Abstract

Analytic expressions for four-wave-mixing terms in an ideal, lossless wavelength–division-multiplexed soliton system are derived with an asymptotic expansion of the N-soliton solution of the nonlinear Schrödinger equation. The four-wave contributions are shown to grow from a vanishing background and then to decay. Their importance becomes evident in real, nonideal fibers, where they grow by an order of magnitude and equilibrate to a stable value as an effect of periodic amplification.

© 1997 Optical Society of America

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References

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  1. J. P. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. R. B. Jenkins, Ph.D. dissertation (University of Colorado, Boulder, Colorado, 1995).
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    [CrossRef] [PubMed]
  7. P. V. Mamyshev and L. F. Mollenauer, Opt. Lett. 21, 396 (1996).
    [CrossRef] [PubMed]
  8. M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins, and J. R. Sauer, Opt. Lett. 21, 1646 (1996).
    [CrossRef] [PubMed]
  9. S. Chakravarty, M. J. Ablowitz, J. R. Sauer, and R. B. Jenkins, Opt. Lett. 20, 136 (1995).
    [CrossRef] [PubMed]
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1995).
  11. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, 1981).
  12. L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, J. Lightwave Technol. 9, 362 (1991).
    [CrossRef]

1996 (2)

1995 (2)

1992 (2)

1991 (2)

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, Opt. Lett. 16, 1841 (1991).
[CrossRef] [PubMed]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, J. Lightwave Technol. 9, 362 (1991).
[CrossRef]

1986 (1)

Ablowitz, M. J.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1995).

Biondini, G.

Chakravarty, S.

Evangelides, S. G.

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, J. Lightwave Technol. 9, 362 (1991).
[CrossRef]

Gordon, J. P.

Hasegawa, A.

Haus, H. A.

Jenkins, R. B.

Kodama, Y.

Lai, Y.

Mamyshev, P. V.

Mecozzi, A.

Mollenauer, L. F.

Moores, J. D.

Sauer, J. R.

Segur, H.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, 1981).

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Figures (10)

Fig. 1
Fig. 1

Simultaneous, three-soliton collision: A1=A2=A3 =1, Ω1=-11, Ω2=0, Ω3=17, and T1=T2=T3=0.

Fig. 2
Fig. 2

Fourier spectrum relative to the collision shown in Fig. 1.

Fig. 3
Fig. 3

Nine four-wave frequencies generated during the collision shown in Fig. 1. The inset shows the location of the soliton contributions.

Fig. 4
Fig. 4

Nonsimultaneous three-soliton collision: A1=A2 =A3=1, Ω1=-10, Ω2=0, Ω3=10, T1=0, T2=10, and T3=0.

Fig. 5
Fig. 5

Fourier spectrum relative to the collision shown in Fig. 1.

Fig. 6
Fig. 6

Inverse transform of the frequency components around Ω3 during the 1 and 2 collision of Fig. 4. The inset shows a comparison between the FWM component as determined numerically (solid curve) and analytically (dashed curve).

Fig. 7
Fig. 7

Location of the maxima of |qˆ| for the collision process shown in Fig. 4. The O() frequency shifts of the colliding solitons are visible, together with the corresponding frequency oscillations induced in the third channel.

Fig. 8
Fig. 8

Same as Fig. 7, but for a simultaneous collision, i.e., T1=T2=T3=0.

Fig. 9
Fig. 9

Nonsimultaneous, three-soliton collision in the presence of damping and periodic amplification: A1=A2=A3=1, Ω1 =-6, Ω2=0, Ω3=6, T1=17.5, T2=7.5, and T3=-17.5. The loss coefficient and the amplifier distance, expressed in nondimensional units, are Γ=10 and za=0.2, respectively.

Fig. 10
Fig. 10

Frequency spectrum relative to the collision shown in Fig. 9.

Equations (45)

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iqz+½qtt+|q|2q=0,
q(z, t)=j,k=1N(Q-1)jk,
Qjk=exp(-iχj-Sj)+exp(-iχk+Sk)Aj+Ak+i(Ωj-Ωk)
Sj(z, t)=Aj(t-Tj-Ωjz),
χj(z, t)=Ωjt-½(Ωj2-Aj2)z+ϕj.
=maxj,k=1,,Nkj|(Aj+Ak)/(Ωj-Ωk)|
q(z, t)=q(0)(z, t)+q(1)(z, t)+q(2)(z, t)+,
q(0)(z, t)=j=1Nqj(z, t)j=1NAj exp(iχj)sech Sj.
qˆ(z, ω)Fω[q(z, t)]=-+dt exp(-iωt)q(z, t),
qˆj(z, ω)πAjexp(iθj)sechπ2Aj(ω-Ωj),
θj(z, ω)=½(Ωj2+Aj2)z-Ωjωz-Tj(ω-Ωj)+ϕj.
D-1MD-1=j,k=1kjNAjAk exp[i(χj+χk)]sech Sj sech Sk×exp(-iχj-Sj)+exp(-iχk+Sk)Aj+Ak+i(Ωj-Ωk)ij,k=1kjN1Ωj-Ωkqjqk[exp(-iχj)sinh Sj-exp(-iχk)sinh Sk]+j,k=1kjNAj+Ak(Ωk-Ωj)2qjqk[exp(-iχj)cosh Sj+exp(-iχk)cosh Sk].
q(1)(z, t)=2ij,k=1kjNAkΩj-Ωkqj tanh Sk.
-+dt exp(-iωt)sech t tanh(t+Δ)
π sech(½πω)cosh Δ-exp(iΔω)sinh Δ.
qˆ(1)(z, ω)=2πi exp(-iθj)j,k=1kjN1Ωj-Ωk×sech[½π(ω-Ωj)]×cosh δjk-exp[iδjk(ω-Ωj)]sinh δjk,
δjk(z)Sk-Sj=(Ωj-Ωk)z+Tj-Tk
q(2)(z, t)=2j,k=1kjN AkAk+Aj(Ωk-Ωj)2qj+j,k,l=1k,ljNAjAkAl(Ωl-Ωj)(Ωk-Ωj)sech Sk×sech Sl sech Sj×{exp(iχj)cosh Δkl-+exp(iχk)cosh Δkj++exp(iχl)cosh Δlj++exp[i(χk+χl-χj)]},
qFWM(z, t)=¼2j,k,l=1k,ljN1(k-j)(l-j)×exp[i(χk+χl-χj)]×sech Sk sech Sl sech Sj,
qFWM(0, t)=¼2j,k,l=1k,ljN1(k-j)(l-j)×exp[i(Ωk+Ωl-Ωj)t]sech3t.
-+dt exp(-iωt)sech2 t sech(t+Δ)
π sech(½πω)I(Δ, ω),
I(Δ, ω)=[cosh Δ+iω sinh Δ-exp(iωΔ)]/sinh2 Δ.
qˆFWM(0, ω)=π2j,k,l=1k,ljN1(k-j)(l-j)×sech[½π(ω-Ωklj)]×[1+(ω-Ωklj)2],
qˆ(z, ω)π sech[½π(ω-Ω)]×{exp(iθ3)+2×exp(iθ221)sech2[½δ21(z)]},
|qˆ(z, ω)|π sech[½π(ω-Ω)]×{1+2 sech2[½δ21(z)]×cos[Ω2z+δ32(z)(ω-Ω)]},
ω=Ω-½(/π)2δ32(z)sech2[½δ21(z)]×sin[Ω2z+δ32(z)(ω-Ω)],
Fα[sech ζ tanh(ζ+Δ)],Fα[sech2 ζ sech(ζ+Δ)],
Fα[f(ζ)]-+dζ exp(-iαζ)f(ζ).
[1+exp(πα)]-+dζ exp(-iαζ)f(ζ)
=2πi{Res[exp(-iαζ)f(ζ)]ζ=½πi
+Res[exp(-iαζ)f(ζ)]ζ=-Δ+½πi}.
Res[exp(-iαζ)sech ζ tanh(ζ+Δ)]ζ=½πi
=-i exp(½πα)tanh Δ,
Res[exp(-iαζ)sech ζ tanh(ζ+Δ)]ζ=-Δ+½πi
=i exp(½πα+iαΔ)csch Δ.
Fα[sech ζ tanh(ζ+Δ)]
=π sech(½πα)×cosh Δ-exp(iαΔ)sinh Δ.
Res[exp(-iαζ)sech2 ζ sech(ζ+Δ)]ζ=½πi
=-iexp(½πα)sinh2 Δ(cosh Δ+iα sinh Δ),
Res[exp(-iαζ)sech2 ζ sech(ζ+Δ)]ζ=-Δ+½πi
=iexp(½πα)sinh2 Δexp(iαΔ).
Fα[sech2 ζ sech(ζ+Δ)]
=π sech(½πα) 
×cosh Δ+iα sinh Δ-exp(iαΔ)sinh2 Δ.

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