Abstract

We report what is to our knowledge the first clear-cut experimental evidence of the reversibility of modulational instability in dispersive Kerr media. It was possible to perform this experiment with standard telecommunication fiber because we used a specially designed 550-ps square-pulse laser source based on the two-wavelength configuration of a nonlinear optical loop mirror. Our observations demonstrate that reversibility is due to well-balanced and synchronous energy transfer among a significant number of spectral wave components. These results provide what we believe is the first evidence, in the field of nonlinear optics, of the universal Fermi–Pasta–Ulam recurrence phenomenon that has been predicted for a large number of conservative nonlinear systems, including those described by a nonlinear Schrödinger equation that is relevant to the context of the present study.

© 2002 Optical Society of America

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References

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  1. T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water. 1. Theory,” J. Fluid Mech. 27, 417–430 (1967).
    [CrossRef]
  2. T. B. Benjamin, “Instability of periodic wavetrains in nonlinear dispersive systems,” Proc. R. Soc. London Ser. A 299, 59–75 (1967).
    [CrossRef]
  3. B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, “Nonlinear deep-water waves: theory and experiment. 2. Evolution of a continuous wave train,” J. Fluid Mech. 83, 49–74 (1977).
    [CrossRef]
  4. H. C. Yuen, J. Warren, and E. Ferguson, “Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation,” Phys. Fluids 21, 1275–1278 (1978).
    [CrossRef]
  5. E. Fermi, J. Pasta, and H. C. Ulam, “Studies of nonlinear problems,” in Collected Papers of Enrico Fermi, E. Segrè, ed. (U. Chicago Press, Chicago, Ill., 1965), Vol. 2, pp. 977–988.
  6. T. Taniuti and H. Washimi, “Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma,” Phys. Rev. Lett. 21, 209–212 (1968).
    [CrossRef]
  7. F. D. Tappert and C. N. Judice, “Recurrence of nonlinear ion acousitc waves,” Phys. Rev. Lett. 29, 1308–1311 (1972).
    [CrossRef]
  8. P. Henrotay, “Periodic solutions and recurrence for nonlinear Schrödinger equation: a Fourier-mode approach,” J. Mec. 20, 159–168 (1981).
  9. E. Infeld, “Quantitive theory of the Fermi–Pasta–Ulam recurrence in the nonlinear Schrödinger equation,” Phys. Rev. Lett. 47, 717–718 (1981).
    [CrossRef]
  10. 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]
  11. 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]
  12. N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986) [ Teor. Mat. Fiz. 69, 189–194 (1986)].
    [CrossRef]
  13. N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first-order solutions of the nonlinear Schödinger equation,” Theor. Math. Phys. 72, 809–818 (1987) [Teor. Mat. Fiz. 72, 183–196 (1987)].
    [CrossRef]
  14. S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Nonlinear dynamics of polarization-modulation instability in optical fiber,” J. Opt. Soc. Am. B 14, 3403–3411 (1997).
    [CrossRef]
  15. S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulation instability in optical fibers,” Opt. Lett. 16, 986–988 (1991).
    [CrossRef] [PubMed]
  16. 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]
  17. S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
    [CrossRef]
  18. D. Cotter, “Suppression of stimulated Brillouin scattering during transmission of high-power narrowband laser light in monomode fibre,” Electron. Lett. 18, 638–640 (1982).
    [CrossRef]
  19. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
    [CrossRef] [PubMed]
  20. N. Doran and D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett. 13, 56–58 (1988).
    [CrossRef] [PubMed]
  21. K. J. Blow, N. J. Doran, B. K. Nayar, and B. Nelson, “Two-wavelength operation of the nonlinear fiber loop mirror,” Opt. Lett. 15, 248–250 (1990).
    [CrossRef] [PubMed]
  22. M. Jinno and T. Matsumoto, “Nonlinear Sagnac interferometer switch and its applications,” IEEE J. Quantum Electron. 28, 875–882 (1992).
    [CrossRef]
  23. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  24. G. Cappellini and S. Trillo, “Energy conversion in degenerate four-photon mixing in birefringent fibers,” Opt. Lett. 16, 895–897 (1991).
    [CrossRef] [PubMed]
  25. S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
    [CrossRef] [PubMed]
  26. S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20, 866–868 (1995).
    [CrossRef] [PubMed]

1997

1995

1994

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]

1992

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

M. Jinno and T. Matsumoto, “Nonlinear Sagnac interferometer switch and its applications,” IEEE J. Quantum Electron. 28, 875–882 (1992).
[CrossRef]

1991

1990

1988

N. Doran and D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett. 13, 56–58 (1988).
[CrossRef] [PubMed]

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[CrossRef] [PubMed]

1987

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first-order solutions of the nonlinear Schödinger equation,” Theor. Math. Phys. 72, 809–818 (1987) [Teor. Mat. Fiz. 72, 183–196 (1987)].
[CrossRef]

1986

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

1982

D. Cotter, “Suppression of stimulated Brillouin scattering during transmission of high-power narrowband laser light in monomode fibre,” Electron. Lett. 18, 638–640 (1982).
[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]

1981

P. Henrotay, “Periodic solutions and recurrence for nonlinear Schrödinger equation: a Fourier-mode approach,” J. Mec. 20, 159–168 (1981).

E. Infeld, “Quantitive theory of the Fermi–Pasta–Ulam recurrence in the nonlinear Schrödinger equation,” Phys. Rev. Lett. 47, 717–718 (1981).
[CrossRef]

1978

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

1977

B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, “Nonlinear deep-water waves: theory and experiment. 2. Evolution of a continuous wave train,” J. Fluid Mech. 83, 49–74 (1977).
[CrossRef]

1972

F. D. Tappert and C. N. Judice, “Recurrence of nonlinear ion acousitc waves,” Phys. Rev. Lett. 29, 1308–1311 (1972).
[CrossRef]

1968

T. Taniuti and H. Washimi, “Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma,” Phys. Rev. Lett. 21, 209–212 (1968).
[CrossRef]

1967

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

T. B. Benjamin, “Instability of periodic wavetrains in nonlinear dispersive systems,” Proc. R. Soc. London Ser. A 299, 59–75 (1967).
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first-order solutions of the nonlinear Schödinger equation,” Theor. Math. Phys. 72, 809–818 (1987) [Teor. Mat. Fiz. 72, 183–196 (1987)].
[CrossRef]

Benjamin, T. B.

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

T. B. Benjamin, “Instability of periodic wavetrains in nonlinear dispersive systems,” Proc. R. Soc. London Ser. A 299, 59–75 (1967).
[CrossRef]

Bergano, N. S.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[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.

Cappellini, G.

Cotter, D.

D. Cotter, “Suppression of stimulated Brillouin scattering during transmission of high-power narrowband laser light in monomode fibre,” Electron. Lett. 18, 638–640 (1982).
[CrossRef]

Doran, N.

Doran, N. J.

Eleonskii, V. M.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first-order solutions of the nonlinear Schödinger equation,” Theor. Math. Phys. 72, 809–818 (1987) [Teor. Mat. Fiz. 72, 183–196 (1987)].
[CrossRef]

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

Feir, J. E.

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

Ferguson, E.

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

Ferguson, W. E.

B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, “Nonlinear deep-water waves: theory and experiment. 2. Evolution of a continuous wave train,” J. Fluid Mech. 83, 49–74 (1977).
[CrossRef]

Gordon, J. P.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

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]

Harvey, J. D.

Hasegawa, A.

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

Henrotay, P.

P. Henrotay, “Periodic solutions and recurrence for nonlinear Schrödinger equation: a Fourier-mode approach,” J. Mec. 20, 159–168 (1981).

Infeld, E.

E. Infeld, “Quantitive theory of the Fermi–Pasta–Ulam recurrence in the nonlinear Schrödinger equation,” Phys. Rev. Lett. 47, 717–718 (1981).
[CrossRef]

Jinno, M.

M. Jinno and T. Matsumoto, “Nonlinear Sagnac interferometer switch and its applications,” IEEE J. Quantum Electron. 28, 875–882 (1992).
[CrossRef]

Judice, C. N.

F. D. Tappert and C. N. Judice, “Recurrence of nonlinear ion acousitc waves,” Phys. Rev. Lett. 29, 1308–1311 (1972).
[CrossRef]

Kulagin, N. E.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first-order solutions of the nonlinear Schödinger equation,” Theor. Math. Phys. 72, 809–818 (1987) [Teor. Mat. Fiz. 72, 183–196 (1987)].
[CrossRef]

Lake, B. M.

B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, “Nonlinear deep-water waves: theory and experiment. 2. Evolution of a continuous wave train,” J. Fluid Mech. 83, 49–74 (1977).
[CrossRef]

Leonhardt, R.

Matsumoto, T.

M. Jinno and T. Matsumoto, “Nonlinear Sagnac interferometer switch and its applications,” IEEE J. Quantum Electron. 28, 875–882 (1992).
[CrossRef]

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

Murdoch, S. G.

Nayar, B. K.

Nelson, B.

Rungaldier, H.

B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, “Nonlinear deep-water waves: theory and experiment. 2. Evolution of a continuous wave train,” J. Fluid Mech. 83, 49–74 (1977).
[CrossRef]

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]

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]

Tai, K.

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

Taniuti, T.

T. Taniuti and H. Washimi, “Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma,” Phys. Rev. Lett. 21, 209–212 (1968).
[CrossRef]

Tappert, F. D.

F. D. Tappert and C. N. Judice, “Recurrence of nonlinear ion acousitc waves,” Phys. Rev. Lett. 29, 1308–1311 (1972).
[CrossRef]

Tomita, A.

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

Trillo, S.

Wabnitz, S.

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

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[CrossRef] [PubMed]

Warren, J.

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

Washimi, H.

T. Taniuti and H. Washimi, “Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma,” Phys. Rev. Lett. 21, 209–212 (1968).
[CrossRef]

Wood, D.

Yuen, H. C.

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

B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, “Nonlinear deep-water waves: theory and experiment. 2. Evolution of a continuous wave train,” J. Fluid Mech. 83, 49–74 (1977).
[CrossRef]

Electron. Lett.

D. Cotter, “Suppression of stimulated Brillouin scattering during transmission of high-power narrowband laser light in monomode fibre,” Electron. Lett. 18, 638–640 (1982).
[CrossRef]

IEEE J. Quantum Electron.

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]

M. Jinno and T. Matsumoto, “Nonlinear Sagnac interferometer switch and its applications,” IEEE J. Quantum Electron. 28, 875–882 (1992).
[CrossRef]

J. Fluid Mech.

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

B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, “Nonlinear deep-water waves: theory and experiment. 2. Evolution of a continuous wave train,” J. Fluid Mech. 83, 49–74 (1977).
[CrossRef]

J. Lightwave Technol.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

J. Mec.

P. Henrotay, “Periodic solutions and recurrence for nonlinear Schrödinger equation: a Fourier-mode approach,” J. Mec. 20, 159–168 (1981).

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Fluids

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

Phys. Rev. A

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[CrossRef] [PubMed]

Phys. Rev. E

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.

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

E. Infeld, “Quantitive theory of the Fermi–Pasta–Ulam recurrence in the nonlinear Schrödinger equation,” Phys. Rev. Lett. 47, 717–718 (1981).
[CrossRef]

T. Taniuti and H. Washimi, “Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma,” Phys. Rev. Lett. 21, 209–212 (1968).
[CrossRef]

F. D. Tappert and C. N. Judice, “Recurrence of nonlinear ion acousitc waves,” Phys. Rev. Lett. 29, 1308–1311 (1972).
[CrossRef]

Proc. R. Soc. London Ser. A

T. B. Benjamin, “Instability of periodic wavetrains in nonlinear dispersive systems,” Proc. R. Soc. London Ser. A 299, 59–75 (1967).
[CrossRef]

Theor. Math. Phys.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first-order solutions of the nonlinear Schödinger equation,” Theor. Math. Phys. 72, 809–818 (1987) [Teor. Mat. Fiz. 72, 183–196 (1987)].
[CrossRef]

Other

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986) [ Teor. Mat. Fiz. 69, 189–194 (1986)].
[CrossRef]

E. Fermi, J. Pasta, and H. C. Ulam, “Studies of nonlinear problems,” in Collected Papers of Enrico Fermi, E. Segrè, ed. (U. Chicago Press, Chicago, Ill., 1965), Vol. 2, pp. 977–988.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

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Figures (10)

Fig. 1
Fig. 1

Spatiotemporal evolution of MI. (The parameters used here for the simulation are those given in Subsection 2.B for an input pump power of 2 W.)

Fig. 2
Fig. 2

Phase-space portrait of scalar MI. Dotted curves, trajectories of the ideal lossless system described by Eqs. (4), with κ=-2 and α=|as(ξ=0)|2-|aa(ξ=0)|2=0.0023. Solid curve, trajectory corresponding to κ=-2, |as(ξ=0)|2=0.0023, and |aa(ξ=0)|2=0 when losses and Raman scattering are included.

Fig. 3
Fig. 3

Evolution of the pump and Stokes powers as a function of ξ. The computed curves were obtained by numerical simulations of the generalized NLS equation. Solid curves, propagation in an ideal lossless fiber; dotted–dashed curves, evolution in a fiber with a linear attenuation of 0.2 dB/km; dotted curves, evolution in the same lossy fiber with the inclusion of the localized 0.2-dB coupler loss and Raman scattering.

Fig. 4
Fig. 4

Envelope evolution over 1.5 recurrence periods of (a) a square pulse and (b) a Gaussian pulse with equal initial peak power (2.3 W) and duration (∼550 ps).

Fig. 5
Fig. 5

Evolution of the pump and Stokes powers versus ξ for κ=-2 and |as|2=0.0023 when linear attenuation, insertion loss of the 99:1 coupler, and Raman scattering have been taken into account. Input waves are a cw (solid curves) and a square pulse (dashed curves).

Fig. 6
Fig. 6

Experimental setup of the square-pulse generator. Inset, measured output square-pulse profile (solid curve) and profile obtained after deconvolution (dashed curve).

Fig. 7
Fig. 7

Experimental setup: PA, preamplifier with an output power of ∼18 mW; A, Er3+-doped fiber amplifier, 25 m long, pumped by four wavelength-multiplexed laser diodes near 980 nm; SMFs, standard telecommunication single-mode fibers; OSA, optical spectrum analyzer.

Fig. 8
Fig. 8

Square-pulse spectra on a linear scale measured at (a) 1 km and (b) 2 km at a pump power of P0=2.2 W. (c) Relative power of the first sideband, P1/P0, versus P0 at 1 and 2 km. (d) Experimental (crosses) and calculated (solid curves) relative sideband and pump powers versus P0 at 2 km.

Fig. 9
Fig. 9

Square-pulse spectrum on a logarithmic scale at an input pump power P0=2.35 W after 1 km.

Fig. 10
Fig. 10

Evolution of the recorded MI spectrum as a function of scaled longitudinal variable ξ=γP0z: (a) log scale, (b) linear scale.

Equations (13)

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

Ez=-iβ22Et2+iγ|E|2E,
K=±1/2|β2|Ω[-Ω2-sgn(β2)Ωc2]1/2,
-idE0dz=γ[|E0|2+2(|Es|+|Ea|)]E0+2γEsEaE0* exp(+iΔkz),
-idEsdz=γ[|Es|2+2(|Ea|+|E0|)]Es+2γEa*E02 exp(-iΔkz),
-idEadz=γ[|Ea|2+2(|Es|+|E0|)]Ea+2γEs*E02 exp(-iΔkz),
dηdξ=-4ηasaa sin Φ,
dasdξ=ηaa sin Φ,
daadξ=ηas sin Φ,
dΦdξ=κ+[2η-(as2+aa2)]sin Φ+ηasaa+aaas-4asaacos Φ,
u(z, t)=f0(z)+2 n=1 fn(z)cos[n(t-t0)2].
Φclkw(t)2 0LPc(t-Δβz)dz,
Φcntw2PcL,
TI(t)=sin2ΔΦ(t)2,

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