Abstract

A singular perturbation method is used to analyze the effect of sliding-frequency guiding filters on an optical soliton, which has been proposed to be used as a bit carrier in fiber-optics communication systems. We find that there is a broad range of physical parameters, only inside of which would the sliding-frequency filter scheme operate stably. The lower limit (in soliton energy) of this parameter regime was found earlier by Mollenauer et al. [Opt. Lett. 17, 1575 (1992)] and by Kodama et al. [Opt. Lett. 18, 1311 (1993)] and is determined by whether the soliton will continue to stay in synchronization with the array of filters. The upper limit is determined when the comoving dispersive waves that are continually being generated by the filtering are no longer decaying and instead start to grow and generate, finally, a secondary soliton. This upper limit was discovered recently in both experiments and numerical simulations by Mamyshev and Mollenauer [Opt. Lett. 15, 2083 (1994)]. We have found a simple analytical estimate of this upper limit by the use of a singular perturbation method. Our analytical results agree well with the numerical and experimental findings of Mamyshev and Mollenauer.

© 1997 Optical Society of America

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References

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  1. A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443 (1990); “Guiding-center soliton,” Phys. Rev. Lett. 66, 161 (1991).
    [CrossRef] [PubMed]
  2. A. Hasegawa and Y. Kodama, Solitons in Optical Communication (Oxford U. Press, Oxford, 1995).
  3. L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 15, 1203 (1990).
    [CrossRef] [PubMed]
  4. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665 (1986).
    [CrossRef] [PubMed]
  5. A. Mecozzi, J. P. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841 (1991).
    [CrossRef] [PubMed]
  6. Y. Kodama and A. Hasegawa, “Generation of asymptotically stable solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31 (1992).
    [CrossRef] [PubMed]
  7. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575 (1992).
    [CrossRef] [PubMed]
  8. L. F. Mollenauer, E. Lichtman, M. N. Neubelt, and G. T. Harvey, “Demonstrations, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbits, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910 (1993).
    [CrossRef]
  9. Y. Kodama and S. Wabnitz, “Analysis of soliton stability and interactions with sliding filters,” Opt. Lett. 18, 1311 (1993).
    [CrossRef]
  10. P. V. Mamyshev and L. F. Mollenauer, “Stability of soliton propagation with sliding-frequency guiding filters,” Opt. Lett. 15, 2083 (1994).
    [CrossRef]
  11. E. A. Golovchenko, A. N. Pilipetskii, C. R. Menyuk, J. P. Gordon, and L. F. Mollenauer, “Soliton propagation with up- and down-sliding frequency guiding filters,” Opt. Lett. 20, 539 (1995).
    [CrossRef] [PubMed]
  12. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689 (1990).
    [CrossRef] [PubMed]
  13. D. J. Kaup, “Second-order perturbation for solitons in optical fibers,” Phys. Rev. A 44, 4582 (1991).
    [CrossRef] [PubMed]
  14. Y. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
    [CrossRef]
  15. D. J. Kaup, “A perturbation expansion for the Zakharov–Shabat inverse scattering transform,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 31, 121 (1976).
    [CrossRef]

1995 (1)

1994 (1)

P. V. Mamyshev and L. F. Mollenauer, “Stability of soliton propagation with sliding-frequency guiding filters,” Opt. Lett. 15, 2083 (1994).
[CrossRef]

1993 (2)

L. F. Mollenauer, E. Lichtman, M. N. Neubelt, and G. T. Harvey, “Demonstrations, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbits, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910 (1993).
[CrossRef]

Y. Kodama and S. Wabnitz, “Analysis of soliton stability and interactions with sliding filters,” Opt. Lett. 18, 1311 (1993).
[CrossRef]

1992 (2)

1991 (2)

A. Mecozzi, J. P. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841 (1991).
[CrossRef] [PubMed]

D. J. Kaup, “Second-order perturbation for solitons in optical fibers,” Phys. Rev. A 44, 4582 (1991).
[CrossRef] [PubMed]

1990 (3)

1989 (1)

Y. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

1986 (1)

1976 (1)

D. J. Kaup, “A perturbation expansion for the Zakharov–Shabat inverse scattering transform,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 31, 121 (1976).
[CrossRef]

Cohen, L. G.

Evangelides, S. G.

Golovchenko, E. A.

Gordon, J. P.

Harvey, G. T.

L. F. Mollenauer, E. Lichtman, M. N. Neubelt, and G. T. Harvey, “Demonstrations, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbits, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910 (1993).
[CrossRef]

Hasegawa, A.

Haus, H. A.

Kaup, D. J.

D. J. Kaup, “Second-order perturbation for solitons in optical fibers,” Phys. Rev. A 44, 4582 (1991).
[CrossRef] [PubMed]

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

D. J. Kaup, “A perturbation expansion for the Zakharov–Shabat inverse scattering transform,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 31, 121 (1976).
[CrossRef]

Kivshar, Y.

Y. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Kodama, Y.

Lai, Y.

Lichtman, E.

L. F. Mollenauer, E. Lichtman, M. N. Neubelt, and G. T. Harvey, “Demonstrations, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbits, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910 (1993).
[CrossRef]

Malomed, B. A.

Y. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Mamyshev, P. V.

P. V. Mamyshev and L. F. Mollenauer, “Stability of soliton propagation with sliding-frequency guiding filters,” Opt. Lett. 15, 2083 (1994).
[CrossRef]

Mecozzi, A.

Menyuk, C. R.

Mollenauer, L. F.

E. A. Golovchenko, A. N. Pilipetskii, C. R. Menyuk, J. P. Gordon, and L. F. Mollenauer, “Soliton propagation with up- and down-sliding frequency guiding filters,” Opt. Lett. 20, 539 (1995).
[CrossRef] [PubMed]

P. V. Mamyshev and L. F. Mollenauer, “Stability of soliton propagation with sliding-frequency guiding filters,” Opt. Lett. 15, 2083 (1994).
[CrossRef]

L. F. Mollenauer, E. Lichtman, M. N. Neubelt, and G. T. Harvey, “Demonstrations, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbits, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910 (1993).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 15, 1203 (1990).
[CrossRef] [PubMed]

Moores, J. P.

Neubelt, M. J.

Neubelt, M. N.

L. F. Mollenauer, E. Lichtman, M. N. Neubelt, and G. T. Harvey, “Demonstrations, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbits, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910 (1993).
[CrossRef]

Pilipetskii, A. N.

Simpson, J. R.

Wabnitz, S.

Electron. Lett. (1)

L. F. Mollenauer, E. Lichtman, M. N. Neubelt, and G. T. Harvey, “Demonstrations, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbits, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910 (1993).
[CrossRef]

Opt. Lett. (9)

P. V. Mamyshev and L. F. Mollenauer, “Stability of soliton propagation with sliding-frequency guiding filters,” Opt. Lett. 15, 2083 (1994).
[CrossRef]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 15, 1203 (1990).
[CrossRef] [PubMed]

A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443 (1990); “Guiding-center soliton,” Phys. Rev. Lett. 66, 161 (1991).
[CrossRef] [PubMed]

A. Mecozzi, J. P. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841 (1991).
[CrossRef] [PubMed]

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31 (1992).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

Y. Kodama and S. Wabnitz, “Analysis of soliton stability and interactions with sliding filters,” Opt. Lett. 18, 1311 (1993).
[CrossRef]

E. A. Golovchenko, A. N. Pilipetskii, C. R. Menyuk, J. P. Gordon, and L. F. Mollenauer, “Soliton propagation with up- and down-sliding frequency guiding filters,” Opt. Lett. 20, 539 (1995).
[CrossRef] [PubMed]

Phys. Rev. A (2)

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

D. J. Kaup, “Second-order perturbation for solitons in optical fibers,” Phys. Rev. A 44, 4582 (1991).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

Y. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

D. J. Kaup, “A perturbation expansion for the Zakharov–Shabat inverse scattering transform,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 31, 121 (1976).
[CrossRef]

Other (1)

A. Hasegawa and Y. Kodama, Solitons in Optical Communication (Oxford U. Press, Oxford, 1995).

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Figures (3)

Fig. 1
Fig. 1

Stability region in the plane of normalized characteristics: filter strength β2 and the sliding rate β0, obtained by numerical and analytical methods. Filled squares (upper limit) and filled circles (lower limit) present numerical results according to Ref. 10. Solid curves denote limits derived analytically by taking into account only the transient part of the dispersive waves.

Fig. 2
Fig. 2

Stability region in the plane of normalized characteristics: filter strength β2 and sliding rate β0, obtained by numerical and analytical methods. Filled squares (upper limit) and filled circles (lower limit) present numerical results according to Ref. 10. Solid curves denote limits derived analytically by taking into account both the transient and the static parts of the dispersive waves.

Fig. 3
Fig. 3

Ratio Emax/Emin of the allowable soliton energy for stable transmission. Filled squares (from bit-error-rate measurements) and open squares (from the numerical simulation of an isolated pulse) according to Ref. 10. Asterisks are singular perturbation method results according to Fig. 2.

Equations (61)

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iqz+qtt+2q2q*=R,
R=i[δq+β2(t+iβ0z)2q],
β2=ZdZa1(FSR/t0)2r(1-r)2,
β0=4πt0fZd,
δ=δRZd,
iqz+qtt=i[δq+β2(t+iβ0z)2q].
q(z)=q0 exp[δz1-β2β02z13/3],
q=q0+q1+2q2+.
q0=A exp(iα)cosh θ,
θ=2η(t-t¯),
α=-2ξ(t-t¯)+α¯.
A=2η,
t¯=-4ξz+t¯0,
α¯=4(η2+ξ2)z+α¯0,
iz|v+4η2L|v=|F,
L=σ3(θ2-1)+2cosh2 θ(2σ3+iσ2),
|v=exp(-iα)(q1+q2+)exp(iα)(q1*+q2*+).
ηz=-i4ϕe|σ3|Fext,
ξz=14ϕo|σ3|Fext,
2ξ(t¯z+4ξ)+(α¯z-4η2-4ξ2)=14ηχ|σ3|Fext,
t¯z+4ξ=-i8η2θϕe|σ3|Fext,
u|σ3|v=- dθu(θ)˜σ3v(θ)
|ϕe=1cosh θ1-1,
|ϕo=sinh θcosh2 θ11,
|χ=θ tanh θ-1cosh θ11,
|θϕe=θ|ϕe
|ν=1cosh3 θ1-1,
|Fext=exp-iαR-expiαR*,
|Fext=|Fext(1)+|Fext(2)+,
12η|Fext(1)=iδσ3|ϕe+iβ2{8iηξσ3|ϕo+4[η2-(ξ)2]σ3|ϕe-8η2σ3|ν},
|Fext(2)=iδ|v+4iβ2[η2θ2|v-2iηξσ3θ|v-(ξ)2|v],
|F=|Fext+2iηz|χ-2ξz|θϕe+(t¯z+4ξ)[4ξη|ϕe-4iη2|ϕo]+(α¯z-4ξ2-4η2)2η|ϕe-θηξz|σ3v-i θηηz|vθ+(t¯z+4ξ)×[2iη|vθ+2ξ|σ3v]+(α¯z-4ξ2-4η2)|σ3v.
ξ=ξ-(1/2)β0z1.
|v=η - dk(g|ψ, k+g¯|ψ¯, k),
|ψ, k=exp(ikθ)1-2ik exp(-θ)(k+i)2 cosh θ01+exp(ikθ)(k+i)2 cosh2 θ11,
|ψ¯, k=σ1|ψ, k,
g=g(kη+ξ, z),
g¯=g¯(kη-ξ, z).
zg-4iη2(k2+1)g=-1i2πa2ηϕ¯, k|σ3|Fext.
g¯(kη-ξ, z)=g*(-kη+ξ, z).
ϕ, k|=exp(-ikθ)1-2ik exp(θ)(k+i)2 cosh θ1˜0+exp(-ikθ)(k+i)2 cosh2 θ1˜1
ϕ¯, k|=ϕ, k|σ1.
z1η=2δη-8β2η[(ξ)2+η2/3],
z1ξ=-β0/2-163β2η2ξ,
zα¯=4ξ2+4η2,
zt¯=-4ξ.
β2β02δ3<6427.
η6-3δ4β2η4+27β021024β22=0.
ξa=-3β032β2ηa2.
ξ=(1/2)β0z1+ξa
zg-iΩg=G,
Ω(k, z)=4η2(k2+1)-2η(k2+1)(kηz+ξz)-iδ+i4β2(ηk+ξ)2,
G[k, ξ(z)]=-k+i(k-i)cosh[(π/2)k]×δ+43β2[(ηk+ξ)2-4(ξ)2].
gˆz+gˆ[-is2-δ+β2(s+β0z)2]=0,
g=expi 0z Ω[k(z)]dz0zG[k(z)]×exp-i 0z Ω[k(z)]dzdz.
g=i GΩ-i G0Ω0expi 0z Ωdz+O(),
i 0z Ω[k(z)]dz
=4iηa2(k2+1)z+2iηaβ0z2+i32β02z3+i2arctan k-i2arctan[kβ0z/(2ηa)]+[δ-4β2(ξa+kηa)2]z-22β0β2(ξa+ηak)z2-3β02β2z3,
δ=β2/3+9β02/(16β2).
β0<8/27β2,
Re[g(k=0, z)]π/2,0z100,

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