, 3252–3255 (1999).
    [Crossref]

2018 (2)

2017 (4)

J. I. Cardesa, A. Vela-Martín, and J. Jiménez, “The turbulent cascade in five dimensions,” Science 357, 782–784 (2017).
[Crossref] [PubMed]

D. Castelvecchi, “Mysteries of turbulence unravelled,” Nature 548, 382–383 (2017).
[Crossref] [PubMed]

T. P. Butler, S. Slepneva, P. M. McNamara, K. Neuhaus, D. Goulding, M. Leahy, and G. Huyet, “Real-time experimental measurement of swept source vcsel properties relevant to oct imaging,” IEEE Photonics J. 9, 1–10 (2017).
[Crossref]

T. Klein and R. Huber, “High-speed oct light sources and systems,” Biomed. Opt. Express 8, 828–859 (2017).
[Crossref] [PubMed]

2015 (4)

2014 (1)

2013 (3)

S. Slepneva, B. Kelleher, B. O’Shaughnessy, S. Hegarty, A. G. Vladimirov, and G. Huyet, “Dynamics of Fourier domain mode-locked lasers,” Opt. Express 21, 19240–19251 (2013).
[Crossref] [PubMed]

C. M. Eigenwillig, W. Wieser, S. Todor, B. R. Biedermann, T. Klein, C. Jirauschek, and R. Huber, “Picosecond pulses from wavelength-swept continuous-wave fourier domain mode-locked lasers,” Nat. Comm. 4, 1848 (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 fibre laser,” Nat. Photonics 7, 783 (2013).
[Crossref]

2011 (2)

2007 (1)

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709 (2007).
[Crossref]

2006 (2)

M. Kolokolnikov, Th. Nizette, T. Erneux, N. Joly, and S. Bielawski, “The Q-switching instability in passively mode-locked lasers,” Phys. D: Nonlinear Phenom. 219, 13–21 (2006).
[Crossref]

R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier domain mode locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14, 3225–3237 (2006).
[Crossref] [PubMed]

2002 (1)

I. S. Aranson and L. Kramer, “The world of the complex ginzburg-landau equation,” Rev. Mod. Phys. 74, 99–161 (2002).
[Crossref]

1999 (1)

J. Burguete, H. Chaté, F. Daviaud, and N. Mukolobwiez, “Bekki-Nozaki amplitude holes in hydrothermal nonlinear waves,” Phys. Rev. Lett. 82, 3252–3255 (1999).
[Crossref]

1997 (1)

1996 (2)

G. Giacomelli and A. Politi, “Relationship between delayed and spatially extended dynamical systems,” Phys. Rev. Lett. 76, 2686 (1996).
[Crossref] [PubMed]

A. Couairon and J. M. Chomaz, “Global instability in fully nonlinear systems,” Phys. Rev. Lett. 77, 4015 (1996).
[Crossref] [PubMed]

1993 (2)

P. Coullet, T. Frisch, and F. Plaza, “Sources and sinks of wave patterns,” Phys. D: Nonlinear Phenom. 62, 75–79 (1993).
[Crossref]

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[Crossref]

1992 (1)

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex ginzburg-landau equations,” Phys. D: Nonlinear Phenom. 56, 303–367 (1992).
[Crossref]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

1989 (1)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

1985 (3)

P. Huerre and P. A. Monkewitz, “Absolute and convective instabilities in free shear layers,” J. Fluid Mech. 159, 151–168 (1985).
[Crossref]

R. J. Deissler, “Noise-sustained structure, intermittency, and the ginzburg-landau equation,” J. Stat. Phys. 40, 371–395 (1985).
[Crossref]

N. Bekki and K. Nozaki, “Formations of spatial patterns and holes in the generalized ginzburg-landau equation,” Phys. Lett. A 110, 133–135 (1985).
[Crossref]

1984 (1)

K. Nozaki and N. Bekki, “Exact solutions of the generalized ginzburg-landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[Crossref]

1944 (1)

L. D. Landau, “On the problem of turbulence,” Dokl. Akad. Nauk SSSR 44, 339–349 (1944).

1884 (1)

O. Reynolds, “XXIX. An experimental investigation of the circumstances which determine whether the motion of water shall he direct or sinuous, and of the law of resistance in parallel channels,” Philos. Transactions Royal Soc. Lond. 174, 935–982 (1884).

Adler, D. C.

D. C. Adler, W. Wieser, F. Trepanier, J. M. Schmitt, and R. A. Huber, “Extended coherence length fourier domain mode locked lasers at 1310 nm,” Opt. Express 19, 20930–20939 (2011).
[Crossref] [PubMed]

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709 (2007).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, “Nonlinear fiber optics,” in Nonlinear Science at the Dawn of the 21st Century (Springer, 2000), pp. 195–211.
[Crossref]

Aranson, I. S.

I. S. Aranson and L. Kramer, “The world of the complex ginzburg-landau equation,” Rev. Mod. Phys. 74, 99–161 (2002).
[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, 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, 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 fibre laser,” Nat. Photonics 7, 783 (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, 192–196 (2011).
[Crossref] [PubMed]

Bekki, N.

N. Bekki and K. Nozaki, “Formations of spatial patterns and holes in the generalized ginzburg-landau equation,” Phys. Lett. A 110, 133–135 (1985).
[Crossref]

K. Nozaki and N. Bekki, “Exact solutions of the generalized ginzburg-landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[Crossref]

Biedermann, B. R.

C. M. Eigenwillig, W. Wieser, S. Todor, B. R. Biedermann, T. Klein, C. Jirauschek, and R. Huber, “Picosecond pulses from wavelength-swept continuous-wave fourier domain mode-locked lasers,” Nat. Comm. 4, 1848 (2013).
[Crossref]

Bielawski, S.

M. Kolokolnikov, Th. Nizette, T. Erneux, N. Joly, and S. Bielawski, “The Q-switching instability in passively mode-locked lasers,” Phys. D: Nonlinear Phenom. 219, 13–21 (2006).
[Crossref]

Bonesi, M.

Boschert, P.

Burguete, J.

J. Burguete, H. Chaté, F. Daviaud, and N. Mukolobwiez, “Bekki-Nozaki amplitude holes in hydrothermal nonlinear waves,” Phys. Rev. Lett. 82, 3252–3255 (1999).
[Crossref]

Butler, T.

T. Butler, S. Slepneva, B. O’Shaughnessy, B. Kelleher, D. Goulding, S. P. Hegarty, H.-C. Lyu, K. Karnowski, M. Wojtkowski, and G. Huyet, “Single shot, time-resolved measurement of the coherence properties of oct swept source lasers,” Opt. Lett. 40, 2277–2280 (2015).
[Crossref] [PubMed]

D. Goulding, T. Butler, B. Kelleher, S. Slepneva, S. P. Hegarty, and G. Huyet, “Visualisation of the intensity and phase dynamics of semiconductor lasers via electric field reconstructions,” in Nonlinear Dynamics: Materials, Theory and Experiments (Springer, 2016), pp. 3–29.

Butler, T. P.

T. P. Butler, S. Slepneva, P. M. McNamara, K. Neuhaus, D. Goulding, M. Leahy, and G. Huyet, “Real-time experimental measurement of swept source vcsel properties relevant to oct imaging,” IEEE Photonics J. 9, 1–10 (2017).
[Crossref]

Cao, Y. L.

Cardesa, J. I.

J. I. Cardesa, A. Vela-Martín, and J. Jiménez, “The turbulent cascade in five dimensions,” Science 357, 782–784 (2017).
[Crossref] [PubMed]

Castelvecchi, D.

D. Castelvecchi, “Mysteries of turbulence unravelled,” Nature 548, 382–383 (2017).
[Crossref] [PubMed]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Chaté, H.

J. Burguete, H. Chaté, F. Daviaud, and N. Mukolobwiez, “Bekki-Nozaki amplitude holes in hydrothermal nonlinear waves,” Phys. Rev. Lett. 82, 3252–3255 (1999).
[Crossref]

Chen, Y.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709 (2007).
[Crossref]

Chinn, S. R.

Chomaz, J. M.

A. Couairon and J. M. Chomaz, “Global instability in fully nonlinear systems,” Phys. Rev. Lett. 77, 4015 (1996).
[Crossref] [PubMed]

Churkin, D. V.

S. Sugavanam, N. Tarasov, S. Wabnitz, and D. V. Churkin, “Ginzburg–landau turbulence in quasi-cw raman fiber lasers,” Laser Photonics Rev. 9, L35–L39 (2015).
[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 fibre laser,” Nat. Photonics 7, 783 (2013).
[Crossref]

Connolly, J.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709 (2007).
[Crossref]

Couairon, A.

A. Couairon and J. M. Chomaz, “Global instability in fully nonlinear systems,” Phys. Rev. Lett. 77, 4015 (1996).
[Crossref] [PubMed]

Coullet, P.

P. Coullet, T. Frisch, and F. Plaza, “Sources and sinks of wave patterns,” Phys. D: Nonlinear Phenom. 62, 75–79 (1993).
[Crossref]

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

Crawford, M.

Cross, M. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[Crossref]

Daviaud, F.

J. Burguete, H. Chaté, F. Daviaud, and N. Mukolobwiez, “Bekki-Nozaki amplitude holes in hydrothermal nonlinear waves,” Phys. Rev. Lett. 82, 3252–3255 (1999).
[Crossref]

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, 192–196 (2011).
[Crossref] [PubMed]

Deissler, R. J.

R. J. Deissler, “Noise-sustained structure, intermittency, and the ginzburg-landau equation,” J. Stat. Phys. 40, 371–395 (1985).
[Crossref]

Draxinger, W.

Drazin, P. G.

P. G. Drazin and W. H. Reid, “Hydrodynamic Stability” (University of Chicago, 1981).

Drexler, W.

Eibl, M.

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A time-encoded technique for fibre-based hyperspectral broadband stimulated raman microscopy,” Nat. Comm. 6, 6784 (2015).
[Crossref] [PubMed]

Eigenwillig, C. M.

C. M. Eigenwillig, W. Wieser, S. Todor, B. R. Biedermann, T. Klein, C. Jirauschek, and R. Huber, “Picosecond pulses from wavelength-swept continuous-wave fourier domain mode-locked lasers,” Nat. Comm. 4, 1848 (2013).
[Crossref]

Ensher, J.

Erneux, T.

M. Kolokolnikov, Th. Nizette, T. Erneux, N. Joly, and S. Bielawski, “The Q-switching instability in passively mode-locked lasers,” Phys. D: Nonlinear Phenom. 219, 13–21 (2006).
[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 fibre laser,” Nat. Photonics 7, 783 (2013).
[Crossref]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Frisch, T.

P. Coullet, T. Frisch, and F. Plaza, “Sources and sinks of wave patterns,” Phys. D: Nonlinear Phenom. 62, 75–79 (1993).
[Crossref]

Fujimoto, J. G.

Gao, L.

Giacomelli, G.

G. Giacomelli and A. Politi, “Relationship between delayed and spatially extended dynamical systems,” Phys. Rev. Lett. 76, 2686 (1996).
[Crossref] [PubMed]

Gil, L.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

Goulding, D.

T. P. Butler, S. Slepneva, P. M. McNamara, K. Neuhaus, D. Goulding, M. Leahy, and G. Huyet, “Real-time experimental measurement of swept source vcsel properties relevant to oct imaging,” IEEE Photonics J. 9, 1–10 (2017).
[Crossref]

T. Butler, S. Slepneva, B. O’Shaughnessy, B. Kelleher, D. Goulding, S. P. Hegarty, H.-C. Lyu, K. Karnowski, M. Wojtkowski, and G. Huyet, “Single shot, time-resolved measurement of the coherence properties of oct swept source lasers,” Opt. Lett. 40, 2277–2280 (2015).
[Crossref] [PubMed]

D. Goulding, T. Butler, B. Kelleher, S. Slepneva, S. P. Hegarty, and G. Huyet, “Visualisation of the intensity and phase dynamics of semiconductor lasers via electric field reconstructions,” in Nonlinear Dynamics: Materials, Theory and Experiments (Springer, 2016), pp. 3–29.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Hegarty, S.

Hegarty, S. P.

T. Butler, S. Slepneva, B. O’Shaughnessy, B. Kelleher, D. Goulding, S. P. Hegarty, H.-C. Lyu, K. Karnowski, M. Wojtkowski, and G. Huyet, “Single shot, time-resolved measurement of the coherence properties of oct swept source lasers,” Opt. Lett. 40, 2277–2280 (2015).
[Crossref] [PubMed]

D. Goulding, T. Butler, B. Kelleher, S. Slepneva, S. P. Hegarty, and G. Huyet, “Visualisation of the intensity and phase dynamics of semiconductor lasers via electric field reconstructions,” in Nonlinear Dynamics: Materials, Theory and Experiments (Springer, 2016), pp. 3–29.

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, 192–196 (2011).
[Crossref] [PubMed]

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[Crossref]

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex ginzburg-landau equations,” Phys. D: Nonlinear Phenom. 56, 303–367 (1992).
[Crossref]

Hoover, E.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Huber, R.

T. Pfeiffer, M. Petermann, W. Draxinger, C. Jirauschek, and R. Huber, “Ultra low noise fourier domain mode locked laser for high quality megahertz optical coherence tomography,” Biomed. Opt. Express 9, 4130–4148 (2018).
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T. Klein and R. Huber, “High-speed oct light sources and systems,” Biomed. Opt. Express 8, 828–859 (2017).
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C. Jirauschek and R. Huber, “Wavelength shifting of intra-cavity photons: Adiabatic wavelength tuning in rapidly wavelength-swept lasers,” Biomed. Opt. Express 6, 2448–2465 (2015).
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S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A time-encoded technique for fibre-based hyperspectral broadband stimulated raman microscopy,” Nat. Comm. 6, 6784 (2015).
[Crossref] [PubMed]

C. M. Eigenwillig, W. Wieser, S. Todor, B. R. Biedermann, T. Klein, C. Jirauschek, and R. Huber, “Picosecond pulses from wavelength-swept continuous-wave fourier domain mode-locked lasers,” Nat. Comm. 4, 1848 (2013).
[Crossref]

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709 (2007).
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R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier domain mode locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14, 3225–3237 (2006).
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Huber, R. A.

Huerre, P.

P. Huerre and P. A. Monkewitz, “Absolute and convective instabilities in free shear layers,” J. Fluid Mech. 159, 151–168 (1985).
[Crossref]

Huyet, G.

T. P. Butler, S. Slepneva, P. M. McNamara, K. Neuhaus, D. Goulding, M. Leahy, and G. Huyet, “Real-time experimental measurement of swept source vcsel properties relevant to oct imaging,” IEEE Photonics J. 9, 1–10 (2017).
[Crossref]

T. Butler, S. Slepneva, B. O’Shaughnessy, B. Kelleher, D. Goulding, S. P. Hegarty, H.-C. Lyu, K. Karnowski, M. Wojtkowski, and G. Huyet, “Single shot, time-resolved measurement of the coherence properties of oct swept source lasers,” Opt. Lett. 40, 2277–2280 (2015).
[Crossref] [PubMed]

S. Slepneva, B. Kelleher, B. O’Shaughnessy, S. Hegarty, A. G. Vladimirov, and G. Huyet, “Dynamics of Fourier domain mode-locked lasers,” Opt. Express 21, 19240–19251 (2013).
[Crossref] [PubMed]

D. Goulding, T. Butler, B. Kelleher, S. Slepneva, S. P. Hegarty, and G. Huyet, “Visualisation of the intensity and phase dynamics of semiconductor lasers via electric field reconstructions,” in Nonlinear Dynamics: Materials, Theory and Experiments (Springer, 2016), pp. 3–29.

Jiménez, J.

J. I. Cardesa, A. Vela-Martín, and J. Jiménez, “The turbulent cascade in five dimensions,” Science 357, 782–784 (2017).
[Crossref] [PubMed]

Jirauschek, C.

Joly, N.

M. Kolokolnikov, Th. Nizette, T. Erneux, N. Joly, and S. Bielawski, “The Q-switching instability in passively mode-locked lasers,” Phys. D: Nonlinear Phenom. 219, 13–21 (2006).
[Crossref]

Karnowski, K.

Karpf, S.

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A time-encoded technique for fibre-based hyperspectral broadband stimulated raman microscopy,” Nat. Comm. 6, 6784 (2015).
[Crossref] [PubMed]

Kelleher, B.

Klein, T.

T. Klein and R. Huber, “High-speed oct light sources and systems,” Biomed. Opt. Express 8, 828–859 (2017).
[Crossref] [PubMed]

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A time-encoded technique for fibre-based hyperspectral broadband stimulated raman microscopy,” Nat. Comm. 6, 6784 (2015).
[Crossref] [PubMed]

C. M. Eigenwillig, W. Wieser, S. Todor, B. R. Biedermann, T. Klein, C. Jirauschek, and R. Huber, “Picosecond pulses from wavelength-swept continuous-wave fourier domain mode-locked lasers,” Nat. Comm. 4, 1848 (2013).
[Crossref]

Kolokolnikov, M.

M. Kolokolnikov, Th. Nizette, T. Erneux, N. Joly, and S. Bielawski, “The Q-switching instability in passively mode-locked lasers,” Phys. D: Nonlinear Phenom. 219, 13–21 (2006).
[Crossref]

Kramer, L.

I. S. Aranson and L. Kramer, “The world of the complex ginzburg-landau equation,” Rev. Mod. Phys. 74, 99–161 (2002).
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Landau, L. D.

L. D. Landau, “On the problem of turbulence,” Dokl. Akad. Nauk SSSR 44, 339–349 (1944).

L. D. Landau and E. M. Lifdhitz, Fluid mechanics (Butterworth-Heinemann, 1987).

Leahy, M.

T. P. Butler, S. Slepneva, P. M. McNamara, K. Neuhaus, D. Goulding, M. Leahy, and G. Huyet, “Real-time experimental measurement of swept source vcsel properties relevant to oct imaging,” IEEE Photonics J. 9, 1–10 (2017).
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Li, Y.

Lifdhitz, E. M.

L. D. Landau and E. M. Lifdhitz, Fluid mechanics (Butterworth-Heinemann, 1987).

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Lyu, H.-C.

McNamara, P. M.

T. P. Butler, S. Slepneva, P. M. McNamara, K. Neuhaus, D. Goulding, M. Leahy, and G. Huyet, “Real-time experimental measurement of swept source vcsel properties relevant to oct imaging,” IEEE Photonics J. 9, 1–10 (2017).
[Crossref]

Minneman, M. P.

Monkewitz, P. A.

P. Huerre and P. A. Monkewitz, “Absolute and convective instabilities in free shear layers,” J. Fluid Mech. 159, 151–168 (1985).
[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, 192–196 (2011).
[Crossref] [PubMed]

Mukolobwiez, N.

J. Burguete, H. Chaté, F. Daviaud, and N. Mukolobwiez, “Bekki-Nozaki amplitude holes in hydrothermal nonlinear waves,” Phys. Rev. Lett. 82, 3252–3255 (1999).
[Crossref]

Neuhaus, K.

T. P. Butler, S. Slepneva, P. M. McNamara, K. Neuhaus, D. Goulding, M. Leahy, and G. Huyet, “Real-time experimental measurement of swept source vcsel properties relevant to oct imaging,” IEEE Photonics J. 9, 1–10 (2017).
[Crossref]

Nizette, Th.

M. Kolokolnikov, Th. Nizette, T. Erneux, N. Joly, and S. Bielawski, “The Q-switching instability in passively mode-locked lasers,” Phys. D: Nonlinear Phenom. 219, 13–21 (2006).
[Crossref]

Nozaki, K.

N. Bekki and K. Nozaki, “Formations of spatial patterns and holes in the generalized ginzburg-landau equation,” Phys. Lett. A 110, 133–135 (1985).
[Crossref]

K. Nozaki and N. Bekki, “Exact solutions of the generalized ginzburg-landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[Crossref]

O’Shaughnessy, B.

Petermann, M.

Pfeiffer, T.

Plaza, F.

P. Coullet, T. Frisch, and F. Plaza, “Sources and sinks of wave patterns,” Phys. D: Nonlinear Phenom. 62, 75–79 (1993).
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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 fibre laser,” Nat. Photonics 7, 783 (2013).
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Politi, A.

G. Giacomelli and A. Politi, “Relationship between delayed and spatially extended dynamical systems,” Phys. Rev. Lett. 76, 2686 (1996).
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Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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Reid, W. H.

P. G. Drazin and W. H. Reid, “Hydrodynamic Stability” (University of Chicago, 1981).

Reynolds, O.

O. Reynolds, “XXIX. An experimental investigation of the circumstances which determine whether the motion of water shall he direct or sinuous, and of the law of resistance in parallel channels,” Philos. Transactions Royal Soc. Lond. 174, 935–982 (1884).

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

Sattmann, H.

Schmitt, J.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709 (2007).
[Crossref]

Schmitt, J. M.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

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 fibre laser,” Nat. Photonics 7, 783 (2013).
[Crossref]

Slepneva, S.

T. P. Butler, S. Slepneva, P. M. McNamara, K. Neuhaus, D. Goulding, M. Leahy, and G. Huyet, “Real-time experimental measurement of swept source vcsel properties relevant to oct imaging,” IEEE Photonics J. 9, 1–10 (2017).
[Crossref]

T. Butler, S. Slepneva, B. O’Shaughnessy, B. Kelleher, D. Goulding, S. P. Hegarty, H.-C. Lyu, K. Karnowski, M. Wojtkowski, and G. Huyet, “Single shot, time-resolved measurement of the coherence properties of oct swept source lasers,” Opt. Lett. 40, 2277–2280 (2015).
[Crossref] [PubMed]

S. Slepneva, B. Kelleher, B. O’Shaughnessy, S. Hegarty, A. G. Vladimirov, and G. Huyet, “Dynamics of Fourier domain mode-locked lasers,” Opt. Express 21, 19240–19251 (2013).
[Crossref] [PubMed]

D. Goulding, T. Butler, B. Kelleher, S. Slepneva, S. P. Hegarty, and G. Huyet, “Visualisation of the intensity and phase dynamics of semiconductor lasers via electric field reconstructions,” in Nonlinear Dynamics: Materials, Theory and Experiments (Springer, 2016), pp. 3–29.

Smirnov, S. 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 fibre laser,” Nat. Photonics 7, 783 (2013).
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Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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Sugavanam, S.

S. Sugavanam, N. Tarasov, S. Wabnitz, and D. V. Churkin, “Ginzburg–landau turbulence in quasi-cw raman fiber lasers,” Laser Photonics Rev. 9, L35–L39 (2015).
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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 fibre laser,” Nat. Photonics 7, 783 (2013).
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Swanson, E. A.

S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, “Optical coherence tomography using a frequency-tunable optical source,” Opt. Lett. 22, 340–342 (1997).
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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Tang, X. S.

Tarasov, N.

S. Sugavanam, N. Tarasov, S. Wabnitz, and D. V. Churkin, “Ginzburg–landau turbulence in quasi-cw raman fiber lasers,” Laser Photonics Rev. 9, L35–L39 (2015).
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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 fibre laser,” Nat. Photonics 7, 783 (2013).
[Crossref]

Todor, S.

C. M. Eigenwillig, W. Wieser, S. Todor, B. R. Biedermann, T. Klein, C. Jirauschek, and R. Huber, “Picosecond pulses from wavelength-swept continuous-wave fourier domain mode-locked lasers,” Nat. Comm. 4, 1848 (2013).
[Crossref]

Trepanier, F.

Turitsyn, S. K.

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 fibre laser,” Nat. Photonics 7, 783 (2013).
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Turitsyna, E. 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 fibre laser,” Nat. Photonics 7, 783 (2013).
[Crossref]

van Saarloos, W.

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex ginzburg-landau equations,” Phys. D: Nonlinear Phenom. 56, 303–367 (1992).
[Crossref]

Vela-Martín, A.

J. I. Cardesa, A. Vela-Martín, and J. Jiménez, “The turbulent cascade in five dimensions,” Science 357, 782–784 (2017).
[Crossref] [PubMed]

Vladimirov, A. G.

Wabnitz, S.

L. Gao, T. Zhu, S. Wabnitz, Y. Li, X. S. Tang, and Y. L. Cao, “Optical puff mediated laminar-turbulent polarization transition,” Opt. Express 26, 6103–6113 (2018).
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S. Sugavanam, N. Tarasov, S. Wabnitz, and D. V. Churkin, “Ginzburg–landau turbulence in quasi-cw raman fiber lasers,” Laser Photonics Rev. 9, L35–L39 (2015).
[Crossref]

Wieser, W.

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A time-encoded technique for fibre-based hyperspectral broadband stimulated raman microscopy,” Nat. Comm. 6, 6784 (2015).
[Crossref] [PubMed]

C. M. Eigenwillig, W. Wieser, S. Todor, B. R. Biedermann, T. Klein, C. Jirauschek, and R. Huber, “Picosecond pulses from wavelength-swept continuous-wave fourier domain mode-locked lasers,” Nat. Comm. 4, 1848 (2013).
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D. C. Adler, W. Wieser, F. Trepanier, J. M. Schmitt, and R. A. Huber, “Extended coherence length fourier domain mode locked lasers at 1310 nm,” Opt. Express 19, 20930–20939 (2011).
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Wojtkowski, M.

Zabihian, B.

Zhu, T.

Biomed. Opt. Express (3)

Dokl. Akad. Nauk SSSR (1)

L. D. Landau, “On the problem of turbulence,” Dokl. Akad. Nauk SSSR 44, 339–349 (1944).

IEEE Photonics J. (1)

T. P. Butler, S. Slepneva, P. M. McNamara, K. Neuhaus, D. Goulding, M. Leahy, and G. Huyet, “Real-time experimental measurement of swept source vcsel properties relevant to oct imaging,” IEEE Photonics J. 9, 1–10 (2017).
[Crossref]

J. Fluid Mech. (1)

P. Huerre and P. A. Monkewitz, “Absolute and convective instabilities in free shear layers,” J. Fluid Mech. 159, 151–168 (1985).
[Crossref]

J. Phys. Soc. Jpn. (1)

K. Nozaki and N. Bekki, “Exact solutions of the generalized ginzburg-landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
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J. Stat. Phys. (1)

R. J. Deissler, “Noise-sustained structure, intermittency, and the ginzburg-landau equation,” J. Stat. Phys. 40, 371–395 (1985).
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Laser Photonics Rev. (1)

S. Sugavanam, N. Tarasov, S. Wabnitz, and D. V. Churkin, “Ginzburg–landau turbulence in quasi-cw raman fiber lasers,” Laser Photonics Rev. 9, L35–L39 (2015).
[Crossref]

Nat. Comm. (2)

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A time-encoded technique for fibre-based hyperspectral broadband stimulated raman microscopy,” Nat. Comm. 6, 6784 (2015).
[Crossref] [PubMed]

C. M. Eigenwillig, W. Wieser, S. Todor, B. R. Biedermann, T. Klein, C. Jirauschek, and R. Huber, “Picosecond pulses from wavelength-swept continuous-wave fourier domain mode-locked lasers,” Nat. Comm. 4, 1848 (2013).
[Crossref]

Nat. Photonics (2)

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709 (2007).
[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 fibre laser,” Nat. Photonics 7, 783 (2013).
[Crossref]

Nature (1)

D. Castelvecchi, “Mysteries of turbulence unravelled,” Nature 548, 382–383 (2017).
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Opt. Commun. (1)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Philos. Transactions Royal Soc. Lond. (1)

O. Reynolds, “XXIX. An experimental investigation of the circumstances which determine whether the motion of water shall he direct or sinuous, and of the law of resistance in parallel channels,” Philos. Transactions Royal Soc. Lond. 174, 935–982 (1884).

Phys. D: Nonlinear Phenom. (3)

P. Coullet, T. Frisch, and F. Plaza, “Sources and sinks of wave patterns,” Phys. D: Nonlinear Phenom. 62, 75–79 (1993).
[Crossref]

M. Kolokolnikov, Th. Nizette, T. Erneux, N. Joly, and S. Bielawski, “The Q-switching instability in passively mode-locked lasers,” Phys. D: Nonlinear Phenom. 219, 13–21 (2006).
[Crossref]

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex ginzburg-landau equations,” Phys. D: Nonlinear Phenom. 56, 303–367 (1992).
[Crossref]

Phys. Lett. A (1)

N. Bekki and K. Nozaki, “Formations of spatial patterns and holes in the generalized ginzburg-landau equation,” Phys. Lett. A 110, 133–135 (1985).
[Crossref]

Phys. Rev. Lett. (3)

G. Giacomelli and A. Politi, “Relationship between delayed and spatially extended dynamical systems,” Phys. Rev. Lett. 76, 2686 (1996).
[Crossref] [PubMed]

A. Couairon and J. M. Chomaz, “Global instability in fully nonlinear systems,” Phys. Rev. Lett. 77, 4015 (1996).
[Crossref] [PubMed]

J. Burguete, H. Chaté, F. Daviaud, and N. Mukolobwiez, “Bekki-Nozaki amplitude holes in hydrothermal nonlinear waves,” Phys. Rev. Lett. 82, 3252–3255 (1999).
[Crossref]

Rev. Mod. Phys. (2)

I. S. Aranson and L. Kramer, “The world of the complex ginzburg-landau equation,” Rev. Mod. Phys. 74, 99–161 (2002).
[Crossref]

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[Crossref]

Science (3)

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

J. I. Cardesa, A. Vela-Martín, and J. Jiménez, “The turbulent cascade in five dimensions,” Science 357, 782–784 (2017).
[Crossref] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, and C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Other (4)

G. P. Agrawal, “Nonlinear fiber optics,” in Nonlinear Science at the Dawn of the 21st Century (Springer, 2000), pp. 195–211.
[Crossref]

P. G. Drazin and W. H. Reid, “Hydrodynamic Stability” (University of Chicago, 1981).

L. D. Landau and E. M. Lifdhitz, Fluid mechanics (Butterworth-Heinemann, 1987).

D. Goulding, T. Butler, B. Kelleher, S. Slepneva, S. P. Hegarty, and G. Huyet, “Visualisation of the intensity and phase dynamics of semiconductor lasers via electric field reconstructions,” in Nonlinear Dynamics: Materials, Theory and Experiments (Springer, 2016), pp. 3–29.

Supplementary Material (4)

NameDescription
» Visualization 1       Visualisation 1 SM1 visualises the experimental evolution of the laser intensity in the absolute instability regime, corresponding to a small value of detuning and demonstrates a sharp transition from laminar to turbulent regimes.
» Visualization 2       Visualisation 2 SM2 visualises the theoreticall evolution of the laser intensity in the absolute instability regime, corresponding to a small value of detuning and demonstrates a sharp transition from laminar to turbulent regimes.
» Visualization 3       Visualisation 3 SM3 visualises the experimental evolu-tion of the laser intensity in the convective instability regime, corresponding to a high value of detuning. The movie shows the transition between the cw and turbulent regimes and the appeara
» Visualization 4       Visualisation 4 SM4 visualises the theoretical evolu-tion of the laser intensity in the convective instability regime, corresponding to a high value of detuning. The movie shows the transition between the cw and turbulent regimes and the appearan

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Figures (6)

Fig. 1
Fig. 1 Subcritical and supercritical bifurcations in a wavelength swept laser. (a) Experimental set-up of a long cavity wavelength swept laser incorporating a semiconductor optical amplifier (SOA) and a fast Fabry-Pérot tunable filter (FFP-TF) (see the detailed description of the laser components in [27]). (b) The filter wavelength variation in time (top). Experimental (middle) and theoretical (bottom) observations of subcritical and supercritical modulational instabilities. A sequence of subcritical bifurcations occurs as the filter transmission wavelength increases while a single supercritical bifurcation leads to the appearance of a turbulent state as the the filter transmission wavelength decreases. The experimental time trace was recorded for the laser with a 20m cavity length and 1Hz filter modulation frequency. The theoretical time trace was obtained by direct numerical simulation of Eqs. (1) and (2) with a cavity round trip time of 14ns.
Fig. 2
Fig. 2 Plot of filter sweep profile, Δ(t), used for simulations. For each sweep period n the model equations are numerically integrated in the green shaded region using the delayed term from the blue shaded region. The solution in the region where Δ(t) = 0 is calculated analytically and the dashed region where the filter returns to zero is ignored.
Fig. 3
Fig. 3 Evolution of the laser intensity near the transition between the laminar and turbulent regimes for subsequent filter periods (n). Experimental (a) and theoretical (b) 2D diagrams of the laser intensity evolution for 250 subsequent roundtrips in the absolute instability regime corresponding to δ c 1 < δ < δ c 2. Experimental (c) and theoretical (d) 2D diagrams of the laser intensity evolution for the subsequent 500 roundtrips in the convective instability regime corresponding to δ c 2 < δ < δ c 3. The triangular features in (c) and (d) represent the emergence of localised structures from the laminar regime. These structures drift toward the turbulent region with a speed v 1 / p u p and the size of the turbulent spot grows at a rate r 1 / p l o w 1 / p u p, where pup and p l o w are the slopes of the upper and lower boundaries between the laminar state and the localised structures. The colorbars represent the normalised laser intensity. The parameters for simulations are given in Table 1.
Fig. 4
Fig. 4 The detailed analysis of the laser intensity evolution near the subcritical bifurcation point. The laser intensity was recorded at t = n T F, where n is the number of the filter cycles of modulation. Experimental (a) and theoretical (b) observations of the laser intensity demonstrating a sharp transition from laminar to turbulent regimes at a small value of detuning δ c 1 < δ < δ c 2. This corresponds to the absolute instability regime. The animations of the corresponding regime are shown in Visualization 1 (experiment) and Visualization 2 (simulations). Experimental (c) and theoretical (d) observations of creation and drift of turbulent fronts in the convective regime ( δ c 2 < δ < δ c 3). The nucleation of a hole structure that appears suddenly in the cavity is shown in (c) at round trip 220. The animations of the corresponding regime are shown in Visualization 3 (experiment) and Visualization 4 (simulations). The parameters for simulations are given in Table 1.
Fig. 5
Fig. 5 Experimentally observed (a) and, theoretically modelled (b), temporal evolution of the laser intensity (top) and phase (bottom) during the creation of a localised structure. Note that the laser intensity vanishes while the phase exhibits a π-jump as observed with Nozaki-Bekki holes.
Fig. 6
Fig. 6 Creation of three turbulent puffs. Experimentally measured temporal evolution of the laser intensity (a), and real part of the laser field (b) for 250 subsequent filter periods showing the emerging of three Nozaki-Bekki holes that initiate three turbulent regions within the laminar regime.

Tables (1)

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Table 1 Parameter values for simulations

Equations (2)

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A ˙ ( t ) + ( Γ i Δ ( t ) ) A ( t ) = Γ κ e ( 1 i α ) G ( t T ) / 2 A ( t T ) ,
G ˙ ( t ) = γ [ g 0 G ( t ) ( e G ( t ) 1 ) | A ( t ) | 2 ] ,