Abstract

We investigate multi-wave mixing associated with the strongly pump depleted regime of induced modulation instability (MI) in optical fibers. For a complete transfer of pump power into the sideband modes, we theoretically and experimentally demonstrate that it is necessary to use a much lower seeding modulation frequency than the peak MI gain value. Our experiment shows that, at such optimal modulation frequency, a record 95 % of the output pump power is frequency converted into the comb of sidebands, in good quantitative agreement with analytical predictions based on the simplest exact breather solution of the nonlinear Schrodinger ¨ equation.

© 2015 Optical Society of America

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Experimental study of the reversible behavior of modulational instability in optical fibers

Gaetan Van Simaeys, Philippe Emplit, and Marc Haelterman
J. Opt. Soc. Am. B 19(3) 477-486 (2002)

Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev
Opt. Express 17(24) 21497-21508 (2009)

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  1. A. Vedenov and L. I. Rudakov, “Interaction of waves in continuous media,” Sov. Phys. Dokl. 9, 1073 (1965).
  2. V. I. Bespalov and V. I. Talanov, “Filamentary structures of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).
  3. T. B. Benjamin and J. E. Feir, “The disintegration of wavetrains on deep water. Part 1: Theory,” J. Fluid Mech. 27, 417–430 (1967).
    [Crossref]
  4. H. C. Yuen and W. E. Ferguson, “Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation,” Phys. Fluids 21, 1275 (1978).
    [Crossref]
  5. N. N. Akhmediev, V.M. Eleonoskii, and N.E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Sov. Phys. JETP 62, 894 (1985).
  6. N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 69, 1089–1093 (1987).
    [Crossref]
  7. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulation instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
    [Crossref] [PubMed]
  8. V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: The beginning,” Physica D 238, 540–548 (2009).
    [Crossref]
  9. G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
    [Crossref] [PubMed]
  10. G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental study of the reversible behavior of modulational instability in optical fibers,” J. Opt. Soc. Am. B. 3, 477–486 (2002).
    [Crossref]
  11. J. Beeckman, X. Hutsebaut, M. Haelterman, and K. Neyts, “Induced modulation instability and recurrence in nematic liquid crystals,” Opt. Express 18, 11185 (2007).
    [Crossref]
  12. J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation,” Opt. Exp. 17, 21497–21508 (2009).
    [Crossref]
  13. K. Hammani, B. Kibler, C. Finot, P. Morin, J. Fatome, J. M. Dudley, and G. Millot, “Peregrine soliton generation and breakup in standard telecommunications fiber,” Opt. Lett. 36, 112–115 (2011).
    [Crossref] [PubMed]
  14. K. Hammani, B. Wetzel, B. Kibler, J. Fatome, C. Finot, G. Millot, N. Akhmediev, and J. M. Dudley, “Spectral dynamics of modulation instability described using Akhmediev breather theory,” Opt. Lett. 36, 2140–2143 (2011).
    [Crossref] [PubMed]
  15. A. Bendahmane, A. Mussot, P. Szriftgiser, O. Zerkak, G. Genty, J. M. Dudley, and A. Kudlinski, “Experimental dynamics of Akhmediev breathers in a dispersion varying optical fiber,” Opt. Lett. 39, 4490–4493 (2011).
    [Crossref]
  16. M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-Order Modulation Instability in Nonlinear Fiber Optics,” Phys. Rev. Lett. 107, 253901 (2011).
    [Crossref]
  17. A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser, and N. Akhmediev, “Fermi-Pasta-Ulam recurrence in nonlinear fiber optics: the role of reversible and irreversible losses,” Phys. Rev. X 4, 011054 (2014).
  18. V. E. Zakharov and A. A. Gelash, “Nonlinear stage of modulation instability,” Phys. Rev. Lett. 111, 054101 (2013).
    [Crossref] [PubMed]
  19. V. E. Zakharov and A. A. Gelash, “Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability,” Nonlinearity 27, R1–R39 (2014).
    [Crossref]
  20. G. Biondini and E. Fagerstrom, “The integrable nature of modulational instability,” SIAM J. Appl. Math. 75, 136–163 (2015).
    [Crossref]
  21. A. Calini and C. M. Schober, “Homoclinic chaos increases the likelihood of rogue wave formation,” Phys. Lett. A. 298, 335–349 (2000).
    [Crossref]
  22. S. Wabnitz and N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
    [Crossref]
  23. M. J. Ablowitz and B. M. Herbst, “On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation,” SIAM J. Appl. Math. 50, 339–351 (1990).
    [Crossref]
  24. S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulational instability in optical fibers,” Opt. Lett. 16, 986–988 (1991).
    [Crossref] [PubMed]
  25. A. Osborne, “The random and deterministic dynamics of rogue waves in unidirectional, deep-water wave trains,” Marine structures 14, 275–293 (2001).
    [Crossref]
  26. 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, 47–89 (2013).
    [Crossref]
  27. J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nature Photonics 8, 755–764 (2014).
    [Crossref]
  28. M. J. Ablowitz, J. Hammack, D. Henderson, and C.M. Schober, “Modulated periodic Stokes waves in deep water,” Phys. Rev. Lett. 84, 887–890 (2000).
    [Crossref] [PubMed]
  29. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
    [Crossref]
  30. S. Trillo and S. Wabnitz, “Self-injected spatial mode locking and coherent all-optical FM/AM switching based on modulational instability,” Opt. Lett. 16, 1566–1568 (1991)
    [Crossref] [PubMed]
  31. M. E. Marhic, K. K. Y. Wong, M. C. Ho, and L. G. Kazovsky, “92% Pump depletion in a continuous-wave one-pump fiber optical parametric amplifier,” Opt. Lett. 26, 620–622 (2001).
    [Crossref]
  32. P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical parametric amplifiers,” J. Lightwave Technol. 24, 3471–3479 (2006).
    [Crossref]
  33. G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B. 8, 824–840 (1991).
    [Crossref]
  34. M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029 (2011).
    [Crossref]
  35. W. Chen, Z. Song, and Z. Meng, “Periodical spectral holes along fiber dispersion at the second-order modulation instability sideband,” Opt. Quantum Electron. 47, 3427–3434 (2015).
    [Crossref]
  36. G. Cappellini and S. Trillo, “Energy conversion in degenerate four-photon mixing in birefringent fibers,” Opt. Lett. 16, 895–897 (1991).
    [Crossref] [PubMed]
  37. S. Trillo, G. Millot, E. Seve, and S. Wabnitz, “Failure of phase matching concept in large-signal parametric frequency conversion,” Appl. Phys. Lett. 72, 150–152 (1997).
    [Crossref]
  38. G. Millot, E. Seve, S. Wabnitz, and S. Trillo, “Observation of a novel large-signal four-photon instability in optical wave mixing,” Phys. Rev. Lett. 80, 504–507 (1998).
    [Crossref]
  39. E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E. 61, 3139–3150 (2000).
    [Crossref]

2015 (2)

G. Biondini and E. Fagerstrom, “The integrable nature of modulational instability,” SIAM J. Appl. Math. 75, 136–163 (2015).
[Crossref]

W. Chen, Z. Song, and Z. Meng, “Periodical spectral holes along fiber dispersion at the second-order modulation instability sideband,” Opt. Quantum Electron. 47, 3427–3434 (2015).
[Crossref]

2014 (3)

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

A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser, and N. Akhmediev, “Fermi-Pasta-Ulam recurrence in nonlinear fiber optics: the role of reversible and irreversible losses,” Phys. Rev. X 4, 011054 (2014).

V. E. Zakharov and A. A. Gelash, “Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability,” Nonlinearity 27, R1–R39 (2014).
[Crossref]

2013 (2)

V. E. Zakharov and A. A. Gelash, “Nonlinear stage of modulation instability,” Phys. Rev. Lett. 111, 054101 (2013).
[Crossref] [PubMed]

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, 47–89 (2013).
[Crossref]

2011 (5)

2010 (1)

S. Wabnitz and N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
[Crossref]

2009 (2)

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

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: The beginning,” Physica D 238, 540–548 (2009).
[Crossref]

2007 (1)

J. Beeckman, X. Hutsebaut, M. Haelterman, and K. Neyts, “Induced modulation instability and recurrence in nematic liquid crystals,” Opt. Express 18, 11185 (2007).
[Crossref]

2006 (2)

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

P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical parametric amplifiers,” J. Lightwave Technol. 24, 3471–3479 (2006).
[Crossref]

2002 (1)

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental study of the reversible behavior of modulational instability in optical fibers,” J. Opt. Soc. Am. B. 3, 477–486 (2002).
[Crossref]

2001 (3)

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[Crossref] [PubMed]

A. Osborne, “The random and deterministic dynamics of rogue waves in unidirectional, deep-water wave trains,” Marine structures 14, 275–293 (2001).
[Crossref]

M. E. Marhic, K. K. Y. Wong, M. C. Ho, and L. G. Kazovsky, “92% Pump depletion in a continuous-wave one-pump fiber optical parametric amplifier,” Opt. Lett. 26, 620–622 (2001).
[Crossref]

2000 (3)

M. J. Ablowitz, J. Hammack, D. Henderson, and C.M. Schober, “Modulated periodic Stokes waves in deep water,” Phys. Rev. Lett. 84, 887–890 (2000).
[Crossref] [PubMed]

A. Calini and C. M. Schober, “Homoclinic chaos increases the likelihood of rogue wave formation,” Phys. Lett. A. 298, 335–349 (2000).
[Crossref]

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E. 61, 3139–3150 (2000).
[Crossref]

1998 (1)

G. Millot, E. Seve, S. Wabnitz, and S. Trillo, “Observation of a novel large-signal four-photon instability in optical wave mixing,” Phys. Rev. Lett. 80, 504–507 (1998).
[Crossref]

1997 (1)

S. Trillo, G. Millot, E. Seve, and S. Wabnitz, “Failure of phase matching concept in large-signal parametric frequency conversion,” Appl. Phys. Lett. 72, 150–152 (1997).
[Crossref]

1991 (4)

1990 (1)

M. J. Ablowitz and B. M. Herbst, “On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation,” SIAM J. Appl. Math. 50, 339–351 (1990).
[Crossref]

1987 (1)

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 69, 1089–1093 (1987).
[Crossref]

1986 (1)

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

1985 (1)

N. N. Akhmediev, V.M. Eleonoskii, and N.E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Sov. Phys. JETP 62, 894 (1985).

1978 (1)

H. C. Yuen and W. E. Ferguson, “Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation,” Phys. Fluids 21, 1275 (1978).
[Crossref]

1967 (1)

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

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structures of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

1965 (1)

A. Vedenov and L. I. Rudakov, “Interaction of waves in continuous media,” Sov. Phys. Dokl. 9, 1073 (1965).

Ablowitz, M. J.

M. J. Ablowitz, J. Hammack, D. Henderson, and C.M. Schober, “Modulated periodic Stokes waves in deep water,” Phys. Rev. Lett. 84, 887–890 (2000).
[Crossref] [PubMed]

M. J. Ablowitz and B. M. Herbst, “On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation,” SIAM J. Appl. Math. 50, 339–351 (1990).
[Crossref]

Akhmediev, N.

A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser, and N. Akhmediev, “Fermi-Pasta-Ulam recurrence in nonlinear fiber optics: the role of reversible and irreversible losses,” Phys. Rev. X 4, 011054 (2014).

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-Order Modulation Instability in Nonlinear Fiber Optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

K. Hammani, B. Wetzel, B. Kibler, J. Fatome, C. Finot, G. Millot, N. Akhmediev, and J. M. Dudley, “Spectral dynamics of modulation instability described using Akhmediev breather theory,” Opt. Lett. 36, 2140–2143 (2011).
[Crossref] [PubMed]

S. Wabnitz and N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
[Crossref]

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

Akhmediev, N. N.

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 69, 1089–1093 (1987).
[Crossref]

N. N. Akhmediev, V.M. Eleonoskii, and N.E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Sov. Phys. JETP 62, 894 (1985).

Andrekson, P. A.

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, 47–89 (2013).
[Crossref]

Beeckman, J.

J. Beeckman, X. Hutsebaut, M. Haelterman, and K. Neyts, “Induced modulation instability and recurrence in nematic liquid crystals,” Opt. Express 18, 11185 (2007).
[Crossref]

Bendahmane, A.

Benjamin, T. B.

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

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structures of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Biondini, G.

G. Biondini and E. Fagerstrom, “The integrable nature of modulational instability,” SIAM J. Appl. Math. 75, 136–163 (2015).
[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, 47–89 (2013).
[Crossref]

Calini, A.

A. Calini and C. M. Schober, “Homoclinic chaos increases the likelihood of rogue wave formation,” Phys. Lett. A. 298, 335–349 (2000).
[Crossref]

Cappellini, G.

G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B. 8, 824–840 (1991).
[Crossref]

G. Cappellini and S. Trillo, “Energy conversion in degenerate four-photon mixing in birefringent fibers,” Opt. Lett. 16, 895–897 (1991).
[Crossref] [PubMed]

Chen, W.

W. Chen, Z. Song, and Z. Meng, “Periodical spectral holes along fiber dispersion at the second-order modulation instability sideband,” Opt. Quantum Electron. 47, 3427–3434 (2015).
[Crossref]

Coen, S.

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

Dias, F.

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

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

Droques, M.

A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser, and N. Akhmediev, “Fermi-Pasta-Ulam recurrence in nonlinear fiber optics: the role of reversible and irreversible losses,” Phys. Rev. X 4, 011054 (2014).

Dudley, J. M.

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

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-Order Modulation Instability in Nonlinear Fiber Optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029 (2011).
[Crossref]

K. Hammani, B. Kibler, C. Finot, P. Morin, J. Fatome, J. M. Dudley, and G. Millot, “Peregrine soliton generation and breakup in standard telecommunications fiber,” Opt. Lett. 36, 112–115 (2011).
[Crossref] [PubMed]

K. Hammani, B. Wetzel, B. Kibler, J. Fatome, C. Finot, G. Millot, N. Akhmediev, and J. M. Dudley, “Spectral dynamics of modulation instability described using Akhmediev breather theory,” Opt. Lett. 36, 2140–2143 (2011).
[Crossref] [PubMed]

A. Bendahmane, A. Mussot, P. Szriftgiser, O. Zerkak, G. Genty, J. M. Dudley, and A. Kudlinski, “Experimental dynamics of Akhmediev breathers in a dispersion varying optical fiber,” Opt. Lett. 39, 4490–4493 (2011).
[Crossref]

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

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

Eleonoskii, V.M.

N. N. Akhmediev, V.M. Eleonoskii, and N.E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Sov. Phys. JETP 62, 894 (1985).

Emplit, Ph.

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental study of the reversible behavior of modulational instability in optical fibers,” J. Opt. Soc. Am. B. 3, 477–486 (2002).
[Crossref]

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[Crossref] [PubMed]

Erkintalo, M.

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

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-Order Modulation Instability in Nonlinear Fiber Optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029 (2011).
[Crossref]

Fagerstrom, E.

G. Biondini and E. Fagerstrom, “The integrable nature of modulational instability,” SIAM J. Appl. Math. 75, 136–163 (2015).
[Crossref]

Fatome, J.

Feir, J. E.

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

Ferguson, W. E.

H. C. Yuen and W. E. Ferguson, “Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation,” Phys. Fluids 21, 1275 (1978).
[Crossref]

Finot, C.

Gelash, A. A.

V. E. Zakharov and A. A. Gelash, “Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability,” Nonlinearity 27, R1–R39 (2014).
[Crossref]

V. E. Zakharov and A. A. Gelash, “Nonlinear stage of modulation instability,” Phys. Rev. Lett. 111, 054101 (2013).
[Crossref] [PubMed]

Genty, G.

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

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-Order Modulation Instability in Nonlinear Fiber Optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

A. Bendahmane, A. Mussot, P. Szriftgiser, O. Zerkak, G. Genty, J. M. Dudley, and A. Kudlinski, “Experimental dynamics of Akhmediev breathers in a dispersion varying optical fiber,” Opt. Lett. 39, 4490–4493 (2011).
[Crossref]

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029 (2011).
[Crossref]

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

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

Haelterman, M.

J. Beeckman, X. Hutsebaut, M. Haelterman, and K. Neyts, “Induced modulation instability and recurrence in nematic liquid crystals,” Opt. Express 18, 11185 (2007).
[Crossref]

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental study of the reversible behavior of modulational instability in optical fibers,” J. Opt. Soc. Am. B. 3, 477–486 (2002).
[Crossref]

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[Crossref] [PubMed]

Hammack, J.

M. J. Ablowitz, J. Hammack, D. Henderson, and C.M. Schober, “Modulated periodic Stokes waves in deep water,” Phys. Rev. Lett. 84, 887–890 (2000).
[Crossref] [PubMed]

Hammani, K.

Hasegawa, A.

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

Henderson, D.

M. J. Ablowitz, J. Hammack, D. Henderson, and C.M. Schober, “Modulated periodic Stokes waves in deep water,” Phys. Rev. Lett. 84, 887–890 (2000).
[Crossref] [PubMed]

Herbst, B. M.

M. J. Ablowitz and B. M. Herbst, “On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation,” SIAM J. Appl. Math. 50, 339–351 (1990).
[Crossref]

Ho, M. C.

Hutsebaut, X.

J. Beeckman, X. Hutsebaut, M. Haelterman, and K. Neyts, “Induced modulation instability and recurrence in nematic liquid crystals,” Opt. Express 18, 11185 (2007).
[Crossref]

Karlsson, M.

Kazovsky, L. G.

Kibler, B.

K. Hammani, B. Kibler, C. Finot, P. Morin, J. Fatome, J. M. Dudley, and G. Millot, “Peregrine soliton generation and breakup in standard telecommunications fiber,” Opt. Lett. 36, 112–115 (2011).
[Crossref] [PubMed]

K. Hammani, B. Wetzel, B. Kibler, J. Fatome, C. Finot, G. Millot, N. Akhmediev, and J. M. Dudley, “Spectral dynamics of modulation instability described using Akhmediev breather theory,” Opt. Lett. 36, 2140–2143 (2011).
[Crossref] [PubMed]

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-Order Modulation Instability in Nonlinear Fiber Optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

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

Korneev, V. I.

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 69, 1089–1093 (1987).
[Crossref]

Kudlinski, A.

A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser, and N. Akhmediev, “Fermi-Pasta-Ulam recurrence in nonlinear fiber optics: the role of reversible and irreversible losses,” Phys. Rev. X 4, 011054 (2014).

A. Bendahmane, A. Mussot, P. Szriftgiser, O. Zerkak, G. Genty, J. M. Dudley, and A. Kudlinski, “Experimental dynamics of Akhmediev breathers in a dispersion varying optical fiber,” Opt. Lett. 39, 4490–4493 (2011).
[Crossref]

Kulagin, N.E.

N. N. Akhmediev, V.M. Eleonoskii, and N.E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Sov. Phys. JETP 62, 894 (1985).

Kylemark, P.

Marhic, M. E.

Meng, Z.

W. Chen, Z. Song, and Z. Meng, “Periodical spectral holes along fiber dispersion at the second-order modulation instability sideband,” Opt. Quantum Electron. 47, 3427–3434 (2015).
[Crossref]

Millot, G.

K. Hammani, B. Wetzel, B. Kibler, J. Fatome, C. Finot, G. Millot, N. Akhmediev, and J. M. Dudley, “Spectral dynamics of modulation instability described using Akhmediev breather theory,” Opt. Lett. 36, 2140–2143 (2011).
[Crossref] [PubMed]

K. Hammani, B. Kibler, C. Finot, P. Morin, J. Fatome, J. M. Dudley, and G. Millot, “Peregrine soliton generation and breakup in standard telecommunications fiber,” Opt. Lett. 36, 112–115 (2011).
[Crossref] [PubMed]

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E. 61, 3139–3150 (2000).
[Crossref]

G. Millot, E. Seve, S. Wabnitz, and S. Trillo, “Observation of a novel large-signal four-photon instability in optical wave mixing,” Phys. Rev. Lett. 80, 504–507 (1998).
[Crossref]

S. Trillo, G. Millot, E. Seve, and S. Wabnitz, “Failure of phase matching concept in large-signal parametric frequency conversion,” Appl. Phys. Lett. 72, 150–152 (1997).
[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, 47–89 (2013).
[Crossref]

Morin, P.

Mussot, A.

A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser, and N. Akhmediev, “Fermi-Pasta-Ulam recurrence in nonlinear fiber optics: the role of reversible and irreversible losses,” Phys. Rev. X 4, 011054 (2014).

A. Bendahmane, A. Mussot, P. Szriftgiser, O. Zerkak, G. Genty, J. M. Dudley, and A. Kudlinski, “Experimental dynamics of Akhmediev breathers in a dispersion varying optical fiber,” Opt. Lett. 39, 4490–4493 (2011).
[Crossref]

Neyts, K.

J. Beeckman, X. Hutsebaut, M. Haelterman, and K. Neyts, “Induced modulation instability and recurrence in nematic liquid crystals,” Opt. Express 18, 11185 (2007).
[Crossref]

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, 47–89 (2013).
[Crossref]

Osborne, A.

A. Osborne, “The random and deterministic dynamics of rogue waves in unidirectional, deep-water wave trains,” Marine structures 14, 275–293 (2001).
[Crossref]

Ostrovsky, L. A.

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: The beginning,” Physica D 238, 540–548 (2009).
[Crossref]

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, 47–89 (2013).
[Crossref]

Rudakov, L. I.

A. Vedenov and L. I. Rudakov, “Interaction of waves in continuous media,” Sov. Phys. Dokl. 9, 1073 (1965).

Schober, C. M.

A. Calini and C. M. Schober, “Homoclinic chaos increases the likelihood of rogue wave formation,” Phys. Lett. A. 298, 335–349 (2000).
[Crossref]

Schober, C.M.

M. J. Ablowitz, J. Hammack, D. Henderson, and C.M. Schober, “Modulated periodic Stokes waves in deep water,” Phys. Rev. Lett. 84, 887–890 (2000).
[Crossref] [PubMed]

Seve, E.

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E. 61, 3139–3150 (2000).
[Crossref]

G. Millot, E. Seve, S. Wabnitz, and S. Trillo, “Observation of a novel large-signal four-photon instability in optical wave mixing,” Phys. Rev. Lett. 80, 504–507 (1998).
[Crossref]

S. Trillo, G. Millot, E. Seve, and S. Wabnitz, “Failure of phase matching concept in large-signal parametric frequency conversion,” Appl. Phys. Lett. 72, 150–152 (1997).
[Crossref]

Song, Z.

W. Chen, Z. Song, and Z. Meng, “Periodical spectral holes along fiber dispersion at the second-order modulation instability sideband,” Opt. Quantum Electron. 47, 3427–3434 (2015).
[Crossref]

Sunnerud, H.

Szriftgiser, P.

A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser, and N. Akhmediev, “Fermi-Pasta-Ulam recurrence in nonlinear fiber optics: the role of reversible and irreversible losses,” Phys. Rev. X 4, 011054 (2014).

A. Bendahmane, A. Mussot, P. Szriftgiser, O. Zerkak, G. Genty, J. M. Dudley, and A. Kudlinski, “Experimental dynamics of Akhmediev breathers in a dispersion varying optical fiber,” Opt. Lett. 39, 4490–4493 (2011).
[Crossref]

Tai, K.

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

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structures of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Tomita, A.

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

Trillo, S.

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E. 61, 3139–3150 (2000).
[Crossref]

G. Millot, E. Seve, S. Wabnitz, and S. Trillo, “Observation of a novel large-signal four-photon instability in optical wave mixing,” Phys. Rev. Lett. 80, 504–507 (1998).
[Crossref]

S. Trillo, G. Millot, E. Seve, and S. Wabnitz, “Failure of phase matching concept in large-signal parametric frequency conversion,” Appl. Phys. Lett. 72, 150–152 (1997).
[Crossref]

G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B. 8, 824–840 (1991).
[Crossref]

S. Trillo and S. Wabnitz, “Self-injected spatial mode locking and coherent all-optical FM/AM switching based on modulational instability,” Opt. Lett. 16, 1566–1568 (1991)
[Crossref] [PubMed]

G. Cappellini and S. Trillo, “Energy conversion in degenerate four-photon mixing in birefringent fibers,” Opt. Lett. 16, 895–897 (1991).
[Crossref] [PubMed]

S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulational instability in optical fibers,” Opt. Lett. 16, 986–988 (1991).
[Crossref] [PubMed]

Van Simaeys, G.

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental study of the reversible behavior of modulational instability in optical fibers,” J. Opt. Soc. Am. B. 3, 477–486 (2002).
[Crossref]

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[Crossref] [PubMed]

Vedenov, A.

A. Vedenov and L. I. Rudakov, “Interaction of waves in continuous media,” Sov. Phys. Dokl. 9, 1073 (1965).

Wabnitz, S.

S. Wabnitz and N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
[Crossref]

G. Millot, E. Seve, S. Wabnitz, and S. Trillo, “Observation of a novel large-signal four-photon instability in optical wave mixing,” Phys. Rev. Lett. 80, 504–507 (1998).
[Crossref]

S. Trillo, G. Millot, E. Seve, and S. Wabnitz, “Failure of phase matching concept in large-signal parametric frequency conversion,” Appl. Phys. Lett. 72, 150–152 (1997).
[Crossref]

S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulational instability in optical fibers,” Opt. Lett. 16, 986–988 (1991).
[Crossref] [PubMed]

S. Trillo and S. Wabnitz, “Self-injected spatial mode locking and coherent all-optical FM/AM switching based on modulational instability,” Opt. Lett. 16, 1566–1568 (1991)
[Crossref] [PubMed]

Wetzel, B.

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029 (2011).
[Crossref]

K. Hammani, B. Wetzel, B. Kibler, J. Fatome, C. Finot, G. Millot, N. Akhmediev, and J. M. Dudley, “Spectral dynamics of modulation instability described using Akhmediev breather theory,” Opt. Lett. 36, 2140–2143 (2011).
[Crossref] [PubMed]

Wong, K. K. Y.

Yuen, H. C.

H. C. Yuen and W. E. Ferguson, “Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation,” Phys. Fluids 21, 1275 (1978).
[Crossref]

Zakharov, V. E.

V. E. Zakharov and A. A. Gelash, “Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability,” Nonlinearity 27, R1–R39 (2014).
[Crossref]

V. E. Zakharov and A. A. Gelash, “Nonlinear stage of modulation instability,” Phys. Rev. Lett. 111, 054101 (2013).
[Crossref] [PubMed]

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: The beginning,” Physica D 238, 540–548 (2009).
[Crossref]

Zerkak, O.

Appl. Phys. Lett. (1)

S. Trillo, G. Millot, E. Seve, and S. Wabnitz, “Failure of phase matching concept in large-signal parametric frequency conversion,” Appl. Phys. Lett. 72, 150–152 (1997).
[Crossref]

J. Fluid Mech. (1)

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

J. Lightwave Technol. (1)

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

G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B. 8, 824–840 (1991).
[Crossref]

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental study of the reversible behavior of modulational instability in optical fibers,” J. Opt. Soc. Am. B. 3, 477–486 (2002).
[Crossref]

JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structures of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Marine structures (1)

A. Osborne, “The random and deterministic dynamics of rogue waves in unidirectional, deep-water wave trains,” Marine structures 14, 275–293 (2001).
[Crossref]

Nature Photonics (1)

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

Nonlinearity (1)

V. E. Zakharov and A. A. Gelash, “Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability,” Nonlinearity 27, R1–R39 (2014).
[Crossref]

Opt. Commun. (1)

S. Wabnitz and N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
[Crossref]

Opt. Exp. (1)

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

Opt. Express (1)

J. Beeckman, X. Hutsebaut, M. Haelterman, and K. Neyts, “Induced modulation instability and recurrence in nematic liquid crystals,” Opt. Express 18, 11185 (2007).
[Crossref]

Opt. Lett. (7)

Opt. Quantum Electron. (1)

W. Chen, Z. Song, and Z. Meng, “Periodical spectral holes along fiber dispersion at the second-order modulation instability sideband,” Opt. Quantum Electron. 47, 3427–3434 (2015).
[Crossref]

Phys. Fluids (1)

H. C. Yuen and W. E. Ferguson, “Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation,” Phys. Fluids 21, 1275 (1978).
[Crossref]

Phys. Lett. A (1)

M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, “Akhmediev breather evolution in optical fiber for realistic initial conditions,” Phys. Lett. A 375, 2029 (2011).
[Crossref]

Phys. Lett. A. (1)

A. Calini and C. M. Schober, “Homoclinic chaos increases the likelihood of rogue wave formation,” Phys. Lett. A. 298, 335–349 (2000).
[Crossref]

Phys. Rep. (1)

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, 47–89 (2013).
[Crossref]

Phys. Rev. E. (1)

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E. 61, 3139–3150 (2000).
[Crossref]

Phys. Rev. Lett. (6)

G. Millot, E. Seve, S. Wabnitz, and S. Trillo, “Observation of a novel large-signal four-photon instability in optical wave mixing,” Phys. Rev. Lett. 80, 504–507 (1998).
[Crossref]

M. J. Ablowitz, J. Hammack, D. Henderson, and C.M. Schober, “Modulated periodic Stokes waves in deep water,” Phys. Rev. Lett. 84, 887–890 (2000).
[Crossref] [PubMed]

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

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-Order Modulation Instability in Nonlinear Fiber Optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[Crossref] [PubMed]

V. E. Zakharov and A. A. Gelash, “Nonlinear stage of modulation instability,” Phys. Rev. Lett. 111, 054101 (2013).
[Crossref] [PubMed]

Phys. Rev. X (1)

A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser, and N. Akhmediev, “Fermi-Pasta-Ulam recurrence in nonlinear fiber optics: the role of reversible and irreversible losses,” Phys. Rev. X 4, 011054 (2014).

Physica D (1)

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: The beginning,” Physica D 238, 540–548 (2009).
[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]

SIAM J. Appl. Math. (2)

M. J. Ablowitz and B. M. Herbst, “On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation,” SIAM J. Appl. Math. 50, 339–351 (1990).
[Crossref]

G. Biondini and E. Fagerstrom, “The integrable nature of modulational instability,” SIAM J. Appl. Math. 75, 136–163 (2015).
[Crossref]

Sov. Phys. Dokl. (1)

A. Vedenov and L. I. Rudakov, “Interaction of waves in continuous media,” Sov. Phys. Dokl. 9, 1073 (1965).

Sov. Phys. JETP (1)

N. N. Akhmediev, V.M. Eleonoskii, and N.E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Sov. Phys. JETP 62, 894 (1985).

Theor. Math. Phys. (1)

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 69, 1089–1093 (1987).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) Predictions of the TWM truncated model: Distance zd and residual pump power fraction η(zd) corresponding to maximally depleted pump versus modulation frequency ω, for a fixed input signal fraction of 3 % (ηs = 0.03). The dashed vertical line stands for ω P W M o p t from Eq. (4). (b,c,d) Full NLSE computation: (b) Evolution of Fourier modal fraction of power at frequency ω = ω A B o p t 1; (c) Residual pump power fraction vs. ω: AB (Eq. (6), solid line) compared with NLSE simulations (circles and crosses); (d) Corresponding distance zd from NLSE simulations compared with approximation (dashed lines) from Eq. (8).
Fig. 2
Fig. 2 Experimental set-up. C1-PM, polarization maintaining coupler; EDFA, erbium-doped fiber amplifier; EOM, electro-optic modulator; C2, coupler; ISO, isolator; SMF, Corning 1.1 km single mode fiber; OSA, optical spectrum analyzer.
Fig. 3
Fig. 3 Symbols: experiments. MI gain (red triangles; right vertical scale), and residual pump (blue squares) and signal (magenta circles) percentages at the output versus frequency detuning Δf = Ω/2π. Dashed curves: simulations (the same colour code has been used for pump and signal).
Fig. 4
Fig. 4 Observed spectra versus wavelength and distance, as reconstructed from cut-back measurements: (a) Δf = 80.3 GHz; (b) Δf = 55.7 GHz.
Fig. 5
Fig. 5 Measured pump and signal fractions against distance obtained from cut-back measurements, for frequencies Δf = 80.3 GHz (close to peak linear gain) and Δf = 55.7 GHz (optimum frequency for depletion), respectively.

Tables (1)

Tables Icon

Table 1 Results summary

Equations (8)

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

i u z β 2 2 2 u t 2 + γ | u | 2 u = 0 ,
u ( z = 0 , t ) = P [ η 0 + η s exp ( i ω t ) ] .
H = 2 η ( 1 η ) 2 α 2 cos ϕ + ( ω 2 + 1 ) η 3 η 2 / 2.
ω T W M o p t = 1 2 3 2 η s .
u A B p e a k ( t ) = ( ω 2 / 2 1 ) + 1 ω 2 / 4 cos ( ω t ) 1 ω 2 / 4 cos ( ω t ) 1 = = n u ˜ n e i n 2 π ω t .
| u ˜ 0 p e a k | 2 = ( ω 1 ) 2 ,
| u ˜ n p e a k | 2 = ω 2 ( 2 ω 2 + ω ) 2 .
z d = 1 ω 1 ω 2 / 4 ln ( ω ( 1 ω 2 / 4 ) η s ) .

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