Abstract

We discuss the evolution of a perturbed soliton in an anomalously dispersive optical fiber. In so doing, the advantage of using a natural mathematical framework based on inverse scattering theory will be emphasized. Results obtained in this way will be contrasted with these obtained by using an alternative approach commonly known as the associate field formalism. The relative merits of the two techniques will be discussed, and in particular the latter will be shown to fail unless the initial radiation field respects certain symmetry requirements.

© 2001 Optical Society of America

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References

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  1. H. A. Haus, W. S. Wong, and F. I. Khatri, “Continuum generation by perturbation of soliton,” J. Opt. Soc. Am. B 14, 304–312 (1997).
    [CrossRef]
  2. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
    [CrossRef]
  3. J. N. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
    [CrossRef] [PubMed]
  4. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” J. Exp. Theor. Phys. 34, 62–69 (1972).
  5. D. J. Kaup, “Closure of the Zakharov–Shabat eigenstates,” J. Math. Anal. Appl. 54, 849–864 (1976).
    [CrossRef]
  6. M. J. Ablowitz and H. J. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, 1981).
  7. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
    [CrossRef]
  8. J. N. Elgin, “Inverse scattering theory with stochastic initial potentials,” Phys. Lett. A 110A, 441–443 (1985).
    [CrossRef]
  9. M. W. Chbat, P. R. Pruncal, M. N. Islam, C. E. Soccolich, and J. P. Gordon, “Long-range interference effects of soliton reshaping in optical fibers,” J. Opt. Soc. Am. B 10, 1386–1395 (1993).
    [CrossRef]
  10. J. N. Elgin, “Soliton propagation in an optical fiber with third-order dispersion,” Opt. Lett. 17, 1409–1411 (1992).
    [CrossRef] [PubMed]
  11. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
    [CrossRef] [PubMed]
  12. J. N. Elgin, T. Bradec, and S. M. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
    [CrossRef]

1997

1995

J. N. Elgin, T. Bradec, and S. M. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

1993

1992

1986

1985

J. N. Elgin, “Inverse scattering theory with stochastic initial potentials,” Phys. Lett. A 110A, 441–443 (1985).
[CrossRef]

1976

D. J. Kaup, “Closure of the Zakharov–Shabat eigenstates,” J. Math. Anal. Appl. 54, 849–864 (1976).
[CrossRef]

1974

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

1972

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” J. Exp. Theor. Phys. 34, 62–69 (1972).

Bradec, T.

J. N. Elgin, T. Bradec, and S. M. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Chbat, M. W.

Chen, H. H.

Elgin, J. N.

J. N. Elgin, T. Bradec, and S. M. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

J. N. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
[CrossRef] [PubMed]

J. N. Elgin, “Soliton propagation in an optical fiber with third-order dispersion,” Opt. Lett. 17, 1409–1411 (1992).
[CrossRef] [PubMed]

J. N. Elgin, “Inverse scattering theory with stochastic initial potentials,” Phys. Lett. A 110A, 441–443 (1985).
[CrossRef]

Gordon, J. P.

Haus, H. A.

Islam, M. N.

Kaup, D. J.

D. J. Kaup, “Closure of the Zakharov–Shabat eigenstates,” J. Math. Anal. Appl. 54, 849–864 (1976).
[CrossRef]

Kelly, S. M.

J. N. Elgin, T. Bradec, and S. M. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Khatri, F. I.

Lee, Y. C.

Menyuk, C. R.

Pruncal, P. R.

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” J. Exp. Theor. Phys. 34, 62–69 (1972).

Soccolich, C. E.

Wai, P. K. A.

Wong, W. S.

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” J. Exp. Theor. Phys. 34, 62–69 (1972).

J. Exp. Theor. Phys.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” J. Exp. Theor. Phys. 34, 62–69 (1972).

J. Math. Anal. Appl.

D. J. Kaup, “Closure of the Zakharov–Shabat eigenstates,” J. Math. Anal. Appl. 54, 849–864 (1976).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

J. N. Elgin, T. Bradec, and S. M. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Opt. Lett.

Phys. Lett. A

J. N. Elgin, “Inverse scattering theory with stochastic initial potentials,” Phys. Lett. A 110A, 441–443 (1985).
[CrossRef]

Phys. Rev. A

J. N. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
[CrossRef] [PubMed]

Suppl. Prog. Theor. Phys.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Other

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

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Figures (2)

Fig. 1
Fig. 1

Associate field for z=2.5 (solid curve) and z=10 (dashed curve).

Fig. 2
Fig. 2

Radiation field δu, corresponding to the associate field profiles shown in Fig. 1, as computed from Eq. (46).

Equations (69)

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-iuz=122ut2+|u|2u,
us=exp(iz/2)sech t,
u1t+iζu1=iu*u2,
u2t-iζu2=iuu1,
ϕ10exp(-iξt),ϕ¯0-1exp(iξt),t-,
ψ¯10exp(-iξt),ψ01exp(iξt),t+,
Ψ=ψ12(t, ζ)ψ22(t, ζ),Ψ¯=ψ¯12(t, ζ)ψ¯22(t, ζ),
Φ=ϕ22(t, ζ)-ϕ12(t, ζ),Φ¯=ϕ¯22(t, ζ)-ϕ¯12(t, ζ),
-+ΦT(t, ξ)Ψ(t, ξ)dt=-πa2(ξ)δ(ξ-ξ),
-+Φ¯T(t, ξ)Ψ(t, ξ)dt=0,
-+ΦT(t, ξ)Ψ¯(t, ξ)dt=0,
-+Φ¯T(t, ξ)Ψ¯(t, ξ)dt=πa¯2(ξ)δ(ξ-ξ).
ϕ=aψ¯+bψ,
ϕ¯=-a¯ψ+b¯ψ¯.
δuc=δu+iαtus-βt(tus),
α=-+ Imustδudt,
β=-+ Re{usδu}dt.
δˆuc(ω)=-+δuc(t)exp(iωt)dt,
δuc(t)=12π-+δˆuc(ω)exp(-iωt)dω.
b¯·a=-+[δuc, -δuc*]Ψdt,
δucδuc*=-1π-+ba¯Φ¯+b¯aΦdξ.
ψ1=12ξ+iexp(iξt+iz/2)sech t,
ψ2=12ξ+iexp(iξt)(2ξ+i tanh t),
a=2ξ-i2ξ+i,
b¯=δˆu*(ω),ω=2ξ,
b¯=ω2-1ω2+1δˆu+2iωω2+1-+δu(t)tanh t exp(iωt)dt
b¯(ω)=- sech(πω/2)=-δˆu(ω),
b¯(ω)=-3ω2 sechπω2=13δˆu(ω).
δˆu=iω sech(πω/2),
b¯=0.
δˆu=-d2dω2sechπω2,
b¯=d2dω2sechπω2+4ωω2+1ddωsechπω2,
=-δˆu+4ωω2+1ddωsechπω2.
b¯=ω2-1ω2+1δˆu+2ωω2+1-+δu(t)tanh t exp(iωt)dt+2ω2+1-+δu(t)sech2 t exp(iωt)dt,
b¯=iω2+14sechπω2=12δˆu.
b¯=-13ω sechπω2=13δˆu.
δˆu=-iω2 sech(πω/2),
b¯=-i2 sechπω2+12δˆu.
b¯(0, ω)=const×δˆu*(0, ω).
δu(0, t)=t tanh t sech t=-(t sech t)+ sech t
ib¯z=-12ω2b¯,
b¯(z, ω)=b¯(0, ω)exp(iω2z/2),
δuc(z, t)=-1π-+[b(0, ω)χ(z, ω)+b¯(0, ω)χ¯(z, ω)]dω,
χ(z, ω)=(ω+i tanh t)2ω2+1expiωt-iω2z2,
χ¯(z, ω)=sech2 tω2+1exp-iωt+iω2z2+iz.
b¯(0, ω)δˆu(0, ω),
δuc(0, t)=-12π-+ba¯ω+i tanh tω-i2 exp(iωt)+b¯asech tω+i2 exp(-iωt)dω.
-2ft2+2 tanh tft-tanh2 tf+sech2 tf*=δuc.
ifz=122ft2,
f(0, t)= cosh t(ln cosh t+c1t+c2),
δuc(0, t)= sech t-2t(t sech t).
fc(0, t)=-t sinh t,
Y=G/cosh t,
y=g/sinh t.
Y=-Re{δuc(0, t)}cosh t,
y+2sinh t cosh ty=-Im{δuc(0, t)}sinh t.
Re{f(0, t)}=c1t cosh t+β2t sinh t-t cosh t×-t Re{δu(0, t)}(sech t)dt+c2 cosh t+cosh t-tt Re{δu(0, t)}(sech t)dt,
Im{f(0, t)}=c3t sinh t-α2t cosh t-t sinh t×-ttanh2 tsinh tIm{δu(0, t)}dt-c3 cosh t+c4 sinh t+α2sinh t+cosh t×-ttanh2 tsinh tIm{δu(0, t)}dt+sinh t-t(t-coth t)tanh2 tsinh t×Im{δu(0, t)}dt,
c1=β2,
c2=0
c2=--+t Re{δu(0, t)}sech t dt
f(0, t)=12t sech t+exp t2-12t sech t=exp t2.
fˆ(0, ω)=-b(0, ω)1+ω2,
c3=α/2,
c4=0
c4=-+(t tanh t-1)sech t Im{δu(0, t)}dt
iuz+122ut2+|u|2u=-iF
F=3ut3.
ifz=2ft2+i3ft3+i4ust

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