Abstract

The modulational instability in an optical fiber, with loss taken into account, is analytically investigated. It is demonstrated how the linearized equation for perturbation can be solved exactly in terms of Bessel functions, and qualitative differences from previous analytic approaches are pointed out. Finally, the limits imposed by this instability on optical communication systems are discussed. The use of this theory in system design is examined.

© 1995 Optical Society of America

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  1. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).
  2. T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water,” J. Fluid Mech. 27, 410–430 (1967).
    [CrossRef]
  3. N. R. Pereira, A. Sen, and A. Bers, “Nonlinear development of lower hybrid cones,” Phys. Fluids 21, 117–120 (1978).
    [CrossRef]
  4. J. R. Myra and C. S. Liu, “Self-modulation of ion Bernstein waves,” Phys. Fluids 23, 2258–2264 (1980).
    [CrossRef]
  5. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  6. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
    [CrossRef]
  7. A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulation instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
    [CrossRef]
  8. A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. 9, 288–290 (1984).
    [CrossRef] [PubMed]
  9. D. Anderson and M. Lisak, “Modulational instability of coherent optical-fiber transmission signals,” Opt. Lett. 9, 468–470 (1984).
    [CrossRef] [PubMed]
  10. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulation instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
    [CrossRef] [PubMed]
  11. M. N. Islam, S. P. Dijaili, and J. P. Gordon, “Modulation-instability-based fiber interferometer switch near 1.5 μ m,” Opt. Lett. 13, 518–520 (1988).
    [CrossRef] [PubMed]
  12. M. Nakasawa, K. Suzuki, H. Kubota, and H. Haus, “The modulation instability laser—part II: theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
    [CrossRef]
  13. V. E. Zakharov and A. B. Shabat, “Exact theory for two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  14. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [CrossRef] [PubMed]
  15. M. Haelterman and A. P. Sheppard, “Vector soliton associated with polarization modulational instability in the normal-dispersion regime,” Phys. Rev. E 49, 3389–3399 (1994).
    [CrossRef]
  16. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
    [CrossRef]
  17. S. Trillo and S. Wabnitz, “Dynamics of modulational instability in optical fibers,” Opt. Lett. 16, 986–988 (1991).
    [CrossRef] [PubMed]
  18. 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]
  19. H. C. Yuen and W. E. Ferguson, “Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation,” Phys. Fluids 21, 1275–1278 (1978).
    [CrossRef]
  20. N. N. Akhmediev, V. M. Elonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solution,” Sov. Phys. JETP 62, 894–899 (1985).
  21. E. R. Tracy, H. H. Chen, and Y. C. Lee, “Study of quasiperiodic solutions of the nonlinear Schrödinger equation and the nonlinear modulational instability,” Phys. Rev. Lett. 53, 218–221 (1984).
    [CrossRef]
  22. V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
    [CrossRef]
  23. S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
    [CrossRef] [PubMed]
  24. F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
    [CrossRef]
  25. P. K. Shukla and J. Juul Rasmussem, “Modulational instability of short pulses in long optical fibers,” Opt. Lett. 11, 171–173 (1986).
    [CrossRef] [PubMed]
  26. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
    [CrossRef]
  27. M. J. Potasek, “Modulation instability in an extended nonlinear Schrödinger equation,” Opt. Lett. 12, 921–923 (1987).
    [CrossRef] [PubMed]
  28. J. P. Hamaide, Ph. Emplit, and J. M. Gabriagues, “Limitations in long-haul IM/DD optical fibre systems caused by chromatic dispersion and nonlinear Kerr effect,” Electron. Lett. 26, 1451–1453 (1990).
    [CrossRef]
  29. N. Henmi, Y. Aoki, T. Ogata, T. Saito, and S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with lumped amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
    [CrossRef]
  30. A. Hasegawa and K. Tai, “Effects of modulational instability on coherent transmission systems,” Opt. Lett. 14, 512–514 (1989).
    [CrossRef] [PubMed]
  31. C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 10.
  32. The parameters g and ω correspond to the notation I and R/2 of Ref. 30.
  33. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9.6.
  34. T. Sugie, T. Imai, and T. Ito, “Over 350 km CPFSK repeaterless transmission at 2.5 Gbit/s employing high-output power erbium-doped fibre amplifiers,” Electron. Lett. 25, 1577–1578 (1990).
    [CrossRef]
  35. S. Ryu, “Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers,” J. Lightwave Technol. 10, 1450–1457 (1992).
    [CrossRef]
  36. Ref. 31, Chap. 6.

1994 (2)

M. Haelterman and A. P. Sheppard, “Vector soliton associated with polarization modulational instability in the normal-dispersion regime,” Phys. Rev. E 49, 3389–3399 (1994).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

1993 (1)

N. Henmi, Y. Aoki, T. Ogata, T. Saito, and S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with lumped amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[CrossRef]

1992 (1)

S. Ryu, “Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers,” J. Lightwave Technol. 10, 1450–1457 (1992).
[CrossRef]

1991 (3)

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]

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

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

1990 (2)

T. Sugie, T. Imai, and T. Ito, “Over 350 km CPFSK repeaterless transmission at 2.5 Gbit/s employing high-output power erbium-doped fibre amplifiers,” Electron. Lett. 25, 1577–1578 (1990).
[CrossRef]

J. P. Hamaide, Ph. Emplit, and J. M. Gabriagues, “Limitations in long-haul IM/DD optical fibre systems caused by chromatic dispersion and nonlinear Kerr effect,” Electron. Lett. 26, 1451–1453 (1990).
[CrossRef]

1989 (3)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

A. Hasegawa and K. Tai, “Effects of modulational instability on coherent transmission systems,” Opt. Lett. 14, 512–514 (1989).
[CrossRef] [PubMed]

M. Nakasawa, K. Suzuki, H. Kubota, and H. Haus, “The modulation instability laser—part II: theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

1988 (1)

1987 (3)

M. J. Potasek, “Modulation instability in an extended nonlinear Schrödinger equation,” Opt. Lett. 12, 921–923 (1987).
[CrossRef] [PubMed]

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

1986 (2)

P. K. Shukla and J. Juul Rasmussem, “Modulational instability of short pulses in long optical fibers,” Opt. Lett. 11, 171–173 (1986).
[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]

1985 (1)

N. N. Akhmediev, V. M. Elonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solution,” Sov. Phys. JETP 62, 894–899 (1985).

1984 (3)

1982 (1)

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

1980 (3)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulation instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[CrossRef]

J. R. Myra and C. S. Liu, “Self-modulation of ion Bernstein waves,” Phys. Fluids 23, 2258–2264 (1980).
[CrossRef]

1978 (2)

H. C. Yuen and W. E. Ferguson, “Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation,” Phys. Fluids 21, 1275–1278 (1978).
[CrossRef]

N. R. Pereira, A. Sen, and A. Bers, “Nonlinear development of lower hybrid cones,” Phys. Fluids 21, 117–120 (1978).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory for two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1967 (1)

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water,” J. Fluid Mech. 27, 410–430 (1967).
[CrossRef]

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Abdullaev, F. Kh.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9.6.

Agrawal, G. P.

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

Akhmediev, N. N.

N. N. Akhmediev, V. M. Elonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solution,” Sov. Phys. JETP 62, 894–899 (1985).

Anderson, D.

Aoki, Y.

N. Henmi, Y. Aoki, T. Ogata, T. Saito, and S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with lumped amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[CrossRef]

Bender, C. M.

C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 10.

Benjamin, T. B.

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water,” J. Fluid Mech. 27, 410–430 (1967).
[CrossRef]

Bers, A.

N. R. Pereira, A. Sen, and A. Bers, “Nonlinear development of lower hybrid cones,” Phys. Fluids 21, 117–120 (1978).
[CrossRef]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Bischoff, S.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

Brinkman, W. F.

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulation instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[CrossRef]

Cappellini, G.

Cavalcanti, S. B.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

Chen, H. H.

E. R. Tracy, H. H. Chen, and Y. C. Lee, “Study of quasiperiodic solutions of the nonlinear Schrödinger equation and the nonlinear modulational instability,” Phys. Rev. Lett. 53, 218–221 (1984).
[CrossRef]

Christiansen, P. L.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Cressoni, J. C.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

da Cruz, H. R.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

Darmanyan, S. A.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Dijaili, S. P.

Elonskii, V. M.

N. N. Akhmediev, V. M. Elonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solution,” Sov. Phys. JETP 62, 894–899 (1985).

Emplit, Ph.

J. P. Hamaide, Ph. Emplit, and J. M. Gabriagues, “Limitations in long-haul IM/DD optical fibre systems caused by chromatic dispersion and nonlinear Kerr effect,” Electron. Lett. 26, 1451–1453 (1990).
[CrossRef]

Feir, J. E.

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water,” J. Fluid Mech. 27, 410–430 (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, 1275–1278 (1978).
[CrossRef]

Gabriagues, J. M.

J. P. Hamaide, Ph. Emplit, and J. M. Gabriagues, “Limitations in long-haul IM/DD optical fibre systems caused by chromatic dispersion and nonlinear Kerr effect,” Electron. Lett. 26, 1451–1453 (1990).
[CrossRef]

Gordon, J. P.

M. N. Islam, S. P. Dijaili, and J. P. Gordon, “Modulation-instability-based fiber interferometer switch near 1.5 μ m,” Opt. Lett. 13, 518–520 (1988).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Gouveia-Neto, A. S.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

Haelterman, M.

M. Haelterman and A. P. Sheppard, “Vector soliton associated with polarization modulational instability in the normal-dispersion regime,” Phys. Rev. E 49, 3389–3399 (1994).
[CrossRef]

Hamaide, J. P.

J. P. Hamaide, Ph. Emplit, and J. M. Gabriagues, “Limitations in long-haul IM/DD optical fibre systems caused by chromatic dispersion and nonlinear Kerr effect,” Electron. Lett. 26, 1451–1453 (1990).
[CrossRef]

Hasegawa, A.

A. Hasegawa and K. Tai, “Effects of modulational instability on coherent transmission systems,” Opt. Lett. 14, 512–514 (1989).
[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]

A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. 9, 288–290 (1984).
[CrossRef] [PubMed]

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulation instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[CrossRef]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Haus, H.

M. Nakasawa, K. Suzuki, H. Kubota, and H. Haus, “The modulation instability laser—part II: theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

Henmi, N.

N. Henmi, Y. Aoki, T. Ogata, T. Saito, and S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with lumped amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[CrossRef]

Imai, T.

T. Sugie, T. Imai, and T. Ito, “Over 350 km CPFSK repeaterless transmission at 2.5 Gbit/s employing high-output power erbium-doped fibre amplifiers,” Electron. Lett. 25, 1577–1578 (1990).
[CrossRef]

Islam, M. N.

Ito, T.

T. Sugie, T. Imai, and T. Ito, “Over 350 km CPFSK repeaterless transmission at 2.5 Gbit/s employing high-output power erbium-doped fibre amplifiers,” Electron. Lett. 25, 1577–1578 (1990).
[CrossRef]

Juul Rasmussem, J.

Kubota, H.

M. Nakasawa, K. Suzuki, H. Kubota, and H. Haus, “The modulation instability laser—part II: theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

Kulagin, N. E.

N. N. Akhmediev, V. M. Elonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solution,” Sov. Phys. JETP 62, 894–899 (1985).

Lee, Y. C.

E. R. Tracy, H. H. Chen, and Y. C. Lee, “Study of quasiperiodic solutions of the nonlinear Schrödinger equation and the nonlinear modulational instability,” Phys. Rev. Lett. 53, 218–221 (1984).
[CrossRef]

Lisak, M.

Liu, C. S.

J. R. Myra and C. S. Liu, “Self-modulation of ion Bernstein waves,” Phys. Fluids 23, 2258–2264 (1980).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Myra, J. R.

J. R. Myra and C. S. Liu, “Self-modulation of ion Bernstein waves,” Phys. Fluids 23, 2258–2264 (1980).
[CrossRef]

Nakasawa, M.

M. Nakasawa, K. Suzuki, H. Kubota, and H. Haus, “The modulation instability laser—part II: theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

Nakaya, S.

N. Henmi, Y. Aoki, T. Ogata, T. Saito, and S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with lumped amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[CrossRef]

Ogata, T.

N. Henmi, Y. Aoki, T. Ogata, T. Saito, and S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with lumped amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[CrossRef]

Orzag, S. A.

C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 10.

Pereira, N. R.

N. R. Pereira, A. Sen, and A. Bers, “Nonlinear development of lower hybrid cones,” Phys. Fluids 21, 117–120 (1978).
[CrossRef]

Potasek, M. J.

Ryu, S.

S. Ryu, “Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers,” J. Lightwave Technol. 10, 1450–1457 (1992).
[CrossRef]

Saito, T.

N. Henmi, Y. Aoki, T. Ogata, T. Saito, and S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with lumped amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[CrossRef]

Sen, A.

N. R. Pereira, A. Sen, and A. Bers, “Nonlinear development of lower hybrid cones,” Phys. Fluids 21, 117–120 (1978).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory for two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Sheppard, A. P.

M. Haelterman and A. P. Sheppard, “Vector soliton associated with polarization modulational instability in the normal-dispersion regime,” Phys. Rev. E 49, 3389–3399 (1994).
[CrossRef]

Shukla, P. K.

Sørensen, M. P.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9.6.

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Sugie, T.

T. Sugie, T. Imai, and T. Ito, “Over 350 km CPFSK repeaterless transmission at 2.5 Gbit/s employing high-output power erbium-doped fibre amplifiers,” Electron. Lett. 25, 1577–1578 (1990).
[CrossRef]

Sukhotskova, N. A.

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

Suzuki, K.

M. Nakasawa, K. Suzuki, H. Kubota, and H. Haus, “The modulation instability laser—part II: theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

Tai, K.

A. Hasegawa and K. Tai, “Effects of modulational instability on coherent transmission systems,” Opt. Lett. 14, 512–514 (1989).
[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]

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

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]

Tracy, E. R.

E. R. Tracy, H. H. Chen, and Y. C. Lee, “Study of quasiperiodic solutions of the nonlinear Schrödinger equation and the nonlinear modulational instability,” Phys. Rev. Lett. 53, 218–221 (1984).
[CrossRef]

Trillo, S.

Vysloukh, V. A.

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

Wabnitz, S.

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K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

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, 1275–1278 (1978).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory for two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Electron. Lett. (2)

J. P. Hamaide, Ph. Emplit, and J. M. Gabriagues, “Limitations in long-haul IM/DD optical fibre systems caused by chromatic dispersion and nonlinear Kerr effect,” Electron. Lett. 26, 1451–1453 (1990).
[CrossRef]

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[CrossRef]

IEEE J. Quantum Electron. (4)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulation instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[CrossRef]

M. Nakasawa, K. Suzuki, H. Kubota, and H. Haus, “The modulation instability laser—part II: theory,” IEEE J. Quantum Electron. 25, 2045–2052 (1989).
[CrossRef]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

J. Fluid Mech. (1)

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[CrossRef]

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S. Ryu, “Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers,” J. Lightwave Technol. 10, 1450–1457 (1992).
[CrossRef]

N. Henmi, Y. Aoki, T. Ogata, T. Saito, and S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with lumped amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[CrossRef]

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

JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Opt. Commun. (1)

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Opt. Lett. (7)

Phys. Fluids (3)

N. R. Pereira, A. Sen, and A. Bers, “Nonlinear development of lower hybrid cones,” Phys. Fluids 21, 117–120 (1978).
[CrossRef]

J. R. Myra and C. S. Liu, “Self-modulation of ion Bernstein waves,” Phys. Fluids 23, 2258–2264 (1980).
[CrossRef]

H. C. Yuen and W. E. Ferguson, “Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation,” Phys. Fluids 21, 1275–1278 (1978).
[CrossRef]

Phys. Rev. A (1)

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (1)

M. Haelterman and A. P. Sheppard, “Vector soliton associated with polarization modulational instability in the normal-dispersion regime,” Phys. Rev. E 49, 3389–3399 (1994).
[CrossRef]

Phys. Rev. Lett. (4)

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

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

E. R. Tracy, H. H. Chen, and Y. C. Lee, “Study of quasiperiodic solutions of the nonlinear Schrödinger equation and the nonlinear modulational instability,” Phys. Rev. Lett. 53, 218–221 (1984).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

Sov. Phys. JETP (2)

N. N. Akhmediev, V. M. Elonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solution,” Sov. Phys. JETP 62, 894–899 (1985).

V. E. Zakharov and A. B. Shabat, “Exact theory for two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Other (4)

Ref. 31, Chap. 6.

C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 10.

The parameters g and ω correspond to the notation I and R/2 of Ref. 30.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9.6.

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Figures (3)

Fig. 1
Fig. 1

Three-dimensional plot of the exact solution of the case in which g = 5.

Fig. 2
Fig. 2

Gain spectrum a(z; 1, ω)/a0 at the different propagation distances z/Lloss = 1, 2, 5, 10.

Fig. 3
Fig. 3

Comparison of the different expressions for the gain spectrum at g = 10. The oscillating solid curve is the exact solution at z/Lloss = 20, the smoother solid curve is Gapp2(10, ω), and the dashed curve is the WKB gain spectrum cosh[10f(ω)].

Equations (27)

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i u z = - β 2 2 2 u t 2 + σ u 2 u - i u 2 L loss .
u ( z , t ) = P 0 exp ( i σ P 0 z - z 2 L loss ) [ 1 + ( z , t ) ] .
d a d z = - b 2 L D , d b d z = a 2 L D - 2 a L NL exp ( - z / L loss ) .
a ( z ) = [ - 1 4 L D 2 + 1 L D L NL exp ( - z L loss ) ] a ( z ) γ 2 ( z ) a ( z ) a ( 0 ) = 0 ,             a ( 0 ) = a 0 ,
S 0 ( z ) = ± 0 z γ ( z ) d z ,             S 1 ( z ) = - ¼ ln [ γ ( z ) ] .
S 0 ( z 0 ) = ± g ω 2 [ ( 1 ω 2 - 1 ) 1 / 2 - arctan ( 1 ω 2 - 1 ) 1 / 2 ] ± g f ( ω ) ,
ω 2 = Ω 2 Ω co 2 = L NL 4 L D ,             g = 4 L loss L NL ,
x d 2 a d x 2 + d a d x = - g 2 ω 4 a x + g 2 ω 2 a x .
a ( z ; g , ω ) = a 0 π g ω sinh ( g ω 2 ) Im { I i g ω 2 ( g ω ) I - i g ω 2 × [ g ω exp ( - z 2 L loss ) ] } .
a app 1 ( z ; g , ω ) = a 0 Im ( exp ( i g ω 2 z 2 L loss ) × { i + g 2 + i g 2 ω 2 4 [ exp ( - z L loss ) - 1 ] + O ( g 3 w 3 ) } ) ,
a app 2 ( z ; g , ω ) = a 0 ( 1 ω 2 - 1 ) 1 / 4 exp [ g f ( ω ) ] × cos ( i g ω 2 z 2 L loss + ϕ ) × [ 1 + O ( 1 g ω 2 ) ] ,
a app 1 ( z , g , w ) a 0 { 1 + g 2 ω 2 4 [ exp ( - z L loss ) - 1 + z L loss ] - g 2 ω 4 z 2 8 L loss 2 } .
a max 1 ( z , g ) a 0 { 1 + g 2 8 [ exp ( - z / L loss ) - 1 + z / L loss z / L loss ] 2 } .
G app 2 ( g , ω ) ( 1 ω 2 - 1 ) 1 / 4 exp [ g f ( ω ) ] .
a max 2 ( g ) a 0 1.6 exp ( 0.18 g ) .
MI penalty = 10 log [ 1 - 2 a max ( z , g ) ] .
MI penalty = 10 log [ 1 - 2 a 0 1.6 exp ( 0.72 L loss σ P 0 ) ] = 10 log { 1 - 2 a 0 1.6 exp [ 0.72 L loss σ P rec exp ( L L loss ) ] } .
G = [ 1 + 2 ( σ P 0 L loss α ) 2 ] N ,
x d 2 a ( x ) d x 2 + d a ( x ) d x = - ν 2 a ( x ) x + a ( x ) x
W [ I - i ν ( x ) , I i ν ( x ) ] = 2 i sinh ( ν π ) π x .
I i ν ( x ) = 1 π 0 π exp [ x cos ( t ) ] cos ( ν t ) d t - sinh ( ν π ) π 0 exp [ - x cosh ( t ) ] sin ( ν t ) d t - i sinh ( ν π ) π 0 exp [ - x cosh ( t ) ] cos ( ν t ) d t ,
I i ν ( x ) ( x / 2 ) i ν Γ ( 1 + i ν ) ( 1 + 1 4 x 2 1 + i ν ) ,
I i ν [ ν sech ( a ) ] 1 + o ( ν - 1 ) [ 2 π ν tanh ( a ) ] 1 / 2 × exp { ν [ π 2 + tanh ( a ) - a ] - i π 4 } ,
I i ν [ ν sech ( b ) ] 1 + o ( ν - 1 ) [ 2 π ν tanh ( b ) ] 1 / 2 exp ( ν π 2 ) × ( exp { ν [ tan ( b ) - b ] } - i 2 exp { - ν [ tan ( b ) - b ] } ) ,
I i ν ( ν ) 1 + o ( ν - 1 ) 2 π 3 Γ ( 1 3 ) ( 6 ν ) 1 / 3 exp ( ν π 2 - i π 6 ) ,
I i ν [ ν sech ( a ) ] 1 + o ( ν - 1 ) [ 4 π ν / sinh ( 2 a ) ] 1 / 2 × exp { ν [ π 2 + tanh ( a ) - a ] - i π 4 } ,
I i ν [ ν sech ( b ) ] 1 + o ( ν - 1 ) [ 4 π ν / sin ( 2 b ) ] 1 / 2 exp ( ν π 2 ) × ( exp { ν [ tan ( b ) - b ] } + i 2 exp { - ν [ tan ( b ) - b ] } ) .

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