Abstract

A higher-order, multiple-scale asymptotic analysis is made of the perturbed nonlinear Schrödinger equation in a strong dispersion-managed optical transmission system. It is found that the averaged equation with the next-order term included significantly improves the description of the characteristics of dispersion-managed solitons. The derived equation is shown to support a new class of soliton solutions, namely, multihump solitons, which depend on both the map strength and dispersion profile. Numerical evidence of the regions of existence and stability of such new solitons is discussed.

© 2002 Optical Society of America

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References

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  1. G. E. Walrafen and P. N. Krishnan, “Model analysis of the Raman spectrum from fused silica optical fibers,” Appl. Opt. 21, 359–360 (1982).
    [CrossRef] [PubMed]
  2. R. H. Stolen, C. Lee, and R. K. Jain, “Development of the stimulated Raman spectrum in single-mode silica fibers,” J. Opt. Soc. Am. B 1, 652–657 (1984).
    [CrossRef] [PubMed]
  3. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
    [CrossRef]
  4. D. Hollenbeck, “Dynamics of a fiberoptic Raman amplifier,” Ph.D. dissertation (University of Texas at Dallas, Dallas, Tex., 2000).
    [CrossRef]
  5. A. R. Chraplyvy, “Optical power limits in multichannel wavelength-division-multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).
    [CrossRef]
  6. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
    [CrossRef]
  7. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1997).
  8. R. J. Bell and P. Dean, “Atomic vibrations in vitreous silica,” Discuss. Faraday Soc. 50, 55–61 (1970).
    [CrossRef]
  9. A. G. Revesz and G. E. Walrafen, “Structural interpretations for some Raman lines from vitreous silica,” J. Non-Cryst. Solids 54, 323–333 (1983).
    [CrossRef]
  10. A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. Press, New York, 1971), p. 100.
    [CrossRef]
  11. A. Icsevgi and W. E. Lamb, Jr., “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).

1989

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[CrossRef]

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

1984

A. R. Chraplyvy, “Optical power limits in multichannel wavelength-division-multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).
[CrossRef]

R. H. Stolen, C. Lee, and R. K. Jain, “Development of the stimulated Raman spectrum in single-mode silica fibers,” J. Opt. Soc. Am. B 1, 652–657 (1984).
[CrossRef] [PubMed]

1983

A. G. Revesz and G. E. Walrafen, “Structural interpretations for some Raman lines from vitreous silica,” J. Non-Cryst. Solids 54, 323–333 (1983).
[CrossRef]

1982

1970

R. J. Bell and P. Dean, “Atomic vibrations in vitreous silica,” Discuss. Faraday Soc. 50, 55–61 (1970).
[CrossRef]

1969

A. Icsevgi and W. E. Lamb, Jr., “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).

Bell, R. J.

R. J. Bell and P. Dean, “Atomic vibrations in vitreous silica,” Discuss. Faraday Soc. 50, 55–61 (1970).
[CrossRef]

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

Chraplyvy, A. R.

A. R. Chraplyvy, “Optical power limits in multichannel wavelength-division-multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).
[CrossRef]

Dean, P.

R. J. Bell and P. Dean, “Atomic vibrations in vitreous silica,” Discuss. Faraday Soc. 50, 55–61 (1970).
[CrossRef]

Gordon, J. P.

Haus, H. A.

Icsevgi, A.

A. Icsevgi and W. E. Lamb, Jr., “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).

Jain, R. K.

Krishnan, P. N.

Lamb Jr., W. E.

A. Icsevgi and W. E. Lamb, Jr., “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).

Lee, C.

Revesz, A. G.

A. G. Revesz and G. E. Walrafen, “Structural interpretations for some Raman lines from vitreous silica,” J. Non-Cryst. Solids 54, 323–333 (1983).
[CrossRef]

Stolen, R. H.

Tomlinson, W. J.

Walrafen, G. E.

A. G. Revesz and G. E. Walrafen, “Structural interpretations for some Raman lines from vitreous silica,” J. Non-Cryst. Solids 54, 323–333 (1983).
[CrossRef]

G. E. Walrafen and P. N. Krishnan, “Model analysis of the Raman spectrum from fused silica optical fibers,” Appl. Opt. 21, 359–360 (1982).
[CrossRef] [PubMed]

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

Appl. Opt.

Discuss. Faraday Soc.

R. J. Bell and P. Dean, “Atomic vibrations in vitreous silica,” Discuss. Faraday Soc. 50, 55–61 (1970).
[CrossRef]

Electron. Lett.

A. R. Chraplyvy, “Optical power limits in multichannel wavelength-division-multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).
[CrossRef]

IEEE J. Quantum Electron.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

J. Non-Cryst. Solids

A. G. Revesz and G. E. Walrafen, “Structural interpretations for some Raman lines from vitreous silica,” J. Non-Cryst. Solids 54, 323–333 (1983).
[CrossRef]

J. Opt. Soc. Am. B

Phys. Rev.

A. Icsevgi and W. E. Lamb, Jr., “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).

Other

D. Hollenbeck, “Dynamics of a fiberoptic Raman amplifier,” Ph.D. dissertation (University of Texas at Dallas, Dallas, Tex., 2000).
[CrossRef]

A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. Press, New York, 1971), p. 100.
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1997).

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Figures (12)

Fig. 1
Fig. 1

Variation of the rms width of the spectrum in the averaging procedure.

Fig. 2
Fig. 2

DM soliton solution (s=0.56, d0=1.65, θ=0.8, M=0.34, za=0.06) (a) in the spectral domain and (b) in the time domain. The dashed curves represent the initial (Gaussian) waveforms in both (a) and (b).

Fig. 3
Fig. 3

Solid curve is an approximate solution u(t)=u(0)(t)+zau(1)(t) which is calculated from Eqs. (7) and (14). The dashed curve represents a DM soliton obtained from the averaging method on the p-NLS equation; it is indistinguishable from the solid curve. The dotted curve is U0(t) which is the inverse Fourier transform of Uˆ0(ω), which is the leading-order solution of the HO-DMNLS. The waveforms are shown in (a) linear and (b) log scale. In (a) the dashed curve is indistinguishable from the solid curve.

Fig. 4
Fig. 4

Evolution of chirp and peak amplitude of a DM soliton within a DM period. The solid curve and dashed curve denote the approximate solution u=u(0)+zau(1) and leading-order solution u(0), respectively. The circles represent the evolution on the p-NLS equation.

Fig. 5
Fig. 5

θ dependency of single DM solitons that are obtained by using the averaging method on the HO-DMNLS. All of the parameters except θ are the same as those used for Fig. 2.

Fig. 6
Fig. 6

Bi-soliton with τs=3τFWHM solution which is obtained by the averaging method on the HO-DMNLS equation for s=0.56, θ=0.8, d0=1.65, and then M=0.340 (solid curve). The dotted curve shows the same solution which is given by the averaging method on the p-NLS equation.

Fig. 7
Fig. 7

Regions in which a bi-soliton solution with τs=3τFWHM can be found by using the averaging method on (a) the HO-DMNLS and (b) the p-NLS equation. In the bright region in (b) two Gaussian inputs with τs=3τFWHM can propagate without collision or large timing shift over long distances.

Fig. 8
Fig. 8

Antiphase bi-soliton with τs=2τFWHM for s=0.46, θ=0.8, d0=1.65, and M=0.279, which is obtained by the averaging method on the HO-DMNLS equation (solid curve) and on the p-NLS equation (dotted curve).

Fig. 9
Fig. 9

In-phase tri-soliton solution with τs=3τFWHM for s=0.54, θ=0.3, d0=1.65, and M=0.328, which is obtained by the averaging method on the HO-DMNLS equation (solid curve) and on the p-NLS equation (dotted curve).

Fig. 10
Fig. 10

Anti-phase tri-soliton solution with τs=2τFWHM for s=0.37, θ=0.8, d0=1.65, and M=0.225, which is obtained by the averaging method on the HO-DMNLS equation (solid curve) and on the p-NLS equation (dotted curve).

Fig. 11
Fig. 11

(a) In-phase quartic-soliton with τs=3τFWHM for s=0.51, θ=0.6, d0=1.65, and M=0.310 and (b) antiphase quartic-soliton with τs=2τFWHM for s=0.37, θ=0.8, d0=1.65, and M=0.225.

Fig. 12
Fig. 12

(a) Initial waveforms of bi-soliton with weak white noise Un(t) and the pure bi-soliton U0(t) which appears in Fig. 6. (b) Evolution of the energy of the difference |Un-U0| as a function of distance.

Equations (45)

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i uz+d(z)2 2ut2+g(z)|u|2u=0,
u(ζ, Z, t)=u(0)(ζ, Z, t)+zau(1)(ζ, Z, t)+za2u(2)(ζ, Z, t)+O(za3).
O(za-1):i u(0)ζ+Δ(ζ)2 2u(0)t2
=0,
O(1):i u(1)ζ+Δ(ζ)2 2u(1)t2
=-i u(0)Z+d02 2u(0)t2+g(ζ)|u(0)|2u(0),
O(za):i u(2)ζ+Δ(ζ)2 2u(2)t2
=-i u(1)Z+d02 2u(1)t2+g(ζ)[2|u(0)|2u(1)+u(0)2u(1)*].
i uˆ(0)ζ-Δ(ζ)2ω2uˆ(0)=0,
uˆ(0)(ζ, Z, ω)=exp-i ω22C(ζ)Uˆ0(Z, ω),
i uˆ(1)ζ-Δ(ζ)2ω2uˆ(1)
=-exp-i ω22C(ζ)×i Uˆ0Z-d02ω2Uˆ0-g(ζ)Pˆ0(ζ, Z, ω),
i Uˆ0Z-d02ω2Uˆ0+g(ζ)expi ω22C(ζ)Pˆ0(ζ, Z, ω)
=0.
i Uˆ0Z-d02ω2Uˆ0+1(2π)2 -+dω1dω2r0(ω1ω2)
×Uˆ0(ω+ω1)Uˆ0(ω+ω2)Uˆ0*(ω+ω1+ω2)=0,
ζ{iuˆ(1) exp[iC(ζ)ω2/2]}
=g(ζ)exp[iC(ζ)ω2/2]Pˆ0(ζ, Z, ω)-g(ζ)exp[iC(ζ)ω2/2]Pˆ0(ζ, Z, ω).
iuˆ(1) exp[iC(ζ)ω2/2]
=Uˆ1-0ζg(ζ)exp[iC(ζ)ω2/2]Pˆ0(ζ, Z, ω)dζ+ζg(ζ)exp[iC(ζ)ω2/2]Pˆ0(ζ, Z, ω),
Uˆ1(Z, ω)=0ζg(ζ)exp[iC(ζ)ω2/2]Pˆ0(ζ, Z, ω)dζ-12g(ζ)exp[iC(ζ)ω2/2]Pˆ0(ζ, Z, ω).
uˆ(1)(ζ, Z, ω)=i exp-i ω22C(ζ)0ζ expi ω22C(ζ)×g(ζ)Pˆ0(ζ, Z, ω)dζ-0ζ expi ω22C(ζ)g(ζ)×Pˆ0(ζ, Z, ω)dζ-ζ-12×expi ω22C(ζ)g(ζ)Pˆ0(ζ, Z, ω),
uˆ(1)(ζ, Z, ω)
=i exp-i ω22C(ζ) 1(2π)2 -+dΩ1dΩ2
×Uˆ0(ω+Ω1)Uˆ0(ω+Ω2)Uˆ0*(ω+Ω1+Ω2)
×0ζg(ζ)K(ζ, Ω1Ω2)dζ-0ζg(ζ)K(ζ, Ω1Ω2)dζ-ζ-12g(ζ)K(ζ, Ω1Ω2).
i Uˆ0Z-d02ω2Uˆ0+1(2π)2 -+dω1dω2r0(ω1ω2)×Uˆ0(ω+ω1)Uˆ0(ω+ω2)Uˆ0*(ω+ω1+ω2)
=zanˆ1(Z, ω)+O(za2).
ζ{iuˆ(2) exp[iC(ζ)ω2/2]}+nˆ1+exp[iC(ζ)ω2/2]
×i uˆ(1)Z-d02ω2uˆ(1)+g(ζ)Pˆ1(ζ, Z, ω)=0,
nˆ1=-g(ζ)exp[iC(ζ)ω2/2]Pˆ1(ζ, Z, ω).
nˆ1=-i(2π)4 -+dω1dω2dΩ1dΩ2r1(ω1ω2, Ω1Ω2)×[2Uˆ0(ω+ω1)Uˆ0*(ω+ω1+ω2)×Uˆ0(ω+ω2+Ω1)Uˆ0(ω+ω2+Ω2)Uˆ0*(ω+ω2+Ω1+Ω2)-Uˆ0(ω+ω1)Uˆ0(ω+ω2)Uˆ0*(ω+ω1+ω2+Ω1)×Uˆ0*(ω+ω1+ω2-Ω2)Uˆ0(ω+ω1+ω2+Ω1-Ω2)],
r1(x, y)=g(ζ)K(ζ, x)0ζg(ζ)K(ζ, y)dζ-g(ζ)K(ζ, x)0ζg(ζ)K(ζ, y)dζ-ζ-12g(ζ)K(ζ, x)g(ζ)K(ζ, y).
r0(x)=sin(sx)sx,
r1(x, y)=i(2θ-1)2s3x2y2(x+y){sxy[y cos(sx)sin(sy)-x cos(sy)sin(sx)]+(x2-y2)sin(sx)sin(sy)}.
i Uˆ0Z-ω22Uˆ0+1(2π)2 -+dω1dω2r0(ω1ω2)
×Uˆ0(ω+ω1)Uˆ0(ω+ω2)Uˆ0*(ω+ω1+ω2)=zanˆ1,
nˆ1=-i(2π)4 -+dω1dω2dΩ1dΩ2r1(ω1ω2, Ω1Ω2)×[2Uˆ0(ω+ω1)Uˆ0*(ω+ω1+ω2)Uˆ0(ω+ω2+Ω1)×Uˆ0(ω+ω2+Ω2)Uˆ0*(ω+ω2+Ω1+Ω2)-Uˆ0(ω+ω1)Uˆ0(ω+ω2)Uˆ0*(ω+ω1+ω2+Ω1)×Uˆ0*(ω+ω1+ω2-Ω2)Uˆ0(ω+ω1+ω2+Ω1-Ω2)],
r0(x)=sin(Mx)Mx,
r1(x, y)=i(2θ-1)2M3x2y2(x+y){Mxy[y cos(Mx)sin(My)-x cos(My)sin(Mx)]+(x2-y2)sin(Mx)sin(My)},
Uˆ0(Z, ω)=1|d0|Uˆ0Z,ω|d0|.
i Uˆ0Z-d02ω2Uˆ0+g(ζ)expi ω22C(ζ)Pˆ(ζ, Z, ω)
=0,
Uˆ(ω)=exp(-iϕmax)Uˆmax(ω)+exp(-iϕmin)Uˆmin(ω)2,
Uˆave(ω)=E0E Uˆ(ω),

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