Abstract

We report an experimental, numerical and theoretical study of the incoherent regime of supercontinuum generation in a two zero-dispersion wavelengths fiber. By using a simple experimental setup, we show that the phenomenon of spectral broadening inherent to supercontinuum generation can be described as a thermalization process, which is characterized by an irreversible evolution of the optical field towards a thermal equilibrium state. In particular, the thermodynamic equilibrium spectrum predicted by the kinetic wave theory is characterized by a double peak structure, which has been found in quantitative agreement with the numerical simulations without adjustable parameters. We also confirm that stimulated Raman scattering leads to the generation of an incoherent structure in the normal dispersion regime which is reminiscent of a spectral incoherent soliton.

© 2009 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  25. A. Picozzi, “Spontaneous polarization induced by natural thermalization of incoherent light,” Opt. Express 16, 17171–17185 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17171.
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  31. S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415–422 (2003).
    [Crossref]
  32. B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B 81, 337–342 (2005).
    [Crossref]
  33. B. Barviau, B. Kibler, and A. Picozzi, “Influence of self-steepening and higher-order dispersion on wave thermalization” (in preparation).
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    [Crossref]
  35. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).
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  37. C. Montes, “Photon soliton and fine structure due to nonlinear Compton scattering,” Phys. Rev. A 20, 1081 (1979).
    [Crossref]
  38. C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176 (1979).
    [Crossref]
  39. S.L. Musher, A.M. Rubenchik, and V.E. Zakharov, “Weak Langmuir turbulence,” Phys. Reports 252, 177 (1995).
    [Crossref]
  40. A.V. Gorbach and D.V. Skryabin, “Spectral discrete solitons and localization in frequency space,” Opt. Lett. 31, 3309–3311 (2006).
    [Crossref] [PubMed]

2008 (10)

A. Kudlinski, G. Bouwmans, Y. Quiquempois, and A. Mussot, “Experimental demonstration of multiwatt continuous-wave supercontinuum tailoring in photonic crystal fibers,” Appl. Phys. Lett. 92, 141103 (2008).
[Crossref]

B.A. Cumberland, J.C. Travers, S.V. Popov, and J.R. Taylor, “29 W High power CW supercontinuum source,” Opt. Express 16, 5964–5972 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5954.
[Crossref]

J.C. Travers, A.B. Rulkov, B.A. Cumberland, S.V. Popov, and J.R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400 W continuous wave fiber laser,” Opt. Express 16, 14435–14447 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14435.
[Crossref] [PubMed]

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, “Towards a thermodynamic description of supercontinuum generation,” Opt. Lett. 33, 2833–2835 (2008).
[Crossref] [PubMed]

A. Picozzi and S. Rica, “Coherence absorption and condensation induced by thermalization of incoherent nonlinear fields,” Europhys. Lett. 84, 34004 (2008).
[Crossref]

A. Picozzi, “Spontaneous polarization induced by natural thermalization of incoherent light,” Opt. Express 16, 17171–17185 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17171.
[Crossref] [PubMed]

L. Levi, T. Schwartz, O. Manela, M. Segev, and H. Buljan, “Spontaneous pattern formation upon incoherent waves: From modulation-instability to steady-state,” Opt. Express 16, 7818–7831 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-7818.
[Crossref] [PubMed]

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[Crossref] [PubMed]

B. Kibler, C. Finot, G. Gadret, G. Millot, J. Wojcik, M. Szpulak, and W. Urbanczyk, “Second zero dispersion wavelength measurement through soliton self-frequency shift compensation in suspended core fibre,” Electron. Lett. 44, 1370–1371 (2008).
[Crossref]

P.M. Moselund, M.H. Frosz, C.L. Thomsen, and O. Bang, “Back-seeding of higher order gain processes in picosecond supercontinuum generation,” Opt. Express 16, 11954–11968 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-11954.
[Crossref] [PubMed]

2007 (7)

2006 (3)

J.M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006).
[Crossref] [PubMed]

A.V. Gorbach and D.V. Skryabin, “Spectral discrete solitons and localization in frequency space,” Opt. Lett. 31, 3309–3311 (2006).
[Crossref] [PubMed]

2005 (2)

B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B 81, 337–342 (2005).
[Crossref]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005).
[Crossref]

2004 (2)

2003 (3)

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415–422 (2003).
[Crossref]

D.V. Skryabin, F. Luan, J.C. Knight, and P.St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref] [PubMed]

A. Efimov, A.J. Taylor, F.G. Omenetto, J.C. Knight, W.J. Wadsworth, and P.St.J. Russell, “Phase-matched third harmonic generation in microstructured fibers,” Opt. Express 11, 2567–2576 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2567.
[Crossref] [PubMed]

2001 (1)

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
[Crossref]

2000 (1)

1995 (2)

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

S.L. Musher, A.M. Rubenchik, and V.E. Zakharov, “Weak Langmuir turbulence,” Phys. Reports 252, 177 (1995).
[Crossref]

1992 (1)

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96–160 (1992).
[Crossref]

1979 (2)

C. Montes, “Photon soliton and fine structure due to nonlinear Compton scattering,” Phys. Rev. A 20, 1081 (1979).
[Crossref]

C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176 (1979).
[Crossref]

1976 (1)

V.E. Zakharov, S.L. Musher, and A.M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80 (1976).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 4th Ed., 2006).

Akhmediev, N

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

Babin, S.A.

Bang, O.

Barviau, B.

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, “Towards a thermodynamic description of supercontinuum generation,” Opt. Lett. 33, 2833–2835 (2008).
[Crossref] [PubMed]

B. Barviau, B. Kibler, and A. Picozzi, “Influence of self-steepening and higher-order dispersion on wave thermalization” (in preparation).

Batrouni, G.

M. Le Bellac, F. Mortessagne, and G. Batrouni, Equilibrium and Nonequilibrium Statistical Thermodynamics (Cambridge Univ. Press, 2004).
[Crossref]

Beaugeois, M.

Biancalana, F.

Birks, T.A.

Biven, L.

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
[Crossref]

Bouazaoui, M.

Bouwmans, G.

A. Kudlinski, G. Bouwmans, Y. Quiquempois, and A. Mussot, “Experimental demonstration of multiwatt continuous-wave supercontinuum tailoring in photonic crystal fibers,” Appl. Phys. Lett. 92, 141103 (2008).
[Crossref]

Buljan, H.

Churkin, D.V.

Coen, S.

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, “Towards a thermodynamic description of supercontinuum generation,” Opt. Lett. 33, 2833–2835 (2008).
[Crossref] [PubMed]

G. Genty, S. Coen, and J.M. Dudley, “Fiber supercontinuum sources,” J. Opt. Soc. Am. B 24, 1771–1785 (2007).
[Crossref]

J.M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B 81, 337–342 (2005).
[Crossref]

Connaughton, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005).
[Crossref]

Courvoisier, F.

B. Kibler, P.-A. Lacourt, F. Courvoisier, and J.M. Dudley, “Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect,” Electron. Lett. 43, 967–968 (2007).
[Crossref]

Cumberland, B.A.

Dias, F.

V. Zakharov, F. Dias, and A. Pushkarev, “One dimensional wave turbulence,” Phys. Rep. 398, 1 (2004).
[Crossref]

Dudley, J. M.

B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B 81, 337–342 (2005).
[Crossref]

Dudley, J.M.

B. Kibler, P.-A. Lacourt, F. Courvoisier, and J.M. Dudley, “Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect,” Electron. Lett. 43, 967–968 (2007).
[Crossref]

G. Genty, S. Coen, and J.M. Dudley, “Fiber supercontinuum sources,” J. Opt. Soc. Am. B 24, 1771–1785 (2007).
[Crossref]

J.M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Dyachenko, S.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96–160 (1992).
[Crossref]

Efimov, A.

Falkovich, G.

V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, Berlin, 1992).

Finot, C.

B. Kibler, C. Finot, G. Gadret, G. Millot, J. Wojcik, M. Szpulak, and W. Urbanczyk, “Second zero dispersion wavelength measurement through soliton self-frequency shift compensation in suspended core fibre,” Electron. Lett. 44, 1370–1371 (2008).
[Crossref]

Frosz, M.H.

Gadret, G.

B. Kibler, C. Finot, G. Gadret, G. Millot, J. Wojcik, M. Szpulak, and W. Urbanczyk, “Second zero dispersion wavelength measurement through soliton self-frequency shift compensation in suspended core fibre,” Electron. Lett. 44, 1370–1371 (2008).
[Crossref]

Genty, G.

G. Genty, S. Coen, and J.M. Dudley, “Fiber supercontinuum sources,” J. Opt. Soc. Am. B 24, 1771–1785 (2007).
[Crossref]

J.M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Gorbach, A.V.

A.V. Gorbach and D.V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Phot. 1, 653–657 (2007).
[Crossref]

A.V. Gorbach and D.V. Skryabin, “Spectral discrete solitons and localization in frequency space,” Opt. Lett. 31, 3309–3311 (2006).
[Crossref] [PubMed]

Hénon, M.

C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176 (1979).
[Crossref]

Ismagulov, A.E.

Jauslin, H. R.

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006).
[Crossref] [PubMed]

Jauslin, H.R.

S. Lagrange, H.R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction,” Europhys. Lett. 79, 64001 (2007).
[Crossref]

Joly, N.

Josserand, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005).
[Crossref]

Kablukov, S.I.

Karlsson, M.

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

Kibler, B.

B. Kibler, C. Finot, G. Gadret, G. Millot, J. Wojcik, M. Szpulak, and W. Urbanczyk, “Second zero dispersion wavelength measurement through soliton self-frequency shift compensation in suspended core fibre,” Electron. Lett. 44, 1370–1371 (2008).
[Crossref]

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, “Towards a thermodynamic description of supercontinuum generation,” Opt. Lett. 33, 2833–2835 (2008).
[Crossref] [PubMed]

B. Kibler, P.-A. Lacourt, F. Courvoisier, and J.M. Dudley, “Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect,” Electron. Lett. 43, 967–968 (2007).
[Crossref]

B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B 81, 337–342 (2005).
[Crossref]

B. Barviau, B. Kibler, and A. Picozzi, “Influence of self-steepening and higher-order dispersion on wave thermalization” (in preparation).

Knight, J.C.

Kudlinski, A.

A. Kudlinski, G. Bouwmans, Y. Quiquempois, and A. Mussot, “Experimental demonstration of multiwatt continuous-wave supercontinuum tailoring in photonic crystal fibers,” Appl. Phys. Lett. 92, 141103 (2008).
[Crossref]

L’vov, V.

V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, Berlin, 1992).

Lacourt, P.-A.

B. Kibler, P.-A. Lacourt, F. Courvoisier, and J.M. Dudley, “Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect,” Electron. Lett. 43, 967–968 (2007).
[Crossref]

Lagrange, S.

S. Lagrange, H.R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction,” Europhys. Lett. 79, 64001 (2007).
[Crossref]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006).
[Crossref] [PubMed]

Le Bellac, M.

M. Le Bellac, F. Mortessagne, and G. Batrouni, Equilibrium and Nonequilibrium Statistical Thermodynamics (Cambridge Univ. Press, 2004).
[Crossref]

Levi, L.

Luan, F.

D.V. Skryabin, F. Luan, J.C. Knight, and P.St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).

Manela, O.

Millot, G.

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[Crossref] [PubMed]

B. Kibler, C. Finot, G. Gadret, G. Millot, J. Wojcik, M. Szpulak, and W. Urbanczyk, “Second zero dispersion wavelength measurement through soliton self-frequency shift compensation in suspended core fibre,” Electron. Lett. 44, 1370–1371 (2008).
[Crossref]

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415–422 (2003).
[Crossref]

Montes, C.

C. Montes, “Photon soliton and fine structure due to nonlinear Compton scattering,” Phys. Rev. A 20, 1081 (1979).
[Crossref]

C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176 (1979).
[Crossref]

Mortessagne, F.

M. Le Bellac, F. Mortessagne, and G. Batrouni, Equilibrium and Nonequilibrium Statistical Thermodynamics (Cambridge Univ. Press, 2004).
[Crossref]

Moselund, P.M.

Musher, S.L.

S.L. Musher, A.M. Rubenchik, and V.E. Zakharov, “Weak Langmuir turbulence,” Phys. Reports 252, 177 (1995).
[Crossref]

V.E. Zakharov, S.L. Musher, and A.M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80 (1976).

Mussot, A.

A. Kudlinski, G. Bouwmans, Y. Quiquempois, and A. Mussot, “Experimental demonstration of multiwatt continuous-wave supercontinuum tailoring in photonic crystal fibers,” Appl. Phys. Lett. 92, 141103 (2008).
[Crossref]

A. Mussot, M. Beaugeois, M. Bouazaoui, and T. Sylvestre, “Tailoring CW supercontinuum generation in microstructured fibers with two-zero dispersion wavelengths,” Opt. Express 15, 11553–11563 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-18-11553.
[Crossref] [PubMed]

Nazarenko, S.

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
[Crossref]

Newell, A. C.

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
[Crossref]

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96–160 (1992).
[Crossref]

Omenetto, F.G.

Peyraud, J.

C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176 (1979).
[Crossref]

Picozzi, A.

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, “Towards a thermodynamic description of supercontinuum generation,” Opt. Lett. 33, 2833–2835 (2008).
[Crossref] [PubMed]

A. Picozzi, “Spontaneous polarization induced by natural thermalization of incoherent light,” Opt. Express 16, 17171–17185 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17171.
[Crossref] [PubMed]

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[Crossref] [PubMed]

A. Picozzi and S. Rica, “Coherence absorption and condensation induced by thermalization of incoherent nonlinear fields,” Europhys. Lett. 84, 34004 (2008).
[Crossref]

S. Lagrange, H.R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction,” Europhys. Lett. 79, 64001 (2007).
[Crossref]

A. Picozzi, “Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics,” Opt. Express 15, 9063–9083 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-9063.
[Crossref] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006).
[Crossref] [PubMed]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005).
[Crossref]

B. Barviau, B. Kibler, and A. Picozzi, “Influence of self-steepening and higher-order dispersion on wave thermalization” (in preparation).

Pitois, S.

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[Crossref] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006).
[Crossref] [PubMed]

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415–422 (2003).
[Crossref]

Podivilov, E.V.

Pomeau, Y.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005).
[Crossref]

Popov, S.V.

Pushkarev, A.

V. Zakharov, F. Dias, and A. Pushkarev, “One dimensional wave turbulence,” Phys. Rep. 398, 1 (2004).
[Crossref]

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96–160 (1992).
[Crossref]

Quiquempois, Y.

A. Kudlinski, G. Bouwmans, Y. Quiquempois, and A. Mussot, “Experimental demonstration of multiwatt continuous-wave supercontinuum tailoring in photonic crystal fibers,” Appl. Phys. Lett. 92, 141103 (2008).
[Crossref]

Ranka, J.K.

Rica, S.

A. Picozzi and S. Rica, “Coherence absorption and condensation induced by thermalization of incoherent nonlinear fields,” Europhys. Lett. 84, 34004 (2008).
[Crossref]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005).
[Crossref]

Rubenchik, A.M.

S.L. Musher, A.M. Rubenchik, and V.E. Zakharov, “Weak Langmuir turbulence,” Phys. Reports 252, 177 (1995).
[Crossref]

V.E. Zakharov, S.L. Musher, and A.M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80 (1976).

Rulkov, A.B.

Russell, P.St.J.

Schwartz, T.

Segev, M.

Skryabin, D.V.

A.V. Gorbach and D.V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Phot. 1, 653–657 (2007).
[Crossref]

A.V. Gorbach and D.V. Skryabin, “Spectral discrete solitons and localization in frequency space,” Opt. Lett. 31, 3309–3311 (2006).
[Crossref] [PubMed]

D.V. Skryabin, F. Luan, J.C. Knight, and P.St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref] [PubMed]

Stentz, A.J.

Sylvestre, T.

Szpulak, M.

B. Kibler, C. Finot, G. Gadret, G. Millot, J. Wojcik, M. Szpulak, and W. Urbanczyk, “Second zero dispersion wavelength measurement through soliton self-frequency shift compensation in suspended core fibre,” Electron. Lett. 44, 1370–1371 (2008).
[Crossref]

Taylor, A.J.

Taylor, J.R.

Thomsen, C.L.

Travers, J.C.

Urbanczyk, W.

B. Kibler, C. Finot, G. Gadret, G. Millot, J. Wojcik, M. Szpulak, and W. Urbanczyk, “Second zero dispersion wavelength measurement through soliton self-frequency shift compensation in suspended core fibre,” Electron. Lett. 44, 1370–1371 (2008).
[Crossref]

Wadsworth, W.J.

Windeler, R.S.

Wojcik, J.

B. Kibler, C. Finot, G. Gadret, G. Millot, J. Wojcik, M. Szpulak, and W. Urbanczyk, “Second zero dispersion wavelength measurement through soliton self-frequency shift compensation in suspended core fibre,” Electron. Lett. 44, 1370–1371 (2008).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).

Zakharov, V.

V. Zakharov, F. Dias, and A. Pushkarev, “One dimensional wave turbulence,” Phys. Rep. 398, 1 (2004).
[Crossref]

V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, Berlin, 1992).

Zakharov, V. E.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96–160 (1992).
[Crossref]

Zakharov, V.E.

S.L. Musher, A.M. Rubenchik, and V.E. Zakharov, “Weak Langmuir turbulence,” Phys. Reports 252, 177 (1995).
[Crossref]

V.E. Zakharov, S.L. Musher, and A.M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80 (1976).

Appl. Phys. B (1)

B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B 81, 337–342 (2005).
[Crossref]

Appl. Phys. Lett. (1)

A. Kudlinski, G. Bouwmans, Y. Quiquempois, and A. Mussot, “Experimental demonstration of multiwatt continuous-wave supercontinuum tailoring in photonic crystal fibers,” Appl. Phys. Lett. 92, 141103 (2008).
[Crossref]

Electron. Lett. (2)

B. Kibler, P.-A. Lacourt, F. Courvoisier, and J.M. Dudley, “Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect,” Electron. Lett. 43, 967–968 (2007).
[Crossref]

B. Kibler, C. Finot, G. Gadret, G. Millot, J. Wojcik, M. Szpulak, and W. Urbanczyk, “Second zero dispersion wavelength measurement through soliton self-frequency shift compensation in suspended core fibre,” Electron. Lett. 44, 1370–1371 (2008).
[Crossref]

Europhys. Lett. (2)

A. Picozzi and S. Rica, “Coherence absorption and condensation induced by thermalization of incoherent nonlinear fields,” Europhys. Lett. 84, 34004 (2008).
[Crossref]

S. Lagrange, H.R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction,” Europhys. Lett. 79, 64001 (2007).
[Crossref]

J. Opt. Soc. Am. B (2)

Nat. Phot. (1)

A.V. Gorbach and D.V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Phot. 1, 653–657 (2007).
[Crossref]

Opt. Commun. (1)

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415–422 (2003).
[Crossref]

Opt. Express (9)

P.M. Moselund, M.H. Frosz, C.L. Thomsen, and O. Bang, “Back-seeding of higher order gain processes in picosecond supercontinuum generation,” Opt. Express 16, 11954–11968 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-11954.
[Crossref] [PubMed]

A. Picozzi, “Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics,” Opt. Express 15, 9063–9083 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-9063.
[Crossref] [PubMed]

A. Picozzi, “Spontaneous polarization induced by natural thermalization of incoherent light,” Opt. Express 16, 17171–17185 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17171.
[Crossref] [PubMed]

L. Levi, T. Schwartz, O. Manela, M. Segev, and H. Buljan, “Spontaneous pattern formation upon incoherent waves: From modulation-instability to steady-state,” Opt. Express 16, 7818–7831 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-7818.
[Crossref] [PubMed]

A. Efimov, A.J. Taylor, F.G. Omenetto, J.C. Knight, W.J. Wadsworth, and P.St.J. Russell, “Phase-matched third harmonic generation in microstructured fibers,” Opt. Express 11, 2567–2576 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2567.
[Crossref] [PubMed]

W.J. Wadsworth, N. Joly, J.C. Knight, T.A. Birks, F. Biancalana, and P.St.J. Russell, “Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibres,” Opt. Express 12, 299–309 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-2-299.
[Crossref] [PubMed]

B.A. Cumberland, J.C. Travers, S.V. Popov, and J.R. Taylor, “29 W High power CW supercontinuum source,” Opt. Express 16, 5964–5972 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5954.
[Crossref]

J.C. Travers, A.B. Rulkov, B.A. Cumberland, S.V. Popov, and J.R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400 W continuous wave fiber laser,” Opt. Express 16, 14435–14447 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14435.
[Crossref] [PubMed]

A. Mussot, M. Beaugeois, M. Bouazaoui, and T. Sylvestre, “Tailoring CW supercontinuum generation in microstructured fibers with two-zero dispersion wavelengths,” Opt. Express 15, 11553–11563 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-18-11553.
[Crossref] [PubMed]

Opt. Lett. (3)

Phys. Fluids (1)

C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176 (1979).
[Crossref]

Phys. Rep. (1)

V. Zakharov, F. Dias, and A. Pushkarev, “One dimensional wave turbulence,” Phys. Rep. 398, 1 (2004).
[Crossref]

Phys. Reports (1)

S.L. Musher, A.M. Rubenchik, and V.E. Zakharov, “Weak Langmuir turbulence,” Phys. Reports 252, 177 (1995).
[Crossref]

Phys. Rev. A (2)

C. Montes, “Photon soliton and fine structure due to nonlinear Compton scattering,” Phys. Rev. A 20, 1081 (1979).
[Crossref]

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

Phys. Rev. Lett. (3)

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[Crossref] [PubMed]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005).
[Crossref]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006).
[Crossref] [PubMed]

Physica D (2)

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
[Crossref]

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96–160 (1992).
[Crossref]

Rev. Mod. Phys. (1)

J.M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Science (1)

D.V. Skryabin, F. Luan, J.C. Knight, and P.St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

V.E. Zakharov, S.L. Musher, and A.M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80 (1976).

Other (5)

V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, Berlin, 1992).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 4th Ed., 2006).

B. Barviau, B. Kibler, and A. Picozzi, “Influence of self-steepening and higher-order dispersion on wave thermalization” (in preparation).

M. Le Bellac, F. Mortessagne, and G. Batrouni, Equilibrium and Nonequilibrium Statistical Thermodynamics (Cambridge Univ. Press, 2004).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Calculated dispersion curve with the two zero dispersion wavelengths located at 1033 and 1209 nm. The scanning electron microscope image of the fiber cross-section is shown in the inset. (b) Calculated modulational instability gain (m-1) bands as a function of pump wavelength for the dispersion curve (a) and for an input peak power of 3.5 kW.

Fig. 2.
Fig. 2.

(a) Experimental results using a logarithmic intensity scale (dB) to illustrate the spectral evolution as a function of propagation distance in our 40-m-long PCF, for an input peak power about 3.5 kW. (b) Experimental output spectra obtained after 5 and 40 m of propagation. (c) Experimental spectra recorded after 40 m propagation in the PCF, as a function of the input pulse peak power. (d) Experimental output spectra at P0 = 0.5 kW and P0 = 3.5 kW. The white dashed lines show the two zero dispersion wavelengths of the optical fiber. ‘S’ indicates the position of the spectral incoherent soliton. Note in Fig. 2(a) the saturation of the spectral broadening in the high-frequency edge of the SC spectrum. (The video bandwidth of the spectrum analyzer was set to 100 Hz.)

Fig. 3.
Fig. 3.

(a) Numerical simulations of the GNLSE Eq.(1) with the dispersion curve of Fig. 1(a) and for an input power of 3.5 kW. The simulations have been realized with an initial Gaussian pulse of 60 ps, i.e. ~10 times shorter than the experimental pulses. (b) Numerical spectrum for both propagation lengths 5 and 40 m. (c-d) Same as in (a-b), except that the initial condition refers to a continuous wave and that the numerical simulation neglects the Raman, shock and loss terms, i.e., Eq.(1) for τs = α ^ = fR = 0. The white dashed lines show both fiber ZDWs. ‘S’ indicates the position of the spectral incoherent soliton. Note the development of the double peak structure in the evolution of the spectrum, a feature which constitutes a key signature of wave thermalization (see Sec. 5).

Fig. 4.
Fig. 4.

Comparison of the theoretical, numerical and experimental spectra in logarithmic scale. (a) Plot of the equilibrium spectrum neq(ω) given in Eq.(3) without adjustable parameters. (b) Spectrum obtained by solving numerically the NLSE, i.e., equation (1) without Raman, loss and shock terms (τs = α ^ = fR = 0) [see Fig. 3(e) at z = 40 m]. (c) Spectrum obtained by solving numerically the GNLSE equation (1) [see Fig. 3(a) at z = 8 m]. (d) Spectrum recorded in our experiment [see Fig. 2(a) at z = 8 m]. ‘S’ indicates the position of the spectral incoherent soliton. Note the good agreement of the frequencies of the spectral peaks. (e) Evolution of the nonequilibrium entropy during the propagation of the optical field corresponding to the simulation of the NLSE in (b): the process of entropy production saturates once the equilibrium state is reached, as described by the H-theorem of entropy growth.

Fig. 5.
Fig. 5.

(a) Numerical simulations of the NLSE showing the evolution of the spectra of the field (in logarithmic intensity) as a function of the input peak power in our 40-m-long PCF. (b) Numerical output spectra recorded for input peak powers of 0.5 and 3.5 kW. The white dashed lines show both fiber ZDWs. (c) Comparison of the thermodynamic equilibrium spectra predicted by the kinetic theory [Eq.(3)] with the numerical spectra of the NLSE. We note an appreciable discrepancy between theory and numerics at small power, which is due to the short (effective) nonlinear propagation length. Conversely, at higher powers, 40 m of propagation becomes sufficient for the field to reach thermal equilibrium.

Fig. 6.
Fig. 6.

(a) Experimental output spectrum obtained after 17 m of propagation in our PCF for an input peak power of 3.5 kW (the video bandwidth of the spectrum analyzer was set to 1 kHz). The left inset shows in particular the output spectrum of the spectral incoherent soliton, which reveals high intensity fluctuations. (b-c): Numerical spectrogram showing the temporal distribution of the spectral power after 17 m, obtained by solving the GNLSE, for a 60 ps input pulse (b), and a CW input (c). The white dashed lines show both fiber ZDWs. ‘S’ indicates the position of the spectral incoherent soliton. The reference pulse used to compute the spectrogram is a 20-fs sech pulse.

Equations (3)

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

A z = i m 2 i m β m m ! m A t m + [ 1 + i τ s t ] ( A ( z , t ) + R ( t ) A ( z , t t ) 2 dt ) α ̂ A .
k ( ω ) = m 2 β m m ! ω m .
n eq ( ω ) = T k ( ω ) + λω μ ,

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