Abstract

Sliding-frequency filters are a standard means of suppressing the Gordon–Haus jitter in long-haul soliton transmission. For narrow solitons, the frequency sliding can be naturally provided by the Raman effect. The passbands of the sliding-frequency filters can then stay closer to the soliton’s central frequency, thereby reducing filter losses, which in turn reduces the necessary compensatory gain and, consequently, the timing jitter. We analyze the dynamics of solitons in a system with sliding-frequency filters, Raman-induced self-frequency downshift, losses, periodic amplification, and third-order dispersion. An optimum mode of operation, based on an analytic approximation, is found and checked against numerical simulations.

© 1998 Optical Society of America

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References

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  1. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
    [CrossRef] [PubMed]
  2. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, London, 1995).
  3. J. P. Gordon, Opt. Lett. 11, 662 (1986).
    [CrossRef] [PubMed]
  4. B. A. Malomed, Opt. Commun. 61, 192 (1987).
    [CrossRef]
  5. J. P. Gordon and H. Haus, Opt. Lett. 11, 665 (1986).
    [CrossRef] [PubMed]
  6. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, Opt. Lett. 16, 841 (1991); Y. Kodama and A. Hasegawa, Opt. Lett. 17, 31 (1992).
    [CrossRef] [PubMed]
  7. N. J. Smith and N. J. Doran, J. Opt. Soc. Am. B 12, 1117 (1995).
    [CrossRef]
  8. E. A. Golovchenko, A. N. Pilipetskii, C. R. Menyuk, J. P. Gordon, and L. F. Mollenauer, Opt. Lett. 20, 539 (1995).
    [CrossRef] [PubMed]
  9. B. A. Malomed, J. Opt. Soc. Am. B 13, 677 (1996).
    [CrossRef]
  10. B. A. Malomed, Phys. Rev. E 47, 2874 (1993).
    [CrossRef]
  11. B. A. Malomed, J. Opt. Soc. Am. B 11, 1261 (1994).
    [CrossRef]
  12. S. Burtsev and D. J. Kaup, J. Opt. Soc. Am. B 14, 627 (1997).
    [CrossRef]
  13. P. V. Mamyshev and L. F. Mollenauer, Opt. Lett. 19, 2083 (1994).
    [CrossRef] [PubMed]
  14. A. Mecozzi, Opt. Lett. 20, 1859 (1995); A. Mecozzi, M. Midrio, and M. Romagnoli, Opt. Lett. 21, 402 (1996).
    [CrossRef] [PubMed]
  15. J. K. Lucek and K. J. Blow, Phys. Rev. A 45, 6666 (1992).
    [CrossRef] [PubMed]
  16. D. J. Kaup and B. A. Malomed, J. Opt. Soc. Am. B 12, 1656 (1995).
    [CrossRef]
  17. K. J. Blow, N. J. Doran, and D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988).
    [CrossRef]
  18. Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
    [CrossRef]
  19. P. K. A. Wai, H. H. Chen, and Y. C. Lee, Phys. Rev. A 41, 426 (1990).
    [CrossRef] [PubMed]
  20. M. Desaix, D. Anderson, and M. Lisak, Opt. Lett. 15, 18 (1990); P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, Opt. Lett. 12, 628 (1987).
    [CrossRef] [PubMed]
  21. J. Satsuma and N. Yajima, Prog. Theor. Phys. Suppl. 55, 284 (1974).
    [CrossRef]
  22. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989).
  23. K. J. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
    [CrossRef]
  24. B. A. Malomed and R. S. Tasgal, in Nonlinear Guided Waves and Their Applications, Vol. 15 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 200.

1997

1996

1995

1994

1993

B. A. Malomed, Phys. Rev. E 47, 2874 (1993).
[CrossRef]

1992

1990

P. K. A. Wai, H. H. Chen, and Y. C. Lee, Phys. Rev. A 41, 426 (1990).
[CrossRef] [PubMed]

1989

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

K. J. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[CrossRef]

1988

1987

B. A. Malomed, Opt. Commun. 61, 192 (1987).
[CrossRef]

1986

1974

J. Satsuma and N. Yajima, Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Blow, K. J.

J. K. Lucek and K. J. Blow, Phys. Rev. A 45, 6666 (1992).
[CrossRef] [PubMed]

K. J. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[CrossRef]

K. J. Blow, N. J. Doran, and D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988).
[CrossRef]

Burtsev, S.

Chen, H. H.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, Phys. Rev. A 41, 426 (1990).
[CrossRef] [PubMed]

Doran, N. J.

Evangelides, S. G.

Golovchenko, E. A.

Gordon, J. P.

Haus, H.

Kaup, D. J.

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Lee, Y. C.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, Phys. Rev. A 41, 426 (1990).
[CrossRef] [PubMed]

Lucek, J. K.

J. K. Lucek and K. J. Blow, Phys. Rev. A 45, 6666 (1992).
[CrossRef] [PubMed]

Malomed, B. A.

Mamyshev, P. V.

Menyuk, C. R.

Mollenauer, L. F.

Pilipetskii, A. N.

Satsuma, J.

J. Satsuma and N. Yajima, Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Smith, N. J.

Wai, P. K. A.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, Phys. Rev. A 41, 426 (1990).
[CrossRef] [PubMed]

Wood, D.

K. J. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[CrossRef]

K. J. Blow, N. J. Doran, and D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

IEEE J. Quantum Electron.

K. J. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

B. A. Malomed, Opt. Commun. 61, 192 (1987).
[CrossRef]

Opt. Lett.

Phys. Rev. A

J. K. Lucek and K. J. Blow, Phys. Rev. A 45, 6666 (1992).
[CrossRef] [PubMed]

P. K. A. Wai, H. H. Chen, and Y. C. Lee, Phys. Rev. A 41, 426 (1990).
[CrossRef] [PubMed]

Phys. Rev. E

B. A. Malomed, Phys. Rev. E 47, 2874 (1993).
[CrossRef]

Prog. Theor. Phys. Suppl.

J. Satsuma and N. Yajima, Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Rev. Mod. Phys.

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Other

M. Desaix, D. Anderson, and M. Lisak, Opt. Lett. 15, 18 (1990); P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, Opt. Lett. 12, 628 (1987).
[CrossRef] [PubMed]

A. Mecozzi, Opt. Lett. 20, 1859 (1995); A. Mecozzi, M. Midrio, and M. Romagnoli, Opt. Lett. 21, 402 (1996).
[CrossRef] [PubMed]

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, Opt. Lett. 16, 841 (1991); Y. Kodama and A. Hasegawa, Opt. Lett. 17, 31 (1992).
[CrossRef] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989).

B. A. Malomed and R. S. Tasgal, in Nonlinear Guided Waves and Their Applications, Vol. 15 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 200.

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, London, 1995).

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Figures (5)

Fig. 1
Fig. 1

Parts (a) and (b) show, respectively, the optimal filter strength σ and the optimal ramp rate Ω, both plotted versus the soliton width. In (a) and for the black curves in (b), the data come from numerical optimization of Eq. (7) for the radiation energy; the gray curves in (b) were obtained by plugging the optimal filter strengths from Fig. 1(a) into the analytic formula (10) or the ramp rate.

Fig. 2
Fig. 2

Comparison of the analytic predictions (solid curve) and results of PDE simulations (points). The numerical results were obtained by adjusting the ramp rate Ω until the soliton’s width reached a steady state with the center of the SFF’s passband placed at the soliton’s central frequency (which is not the optimal operation mode, but is close to it). The coefficients were fixed (=0.05, γ0=0.001, za=5.0, σ=0.05) except for the amplification (γ1), which was varied from 0.1 to 0.5, producing a range of widths.

Fig. 3
Fig. 3

Propagation of the pulse along the fiber in direct simulations of Eq. (1). For the parameters of the equation, the following values were taken: Ω=-0.1, σ=0.03, za=5.0, γ1 =0.155, γ0=0.005, and =0.045.

Fig. 4
Fig. 4

Variation of the soliton’s central frequency between the amplifiers that underlies the GH jitter. The passive loss is γ0 =0.046, the Raman coefficient is =0.006, and the total soliton energy varies around 1.65; this corresponds approximately to a 1.2-ps soliton in the reduced-dispersion (-0.05 ps2/km) fiber. The amplifier spacing is za=1, and the ramp rate is Ω=-0.1 in all cases. The top curve (gray) is the highest amplification and filter strength, γ1=0.25, σ=0.28. The middle curve (black) is the predicted optimum, γ1=0.14, σ=0.09. The bottom curve is the lowest amplification and filtering, γ1=0.117, σ=0.05.

Fig. 5
Fig. 5

Two simulations, with (a) zero (k=0) and (b) large (k=0.5) third-order dispersion. The other parameters are =0.006, γ0=0.046, γ1=0.14, za=1.0, σ=0.09, Ω=-0.1, and no noise in the system, and runs are for 25 amplification periods. Note that, aside from the changes from the soliton’s moving into a higher-dispersion range (owing to the large k and frequency shift), neither the soliton’s shape nor its instantaneous dynamics are significantly affected by third-order dispersion.

Equations (19)

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iuz+12 uττ+|u|2u=i k6 uτττ+(|u|2)τu+i-γ0+γ1n=-+δ(z-nza)u+iσ(τ+iΩz)2u.
γdist=γ0-γ1za.
u=η sech[η(τ-y)]exp{-i[a+b(τ-y)]}.
ddz η=-2ηγ0+ση23+(b-Ωz)2,
ddz (b-Ωz)=-Ω-815 η4-43 η2σ(b-Ωz),
ddz I(ω)=-2I(ω)[γ0+σ(ω-Ωz)2].
ηpost-amplifier=(1+2γ1)ηpreamplifier.
I(ω)=-1π ln1-sin2(πγ1)sech2[πη-1(ω-b)].
Eradiation/Esoliton
=12η m=0-(-1/π)ln{1-sin2(πγ1)sech2[π(ω-b)/η]}×exp{[2γ0+4σ(η2/3+c2)]mzα-23 σΩ (ω-Ωmzα)3}dω,
cb-Ωz.
Iestimate=(γ1)2(2π)exp{-[π(ω-b)/η]2},
Eradiation/Esoliton
=γ0+σ13 η2+c22za0dz×exp2γ0+4σ13 η2+c2z-12 ln[1+2η2σ/π2z]-2σzc-12 Ωz2×[1+2(η2σ/π2)z]-1-16 σΩ2z3.
ddz ln(2η)=-2γ0+ση23+c2,
ddz c=-Ω-815 η4-43 η2σc.
c=-Ω+(8/15)η4(4/3)ση2.
Ωoptimalsimplestmodel=-827 σ2η6+815 2η8.
λ=-43 ση2±6η2 Ω2-815 η42.

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