Abstract

The combined effect of lumped filtering and the gain–loss periodicity in a wavelength-division-multiplexed soliton system is evaluated. We find that even with an ideal dispersion tapering that cancels the effect of the gain–loss dynamics, lumped filtering gives a permanent timing displacement after two-soliton collision. The implications on wavelength-division-multiplexed soliton transmission with a large number of channels is discussed. A statistical approach is used to give an estimate of the timing jitter induced by two-soliton collisions in soliton wavelength-division-multiplexed systems.

© 1998 Optical Society of America

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References

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  1. A. Mecozzi and H. A. Haus, “Effect of filters on soliton interactions in wavelength division multiplexing systems,” Opt. Lett. 17, 988–990 (1992).
    [CrossRef] [PubMed]
  2. L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–369 (1991).
    [CrossRef]
  3. M. Midrio, P. Franco, F. Matera, M. Romagnoli, and M. Settembre, “Wavelength division multiplexed soliton transmission with filtering,” Opt. Commun. 112, 283–288 (1994).
    [CrossRef]
  4. L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7×10 Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–473 (1996).
    [CrossRef]
  5. A. Hasegawa and Y. Kodama, Solitons in Optical Communications, A. Hasegawa, M. Lapp, J. Nishizawa, B. B. Snavely, H. Stark, A. C. Tam, and T. Wilson, eds., Vol. 7 of the Oxford Series in Optical and Imaging Sciences (Clarendon Press, Oxford, 1995).
  6. A. Hasegawa, S. Kumar, and Y. Kodama, “Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems,” Opt. Lett. 21, 39–41 (1996).
    [CrossRef] [PubMed]
  7. 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, 539–541 (1995).
    [CrossRef] [PubMed]
  8. P. V. Mamyshev and L. F. Mollenauer, “Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission,” Opt. Lett. 21, 396–398 (1996).
    [CrossRef] [PubMed]
  9. M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins, and J. R. Sauer, “Four-wave mixing in wavelength-division-multiplexed soliton systems: damping and amplification,” Opt. Lett. 21, 1646–1648 (1996).
    [CrossRef] [PubMed]
  10. P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
    [CrossRef] [PubMed]
  11. 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]
  12. P. K. 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]
  13. L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton–soliton collision,” Opt. Lett. 20, 2060–2062 (1995).
    [CrossRef] [PubMed]

1996 (5)

1995 (2)

1994 (1)

M. Midrio, P. Franco, F. Matera, M. Romagnoli, and M. Settembre, “Wavelength division multiplexed soliton transmission with filtering,” Opt. Commun. 112, 283–288 (1994).
[CrossRef]

1992 (2)

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]

A. Mecozzi and H. A. Haus, “Effect of filters on soliton interactions in wavelength division multiplexing systems,” Opt. Lett. 17, 988–990 (1992).
[CrossRef] [PubMed]

1991 (2)

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–369 (1991).
[CrossRef]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[CrossRef] [PubMed]

Ablowitz, M. J.

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]

Biondini, G.

Chakravarty, S.

Chen, H. H.

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]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–369 (1991).
[CrossRef]

Franco, P.

M. Midrio, P. Franco, F. Matera, M. Romagnoli, and M. Settembre, “Wavelength division multiplexed soliton transmission with filtering,” Opt. Commun. 112, 283–288 (1994).
[CrossRef]

Golovchenko, E. A.

Gordon, J. P.

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, 539–541 (1995).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton–soliton collision,” Opt. Lett. 20, 2060–2062 (1995).
[CrossRef] [PubMed]

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]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–369 (1991).
[CrossRef]

Hasegawa, A.

Haus, H. A.

Heismann, F.

Jenkins, R. B.

Kodama, Y.

Kumar, S.

Mamyshev, P. V.

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7×10 Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–473 (1996).
[CrossRef]

P. V. Mamyshev and L. F. Mollenauer, “Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission,” Opt. Lett. 21, 396–398 (1996).
[CrossRef] [PubMed]

Matera, F.

M. Midrio, P. Franco, F. Matera, M. Romagnoli, and M. Settembre, “Wavelength division multiplexed soliton transmission with filtering,” Opt. Commun. 112, 283–288 (1994).
[CrossRef]

Mecozzi, A.

Menyuk, C. R.

Midrio, M.

M. Midrio, P. Franco, F. Matera, M. Romagnoli, and M. Settembre, “Wavelength division multiplexed soliton transmission with filtering,” Opt. Commun. 112, 283–288 (1994).
[CrossRef]

Mollenauer, L. F.

P. V. Mamyshev and L. F. Mollenauer, “Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission,” Opt. Lett. 21, 396–398 (1996).
[CrossRef] [PubMed]

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7×10 Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–473 (1996).
[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, 539–541 (1995).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton–soliton collision,” Opt. Lett. 20, 2060–2062 (1995).
[CrossRef] [PubMed]

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]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–369 (1991).
[CrossRef]

Neubelt, M. J.

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7×10 Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–473 (1996).
[CrossRef]

Pilipetskii, A. N.

Romagnoli, M.

M. Midrio, P. Franco, F. Matera, M. Romagnoli, and M. Settembre, “Wavelength division multiplexed soliton transmission with filtering,” Opt. Commun. 112, 283–288 (1994).
[CrossRef]

Sauer, J. R.

Settembre, M.

M. Midrio, P. Franco, F. Matera, M. Romagnoli, and M. Settembre, “Wavelength division multiplexed soliton transmission with filtering,” Opt. Commun. 112, 283–288 (1994).
[CrossRef]

Wai, P. K.

P. K. 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]

Wai, P. K. A.

Electron. Lett. (1)

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7×10 Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–473 (1996).
[CrossRef]

J. Lightwave Technol. (3)

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–369 (1991).
[CrossRef]

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]

P. K. 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]

Opt. Commun. (1)

M. Midrio, P. Franco, F. Matera, M. Romagnoli, and M. Settembre, “Wavelength division multiplexed soliton transmission with filtering,” Opt. Commun. 112, 283–288 (1994).
[CrossRef]

Opt. Lett. (7)

Other (1)

A. Hasegawa and Y. Kodama, Solitons in Optical Communications, A. Hasegawa, M. Lapp, J. Nishizawa, B. B. Snavely, H. Stark, A. C. Tam, and T. Wilson, eds., Vol. 7 of the Oxford Series in Optical and Imaging Sciences (Clarendon Press, Oxford, 1995).

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Figures (10)

Fig. 1
Fig. 1

Residual timing displacement in soliton units for two colliding solitons versus the ratio of the collision length to the amplifier spacing when in-line control filters are used. The solid curve corresponds to a normalized filter strength η=0.2 and the dashed curve to η=0.6. The dotted–dashed curve represents comparison with the normalized timing shift |δT|=Ω-2 for the lossless case without filters.

Fig. 2
Fig. 2

Residual timing displacement in soliton units for two colliding solitons versus the ratio of the collision length to the amplifier spacing when in-line control filters are used and with exponential tapering of dispersion. The solid curve is for the center of the collision in the middle of two consecutive amplifiers, and the dashed curve is for the center of the collision at an amplifier position. The dotted–dashed curve represents comparison with the normalized timing shift |δT|=Ω-2 for the lossless case without filters.

Fig. 3
Fig. 3

Residual timing displacement in soliton units for two colliding solitons versus the ratio of the collision length to the amplifier spacing when in-line control filters are used. The solid curve corresponds to filters placed every ten amplifiers, and the dashed curve is for filters placed every five amplifiers. The normalized average filter strength is η=0.2. The other parameters are the same as those of Fig. 1. The dotted–dashed curve represents comparison with the normalized timing shift |δT|=Ω-2 for the lossless case without filters.

Fig. 4
Fig. 4

Timing shift per collision in picoseconds versus the phase of the collision for constant dispersion.

Fig. 5
Fig. 5

Timing shift per collision in picoseconds versus the phase of the collision for exponential tapering of the dispersion. The solid curve is for equal filter spacing and Lper=Lf=50 km; the dashed curve is for Lper=100 km and a filter spacing of 33.3 and 66.6 km that is periodically repeated. The dotted–dashed line is the constant value of the timing shift with exponential tapering and no filters.

Fig. 6
Fig. 6

Timing shift per collision in picoseconds versus the phase of the collision for stepwise approximation of the optimum exponential tapering of dispersion. The solid curve is a three-step approximation, the dashed curve a four-step approximation, and the dotted–dashed curve a 100-step approximation. The length of the steps and the values of the dispersion are chosen by the algorithm of Ref. 6.

Fig. 7
Fig. 7

Average number of collisions for the channel at a lower frequency of the WDM spectrum versus the total number of channels.

Fig. 8
Fig. 8

Root-mean-square timing jitter in picoseconds versus the number of channels for constant dispersion. The dots are for the channels at longer or shorter wavelength, and the circles are for the channel at the central frequency of the WDM comb.

Fig. 9
Fig. 9

Root-mean-square timing jitter in picoseconds versus the number of channels for exponential tapering of dispersion and for the channels at longer or shorter wavelengths. The dots are for equal filter spacing and Lper=Lf=50 km, and the circles are for Lper=100 km and a filter spacing of 33.3 and 66.6 km that is periodically repeated.

Fig. 10
Fig. 10

Root-mean-square timing jitter in picoseconds versus the number of channels for a stepwise dispersion approximating the optimum exponential one for the channel at longer, or shorter, wavelengths of the WDM spectrum. The circles are for a three-step approximation, and the dots are for a four-step approximation. The length of the steps and the values of the dispersion have been chosen by the algorithm of Ref. 6.

Equations (64)

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qz=η(z)2 2qt2+i2 2qt2+iG(z)|q|2q,
G(z)=ΓLA1-exp(-ΓLA) exp(-Γzsz)
qz=η(z)2 2qt2+i Δ(z)2 2qt2+iG(z)|q|2q,
ζ=0zΔ(z)dz.
ζ(zA)=0zAΔ(z)dz=zA.
qζ=η(ζ)2 2qt2+i2 2qt2+iG(ζ)|q|2q,
G(ζ)=G(z)Δ(z),
η(ζ)=η(z)Δ(z),
dδΩdζ=-23 η(ζ)δΩ+C(ζ),
C(ζ)=G(ζ)2Ω ζ dt×sech2(t+Ωζ)sech2(t-Ωζ).
δΩ(ζ)=-ζdζ exp-23 ζζη(ζ)dζC(ζ).
δT=--dζδΩ(ζ).
-dζ-ζdζ=-dζζdζ,
δT=-0dσ-dζC(ζ)exp-23 ηzfN(σ, ζ),
N(σ, ζ)=1ηzf 0ση(ζ+ζ)dζ.
η(z)=k=-ηzfδ(z-kzf),
η(ζ)dζ=η(z)dz,
N(σ, ζ)=n,
nzf-ζσ<(n+1)zf-ζ.
0 dσ exp-23 ηzfN(σ, ζ)
=-ζ+zfn=0 exp-23 ηnzf=-ζ+zf1-exp-23 ηzf-1.
δT=-zf1-exp-23 ηzf-1I1+I2,
I1=-dζC(ζ),
I2=-dζS(ζ)C(ζ),
Ij=-dζ Aj(ζ)2Ω ζ dt sech2(t+Ωζ)sech2(t-Ωζ),
Ij=16zperπ3 Im n=0an,j n3y4sinh2(ny),
an,1=1zper 0zperdζG(ζ)exp(-iknζ),
an,2=1zper 0zperdζζG(ζ)exp(-iknζ)=izper k 0zperdζG(ζ)exp(-ikζ)k=kn,
kn=2πnzper,
an,1=1zper 0zperdzG(z)exp(-iknζ),
an,1=f(n),
an,2=g(n),
f(n)=ΓLAΓLA+in2π,
g(n)=zAΓLA [1-(1+ΓLA+in2π)exp(-ΓLA)][1-exp(-ΓLA)](ΓLA+in2π)2.
an,1=ΓLA1-exp(-ΓLA) j=0N-1 ej-ej+1ΓLA+in2πΔj+1,
an,2=zAΓLA1-exp(-ΓLA) j=0N-1 1(ΓLA+in2πΔj+1)2×Δj+1+ϕj2π (ΓLA+in2πΔj+1)ej-Δj+1+ϕj+12π (ΓLA+in2πΔj+1)ej+1,
ej=exp(-ΓLj-inϕj),
ϕj=0j=02πk=1j Δk(Lk-Lk-1)LAj1.
an,1=0,
an,2=zf/2,n=0izf/(2πn),n1.
an,2=izper2πn 1-L1Lper 1-exp-2iπn L1Lper.
an,1=fnM
an,2=gnM+(M-1)zA2 fnM
an,2=-zA1-exp(-i2πn/M)×1-exp(-ΓLA-i2πn/M)1-exp(-ΓLA) fnM
exp[iϕa(n)]=expi 2πnzper 0zadzΔ(z),
Ij=16zperπ3 Im n=0 exp[iϕa(n)]an,j n3y4sinh2(ny).
zcoll=1.7627Ω-1.
δTc=ichannelsckSi,kδTi,k,
δTc=12 ichannelsckδTi,k,
δTc2-δTc2=14 ichannelsckδTi,k2.
P(T)=12N i=0NNiδ(T-iδT).
kδTi,k=Nc,ik δTi,kNc,i,
kδTi,k2=Nc,ik δTi,k2Nc,i,
k δTi,kNc,iδT(2Ωc,i)¯=0,
k δTi,k2Nc,iδT(2Ωc,i)2¯=128zA2π6 n=13zAan,12η-an,22×n6y(2Ωc,i)8sinh4[ny(2Ωc,i)],
Nc,i(2Ωc,i)=2Ωc,iztotT,
kδTi,k2=Nc,i(2Ωc,i)δT(2Ωc,i)2¯.
δTc,tot2=14 ichannelscNc,i(2Ωc,i)δT(2Ωc,i)2¯.
δTc=δTc,tot21/2.
Ncoll=i=2N N1,i2=N(N-1)(2Ω1,2)ztot4T,
δTtot=[δTc,tot2+δTGH2]1/2.
Ecoll=erfctw2δTc.
uz=η(z)+i2 2ut2+iG(z)(|u|2+|v|2)u,
vz=η(z)+i2 2vt2+iG(z)(|u|2+|v|2),

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