Abstract

In this study we examine the physical mechanism that prevents the use of short optical pulses in high bit-rate transmission systems. Continuum-induced soliton–soliton interaction is the cause of long-range as well as enhanced direct soliton–soliton interaction. We show, by means of a purely analytical approach, that soliton–continuum interaction causes position wiggling of the soliton and that when this effect is applied to a random sequence of solitons it becomes an additional cause of timing jitter.

© 1999 Optical Society of America

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References

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  1. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91 (1992).
    [CrossRef]
  2. H. A. Haus, W. S. Wong, and F. I. Kathri, “Continuum generation by perturbation of soliton,” J. Opt. Soc. Am. B 14, 304 (1997).
    [CrossRef]
  3. V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
    [CrossRef]
  4. 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 (1991).
    [CrossRef]
  5. B. A. Malomed, “Bound states of envelope solitons,” Phys. Rev. E 47, 2874 (1993).
    [CrossRef]
  6. B. A. Malomed, “Propagation of a soliton in a nonlinear waveguide with dissipation and pumping,” Opt. Commun. 61, 192 (1987).
    [CrossRef]
  7. J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170 (1991).
    [CrossRef]
  8. S. Kumar, A. Hasegawa, and Y. Kodama, “Adiabatic soliton transmission in fibers with lumped amplifiers: analysis,” J. Opt. Soc. Am. B 14, 888 (1997).
    [CrossRef]

1997 (2)

1993 (1)

B. A. Malomed, “Bound states of envelope solitons,” Phys. Rev. E 47, 2874 (1993).
[CrossRef]

1992 (1)

1991 (2)

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170 (1991).
[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 (1991).
[CrossRef]

1987 (1)

B. A. Malomed, “Propagation of a soliton in a nonlinear waveguide with dissipation and pumping,” Opt. Commun. 61, 192 (1987).
[CrossRef]

1981 (1)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

Evangelides, S. G.

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 (1991).
[CrossRef]

Gordon, J. P.

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91 (1992).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170 (1991).
[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 (1991).
[CrossRef]

Hasegawa, A.

Haus, H. A.

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

Kathri, F. I.

Kodama, Y.

Kumar, S.

Malomed, B. A.

B. A. Malomed, “Bound states of envelope solitons,” Phys. Rev. E 47, 2874 (1993).
[CrossRef]

B. A. Malomed, “Propagation of a soliton in a nonlinear waveguide with dissipation and pumping,” Opt. Commun. 61, 192 (1987).
[CrossRef]

Mollenauer, L. F.

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 (1991).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170 (1991).
[CrossRef]

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

Wong, W. S.

J. Lightwave Technol. (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 (1991).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170 (1991).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

B. A. Malomed, “Propagation of a soliton in a nonlinear waveguide with dissipation and pumping,” Opt. Commun. 61, 192 (1987).
[CrossRef]

Phys. Rev. E (1)

B. A. Malomed, “Bound states of envelope solitons,” Phys. Rev. E 47, 2874 (1993).
[CrossRef]

Physica D (1)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Growth of the continuum generated by a soliton propagating in a periodically amplified system. The amplifier spacing was za=1 (Za=ZD=50 km), and Γ=10. The solid curve is obtained by integration of Eq. (2), whereas the dashed curve is the total field obtained with the analytical formula for the continuum given by Gordon in Ref. 1.

Fig. 2
Fig. 2

Plot of the continuum energy E as a function of the normalized amplifier spacing za for the case of Za=10 km, group-velocity dispersion β2=-20 ps2/km, and a propagation distance z=100 (Z=1000 km). The energy is computed by integration of Eq. (3b).

Fig. 3
Fig. 3

Deviation from the initial soliton separation [Δ(z)-Δ0] as a function of the initial separation (Δ0). The solid curve was obtained by numerical solution of Eq. (2); the dashed curve, from the analytical result reported in Eqs. (9). The amplifier spacing was za=1 (Za=ZD=50 km), and Γ=10. The analytical theory accounts only for the soliton–continuum interaction, whereas the simulation accounts for direct soliton–soliton interaction as well. This explains the discrepancy between the curves for Δ0<10.

Fig. 4
Fig. 4

Timing jitter, δq21/2, of the central soliton in the data stream. The initial soliton–soliton separation (time slot) was Δ0=8, whereas the other parameters are the same as those in the previous figures. Both the curves show the effect of the interaction with the continuum emitted by all the solitons in the stream. In particular, the dotted–dashed curve also includes the contribution due to the direct interaction with the adjacent solitons, whereas one obtains the solid curve by always keeping the adjacent time slots empty […010…]. The statistics were evaluated over 500 numerical random pattern realizations, and the number of bits used for each temporal pattern was 2Mmax+1, where Mmax=Int(zTΔω1/Δ0)=12. The analytical evaluation of δq21/2 is obtained from Eq. (10), and the plot exactly coincides with the fully numerical estimation (solid curve). The dotted line indicates the threshold of factor Q=6.1.

Equations (34)

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iQZ-β222QT2+γNL|Q|2Q
=-iα-nδ(Z-nZA)G0Q.
iuz+122ut2+[1+a(z)]|u|2u=0.
iusiz+122usit2+|usi|2usi
=-[1+a(z)](2|usi|2upj+usi2upj*),
iupiz+122upit2+(2|usi|2upi+usi2upi*)
=-a(z)(|usi|2-1)usi.
[Ai(z), Ωi(z), qi(z), ϕi(z)]
[Ai(z), -Ωi(z), -qi(z), ϕi(z)],
up(z, t)=C exp[-i(ω|t|+12ω2z-ϕ)]|t|<ωzup(z, t)=0|t|>ωz,
ω-=Ω±Δω,Δω=4πza-A2,
C=zaπ4(A2+Δω2)sechπ2ΔωΔω1+2πΓza1/2,
ϕ=-Δωq0+tan-12πΓza+2 tan-1AΔω+π2.
R(us, up)=i[1+a(z)]CA-1 sech[A(t-q)]3×[sin(ϕs-ϕp)+3i cos(ϕs-ϕp)].
dAdz=[1+a(z)]A2C- sech3(x)sin(ϕs-ϕp)dx,
dΩdz=3[1+a(z)]A2C- tanh(x)sech3(x)×cos(ϕs-ϕp)dx,
dqdz=-Ω+[1+a(z)]A2C-x sech3(x)×sin(ϕs-ϕp)dx,
dϕsdz=A2-Ω22+qdΩdz+[1+a(z)]AC×- sech3(x)[1-x tanh(x)]×cos(ϕs-ϕp)dx,
dAdz=A2I1C2[1+a(z)]cos2πzaz+Ψ,
dΩdz=-AI2CΔω2[1+a(z)]sin2πzaz+Ψ,
dqdz=-Ω-A2I3C[1+a(z)]cos2πzaz+Ψ,
dϕdz=A2-Ω22+ΩI3C[1+a(z)]cos2πzaz+Ψ+I4CA[1+a(z)]sin2πzaz+Ψ.
Ψ=ΔωΔ0-tan-12πΓza-2 tan-1AΔω,
I1=I2=-dx sech(x)3 cosΔωAx,
I3=-dx sech(x)3x sinΔωAx,
I4=-dx[1-x tanh(x)]sech(x)3 cosΔωAx.
A(z)=A(0),
Ω(Nza)=Ω(0)+b12(N-NC)za,
q(Nza)=q(0)+Ω(0)(N-NC)za+b2(N-NC)za+b1(N-NC)2za2,
ϕ(Nza)=ϕ(0)+A(0)22Nza,
b1=14I2A(0)CΔω sinΔωΔ0-2 tan-1A(0)Δω,
b2=CI1 cosΔωΔ0-2 tan-1A(0)Δω.
Δ0=nπ+2 tan-1A(0)ΔωΔωn=1,, ,
δq(z)21/2=12m=1Mq(z, mΔ0)21/2=π16za21/2[A(0)2+Δω2]sechπ2ΔωI2A(0)1+2πΓza1/2×m=1Mmax sinmΔ0Δω-2 tan-1A(0)Δω2×z-mΔ0Δω41/2,

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