Abstract

We show that, by properly adjusting the relative state of polarization of the pump and of a weak modulation, with a frequency such that at least one of its even harmonics falls within the band of modulation instability, one obtains a fully modulated wave at the second or higher even harmonic of the initial modulation. An application of this principle to the generation of an 80 GHz optical pulse train with high extinction ratio from a 40 GHz weakly modulated pump is experimentally demonstrated using a nonzero dispersion-shifted fiber in the telecom C band.

© 2012 Optical Society of America

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  1. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” Pis’ma Zh. Eksp. Teor. Fiz. 3, 471–476 (1966) [JETP Lett. 3, 307–310 (1966)].
  2. T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water. Part 1: theory,” J. Fluid Mech. 27, 417–430 (1967).
    [CrossRef]
  3. L. A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Zh. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)].
  4. V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Phys. D 238, 540–549 (2009).
    [CrossRef]
  5. N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Zh. Eksp. Teor. Fiz. 89, 1542–1551 (1985) [Sov. Phys. JETP 62, 894–899 (1985)].
  6. N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Mat. Fiz. 69, 189–194 (1986) [Theor. Mat. Phys. 69, 1089 (1986)].
    [CrossRef]
  7. N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Zh. Eksp. Teor. Fiz. 94, 159–170 (1988) [Sov. Phys. JETP 67, 89–95 (1988)].
  8. 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, 824–838 (1991).
    [CrossRef]
  9. S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulational instability in optical fibers,” Opt. Lett. 16, 986–988 (1991).
    [CrossRef]
  10. 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]
  11. S. Wabnitz and N. N. Akhmediev, “Efficient modulation frequency doubling by induced modulation instability,” Opt. Commun. 283, 1152–1154 (2010).
    [CrossRef]
  12. M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
    [CrossRef]
  13. G. Millot, “Multiple four-wave mixing-induced modulational instability in highly birefringent fibers,” Opt. Lett. 26, 1391–1393 (2001).
    [CrossRef]
  14. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505–516 (1973) [Sov. Phys. JETP 38, 248–253 (1974)].
  15. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
    [CrossRef]
  16. G. J. Roske, “Some nonlinear multiphase reactions,” Stud. Appl. Math. 55, 231–238 (1976).
  17. M. G. Forest, S. P. Sheu, and O. C. Wright, “On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems,” Phys. Lett. A 266, 24–33(2000).
    [CrossRef]
  18. M. G. Forest, D. W. McLaughlin, D. J. Muraki, and O. C. Wright, “Nonfocusing instabilities in coupled, integrable nonlinear Schrödinger PDES,” J. Nonlinear Sci. 10, 291–331 (2000).
    [CrossRef]
  19. O. C. Wright and M. G. Forest, “On the Bäcklund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrödinger system,” Phys. D 141, 104–116 (2000).
  20. M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Phys. D 178, 173–189 (2003).
    [CrossRef]
  21. O. C. Wright, “The Darboux transformation of some Manakov systems,” Applied Math. Lett. 16, 647–652 (2003).
    [CrossRef]
  22. O. C. Wright, “Dressing procedure for some homoclinic connections of the Manakov system,” Applied Math. Lett. 19, 1185–1190 (2006).
    [CrossRef]
  23. B. Guo and L. Ling, “Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations,” Chin. Phys. Lett. 28, 110202 (2011).
    [CrossRef]
  24. F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
    [CrossRef]
  25. 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, 112–114 (2011).
    [CrossRef]

2012 (1)

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[CrossRef]

2011 (3)

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. 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. 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, 112–114 (2011).
[CrossRef]

B. Guo and L. Ling, “Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations,” Chin. Phys. Lett. 28, 110202 (2011).
[CrossRef]

2010 (1)

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

2009 (1)

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Phys. D 238, 540–549 (2009).
[CrossRef]

2006 (1)

O. C. Wright, “Dressing procedure for some homoclinic connections of the Manakov system,” Applied Math. Lett. 19, 1185–1190 (2006).
[CrossRef]

2003 (2)

M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Phys. D 178, 173–189 (2003).
[CrossRef]

O. C. Wright, “The Darboux transformation of some Manakov systems,” Applied Math. Lett. 16, 647–652 (2003).
[CrossRef]

2001 (2)

G. Millot, “Multiple four-wave mixing-induced modulational instability in highly birefringent fibers,” Opt. Lett. 26, 1391–1393 (2001).
[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]

2000 (3)

M. G. Forest, S. P. Sheu, and O. C. Wright, “On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems,” Phys. Lett. A 266, 24–33(2000).
[CrossRef]

M. G. Forest, D. W. McLaughlin, D. J. Muraki, and O. C. Wright, “Nonfocusing instabilities in coupled, integrable nonlinear Schrödinger PDES,” J. Nonlinear Sci. 10, 291–331 (2000).
[CrossRef]

O. C. Wright and M. G. Forest, “On the Bäcklund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrödinger system,” Phys. D 141, 104–116 (2000).

1996 (1)

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[CrossRef]

1991 (2)

1988 (1)

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Zh. Eksp. Teor. Fiz. 94, 159–170 (1988) [Sov. Phys. JETP 67, 89–95 (1988)].

1986 (1)

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Mat. Fiz. 69, 189–194 (1986) [Theor. Mat. Phys. 69, 1089 (1986)].
[CrossRef]

1985 (1)

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Zh. Eksp. Teor. Fiz. 89, 1542–1551 (1985) [Sov. Phys. JETP 62, 894–899 (1985)].

1976 (1)

G. J. Roske, “Some nonlinear multiphase reactions,” Stud. Appl. Math. 55, 231–238 (1976).

1973 (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505–516 (1973) [Sov. Phys. JETP 38, 248–253 (1974)].

1970 (1)

L. A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Zh. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)].

1967 (1)

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

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” Pis’ma Zh. Eksp. Teor. Fiz. 3, 471–476 (1966) [JETP Lett. 3, 307–310 (1966)].

Akhmediev, N. N.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[CrossRef]

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

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Zh. Eksp. Teor. Fiz. 94, 159–170 (1988) [Sov. Phys. JETP 67, 89–95 (1988)].

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Mat. Fiz. 69, 189–194 (1986) [Theor. Mat. Phys. 69, 1089 (1986)].
[CrossRef]

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Zh. Eksp. Teor. Fiz. 89, 1542–1551 (1985) [Sov. Phys. JETP 62, 894–899 (1985)].

Baronio, F.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[CrossRef]

Benjamin, T. B.

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

Berkhoer, L. A. L.

L. A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Zh. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)].

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” Pis’ma Zh. Eksp. Teor. Fiz. 3, 471–476 (1966) [JETP Lett. 3, 307–310 (1966)].

Cappellini, G.

Conforti, M.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[CrossRef]

Degasperis, A.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[CrossRef]

Dudley, J. M.

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, 112–114 (2011).
[CrossRef]

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[CrossRef]

Eleonskii, V. M.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Zh. Eksp. Teor. Fiz. 89, 1542–1551 (1985) [Sov. Phys. JETP 62, 894–899 (1985)].

Emplit, Ph.

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]

Erkintalo, M.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[CrossRef]

Fatome, J.

Feir, J. E.

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

Finot, C.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. 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. 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, 112–114 (2011).
[CrossRef]

Forest, M. G.

M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Phys. D 178, 173–189 (2003).
[CrossRef]

M. G. Forest, D. W. McLaughlin, D. J. Muraki, and O. C. Wright, “Nonfocusing instabilities in coupled, integrable nonlinear Schrödinger PDES,” J. Nonlinear Sci. 10, 291–331 (2000).
[CrossRef]

M. G. Forest, S. P. Sheu, and O. C. Wright, “On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems,” Phys. Lett. A 266, 24–33(2000).
[CrossRef]

O. C. Wright and M. G. Forest, “On the Bäcklund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrödinger system,” Phys. D 141, 104–116 (2000).

Genty, G.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[CrossRef]

Guo, B.

B. Guo and L. Ling, “Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations,” Chin. Phys. Lett. 28, 110202 (2011).
[CrossRef]

Haelterman, M.

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]

Hammani, K.

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, 112–114 (2011).
[CrossRef]

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[CrossRef]

Kibler, B.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. 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. 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, 112–114 (2011).
[CrossRef]

Korneev, V. I.

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Zh. Eksp. Teor. Fiz. 94, 159–170 (1988) [Sov. Phys. JETP 67, 89–95 (1988)].

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Mat. Fiz. 69, 189–194 (1986) [Theor. Mat. Phys. 69, 1089 (1986)].
[CrossRef]

Kulagin, N. E.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Zh. Eksp. Teor. Fiz. 89, 1542–1551 (1985) [Sov. Phys. JETP 62, 894–899 (1985)].

Ling, L.

B. Guo and L. Ling, “Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations,” Chin. Phys. Lett. 28, 110202 (2011).
[CrossRef]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505–516 (1973) [Sov. Phys. JETP 38, 248–253 (1974)].

McLaughlin, D. W.

M. G. Forest, D. W. McLaughlin, D. J. Muraki, and O. C. Wright, “Nonfocusing instabilities in coupled, integrable nonlinear Schrödinger PDES,” J. Nonlinear Sci. 10, 291–331 (2000).
[CrossRef]

Menyuk, C. R.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[CrossRef]

Millot, G.

Mitskevich, N. V.

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Zh. Eksp. Teor. Fiz. 94, 159–170 (1988) [Sov. Phys. JETP 67, 89–95 (1988)].

Morin, P.

Muraki, D. J.

M. G. Forest, D. W. McLaughlin, D. J. Muraki, and O. C. Wright, “Nonfocusing instabilities in coupled, integrable nonlinear Schrödinger PDES,” J. Nonlinear Sci. 10, 291–331 (2000).
[CrossRef]

Ostrovsky, L. A.

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Phys. D 238, 540–549 (2009).
[CrossRef]

Roske, G. J.

G. J. Roske, “Some nonlinear multiphase reactions,” Stud. Appl. Math. 55, 231–238 (1976).

Sheu, S. P.

M. G. Forest, S. P. Sheu, and O. C. Wright, “On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems,” Phys. Lett. A 266, 24–33(2000).
[CrossRef]

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” Pis’ma Zh. Eksp. Teor. Fiz. 3, 471–476 (1966) [JETP Lett. 3, 307–310 (1966)].

Trillo, S.

Van Simaeys, G.

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]

Wabnitz, S.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[CrossRef]

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

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

Wai, P. K. A.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[CrossRef]

Wright, O. C.

O. C. Wright, “Dressing procedure for some homoclinic connections of the Manakov system,” Applied Math. Lett. 19, 1185–1190 (2006).
[CrossRef]

M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Phys. D 178, 173–189 (2003).
[CrossRef]

O. C. Wright, “The Darboux transformation of some Manakov systems,” Applied Math. Lett. 16, 647–652 (2003).
[CrossRef]

M. G. Forest, D. W. McLaughlin, D. J. Muraki, and O. C. Wright, “Nonfocusing instabilities in coupled, integrable nonlinear Schrödinger PDES,” J. Nonlinear Sci. 10, 291–331 (2000).
[CrossRef]

M. G. Forest, S. P. Sheu, and O. C. Wright, “On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems,” Phys. Lett. A 266, 24–33(2000).
[CrossRef]

O. C. Wright and M. G. Forest, “On the Bäcklund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrödinger system,” Phys. D 141, 104–116 (2000).

Zakharov, V. E.

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Phys. D 238, 540–549 (2009).
[CrossRef]

L. A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Zh. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)].

Applied Math. Lett. (2)

O. C. Wright, “The Darboux transformation of some Manakov systems,” Applied Math. Lett. 16, 647–652 (2003).
[CrossRef]

O. C. Wright, “Dressing procedure for some homoclinic connections of the Manakov system,” Applied Math. Lett. 19, 1185–1190 (2006).
[CrossRef]

Chin. Phys. Lett. (1)

B. Guo and L. Ling, “Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations,” Chin. Phys. Lett. 28, 110202 (2011).
[CrossRef]

J. Fluid Mech. (1)

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

J. Lightwave Technol. (1)

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[CrossRef]

J. Nonlinear Sci. (1)

M. G. Forest, D. W. McLaughlin, D. J. Muraki, and O. C. Wright, “Nonfocusing instabilities in coupled, integrable nonlinear Schrödinger PDES,” J. Nonlinear Sci. 10, 291–331 (2000).
[CrossRef]

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

Mat. Fiz. (1)

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Mat. Fiz. 69, 189–194 (1986) [Theor. Mat. Phys. 69, 1089 (1986)].
[CrossRef]

Opt. Commun. (1)

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

Opt. Lett. (3)

Phys. D (3)

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Phys. D 238, 540–549 (2009).
[CrossRef]

O. C. Wright and M. G. Forest, “On the Bäcklund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrödinger system,” Phys. D 141, 104–116 (2000).

M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Phys. D 178, 173–189 (2003).
[CrossRef]

Phys. Lett. A (1)

M. G. Forest, S. P. Sheu, and O. C. Wright, “On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems,” Phys. Lett. A 266, 24–33(2000).
[CrossRef]

Phys. Rev. Lett. (3)

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[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]

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[CrossRef]

Pis’ma Zh. Eksp. Teor. Fiz. (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” Pis’ma Zh. Eksp. Teor. Fiz. 3, 471–476 (1966) [JETP Lett. 3, 307–310 (1966)].

Stud. Appl. Math. (1)

G. J. Roske, “Some nonlinear multiphase reactions,” Stud. Appl. Math. 55, 231–238 (1976).

Zh. Eksp. Teor. Fiz. (4)

N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Zh. Eksp. Teor. Fiz. 94, 159–170 (1988) [Sov. Phys. JETP 67, 89–95 (1988)].

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505–516 (1973) [Sov. Phys. JETP 38, 248–253 (1974)].

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Zh. Eksp. Teor. Fiz. 89, 1542–1551 (1985) [Sov. Phys. JETP 62, 894–899 (1985)].

L. A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Zh. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)].

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Figures (12)

Fig. 1.
Fig. 1.

Surface and contour plots of the evolution with distance of the field amplitude |u|. Initial in-phase amplitude modulation with Ω=Ω1=0.8718, εu=εv=102u0, ϕu=ϕv=0.

Fig. 2.
Fig. 2.

Evolution with distance z of the amplitudes of the pump and its harmonics ui: (a) initial in-phase and parallel modulation, CW pump (blue solid curve), the sideband at frequency shift Ω=+Ω1 from the pump (red dashed curve), its second harmonic at Ω=+2Ω1 (green dotted-dashed curve with a peak near z=13), third harmonic Ω=+3Ω1 (violet dotted–dashed curve), and fourth harmonic Ω=+4Ω1 (pink dotted curve) corresponding to the case in Fig. 1; (b) initial modulation orthogonal to the pump.

Fig. 3.
Fig. 3.

Same as Fig. 1, with a quadrature modulation that is orthogonal to the pump, i.e., with ϕu=ϕv=π/2.

Fig. 4.
Fig. 4.

Surface plot of the amplitude |u| with orthogonal input pump and sidebands ϕu=0, ϕv=π and the different sideband modulation frequencies: (a) Ω=2, (b) Ω=0.5, and (c) Ω=0.4.

Fig. 5.
Fig. 5.

Same as in Fig. 2, with reference to the cases in Fig. 4 with (a) Ω=0.5 and (b) Ω=0.4.

Fig. 6.
Fig. 6.

Contour plots as in Fig. 4, with (a) Ω=0.5 and (b) Ω=0.4.

Fig. 7.
Fig. 7.

Surface plots of the temporal evolution with distance of the field amplitudes |u| and |v|, exhibiting recursive behavior. Input pump in the u mode (u0=1, v0=0) and modulation at frequency Ω=1/2=0.707 in the v mode, εu=0, εv=102, ϕu=ϕv=0. Note also the different vertical scale.

Fig. 8.
Fig. 8.

CW and sideband (a) amplitudes and (b) contour plot in the u mode with Ω=0.4, εu=0, εv=101, ϕu=ϕv=0.

Fig. 9.
Fig. 9.

Experimental setup. EDFA, erbium-doped fiber amplifier; POL, polarizer; OSO, optical sampling oscilloscope; OSA, optical spectrum analyzer.

Fig. 10.
Fig. 10.

(a) Temporal profiles at the input (dashed curve) and output (solid curve) of the fiber when pump and signal waves have parallel polarization states. (b) Experimental spectrum at the output of the fiber for input parallel polarized pump and signal and for a pump power of 19.7 dBm. Note that the output polarizer was oriented parallel to the pump polarization.

Fig. 11.
Fig. 11.

(a) Experimental temporal profiles at the input (dashed curve) and output (solid curve) of the fiber for orthogonal polarized pump and signal waves. (b) Corresponding output experimental spectrum. The output polarizer was oriented parallel to the pump polarization.

Fig. 12.
Fig. 12.

(a) Temporal and spectral profiles obtained from numerical simulations when the pump and signal waves are injected with orthogonal polarizations.

Equations (2)

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iuz+122ut2+(|u|2+|v|2)u=0,ivz+122vt2+(|v|2+|u|2)v=0,
u(z=0,t)=u0+εuexp(iφu)cos(Ωt),v(z=0,t)=v0+εvexp(iφv)cos(Ωt),

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