Abstract

A useful analysis of dispersive (radiative) perturbations of solitons of the nonlinear Schrödinger equation is developed. With reference to the propagation of optical solitons in glass fibers, the analysis is used to treat the collision of a low-intensity wave packet with a soliton, the radiation field created by the local perturbation of a soliton, and finally that created by a spatially periodic perturbation of the parameters of the fiber, or equivalently by a periodic variation in gain and loss that averages to zero. Perturbations whose wavelength is short compared with the soliton period produce exponentially small radiation fields as a result of the need for phase matching.

© 1992 Optical Society of America

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References

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  1. V. E. Zahkarov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].
  2. V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica 3D, 487–502 (1981).
  3. V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Zh. Eksp. Teor. Fiz. 73, 537–559 (1977) [Sov. Phys. JETP 46, 281–291 (1978)].
  4. D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London Ser. A 361, 413–446 (1978).
    [Crossref]
  5. L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
    [Crossref]
  6. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
    [Crossref] [PubMed]
  7. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. 55, 284–306 (1974).
    [Crossref]
  8. L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wave-length division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” IEEE J. Lightwave Technol. 9, 362–367 (1991).
    [Crossref]
  9. A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990); “Guiding-center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
    [Crossref] [PubMed]

1991 (1)

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wave-length division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” IEEE J. Lightwave Technol. 9, 362–367 (1991).
[Crossref]

1990 (1)

1986 (2)

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[Crossref]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[Crossref] [PubMed]

1981 (1)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica 3D, 487–502 (1981).

1978 (1)

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London Ser. A 361, 413–446 (1978).
[Crossref]

1977 (1)

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Zh. Eksp. Teor. Fiz. 73, 537–559 (1977) [Sov. Phys. JETP 46, 281–291 (1978)].

1974 (1)

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

1971 (1)

V. E. Zahkarov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Evangelides, S. G.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wave-length division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” IEEE J. Lightwave Technol. 9, 362–367 (1991).
[Crossref]

Gordon, J. P.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wave-length division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” IEEE J. Lightwave Technol. 9, 362–367 (1991).
[Crossref]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[Crossref] [PubMed]

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[Crossref]

Hasegawa, A.

Haus, H. A.

Islam, M. N.

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[Crossref]

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica 3D, 487–502 (1981).

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Zh. Eksp. Teor. Fiz. 73, 537–559 (1977) [Sov. Phys. JETP 46, 281–291 (1978)].

Kaup, D. J.

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London Ser. A 361, 413–446 (1978).
[Crossref]

Kodama, Y.

Maslov, E. M.

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Zh. Eksp. Teor. Fiz. 73, 537–559 (1977) [Sov. Phys. JETP 46, 281–291 (1978)].

Mollenauer, L. F.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wave-length division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” IEEE J. Lightwave Technol. 9, 362–367 (1991).
[Crossref]

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[Crossref]

Newell, A. C.

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London Ser. A 361, 413–446 (1978).
[Crossref]

Satsuma, J.

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

Shabat, A. B.

V. E. Zahkarov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica 3D, 487–502 (1981).

Yajima, N.

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

Zahkarov, V. E.

V. E. Zahkarov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

IEEE J. Lightwave Technol. (1)

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wave-length division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” IEEE J. Lightwave Technol. 9, 362–367 (1991).
[Crossref]

IEEE J. Quantum Electron. (1)

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[Crossref]

Opt. Lett. (2)

Physica (1)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica 3D, 487–502 (1981).

Proc. R. Soc. London Ser. A (1)

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London Ser. A 361, 413–446 (1978).
[Crossref]

Prog. Theor. Phys. (1)

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

Zh. Eksp. Teor. Fiz. (2)

V. E. Zahkarov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Zh. Eksp. Teor. Fiz. 73, 537–559 (1977) [Sov. Phys. JETP 46, 281–291 (1978)].

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Figures (8)

Fig. 1
Fig. 1

Dispersion curves for a soliton (k = 1/2) and for linear waves (k = −ω2/2). The dashed lines show coupling by a perturbation with a wave number of −3.

Fig. 2
Fig. 2

Numerically simulated soliton propagation in a fiber with a perturbation A(z) = 0.5 cos(9.8z). The input is the unperturbed soliton u = sech(t).

Fig. 3
Fig. 3

Linear plot at 8000 km showing results of the numerical simulation (to the left) and the perturbation theory result of this paper (to the right). The parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Same as Fig. 2 but with A(z) = 0.5 sin(9.8z).

Fig. 5
Fig. 5

Same as Fig. 3 but with A(z) as in Fig. 4.

Fig. 6
Fig. 6

Linear plot of the numerical results for the case A(z) = 0.06 cos(2z) at z = 8Lp (15,680 km).

Fig. 7
Fig. 7

Same as Fig. 2 but with an initial perturbation that cancels the spectrum of the radiation from the input at low frequencies.

Fig. 8
Fig. 8

Same as Fig. 2 but with an initial perturbation that closely matches the quasi-stationary soliton with its radiation field.

Equations (41)

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- i u z = 1 2 2 u t 2 + u * u 2 ,
u s ( z , t ) = A sech ( A t - q ) exp ( - i Ω t + i ϕ ) ,
u = u s + u p ,
- i u p z = 1 2 2 u p t 2 + 2 u s 2 u p + u s 2 u p * .
δ A = Re ( u s * u p ) d t ,
δ ( A Ω ) = - Im ( u s * u p t ) d t ,
δ q = t Re ( u s * u p ) d t ,
A δ ϕ + Ω δ q = - t Im ( u s * u p t ) d t .
- i f z = 1 2 2 f t 2 .
u p ( z , t ) = - 2 f t 2 + 2 γ f t - γ 2 f + u s 2 f * ,
Re ( u s * u p ) = - 2 t 2 Re ( u s * f ) ,
Im ( u s * u p t ) = - 2 t 2 Im ( u s * f t ) ,
u s sech ( t ) exp ( i z / 2 ) .
f ( z , t ) = d ω f ˜ ( ω ) exp ( - i ω t - i ω 2 z 2 ) ,
u p ( z , t ) = d ω u ˜ p ( z , ω ) exp ( - i ω t - i ω 2 z 2 ) .
u ˜ p ( z a , ω ) = ( ω - i ) 2 f ˜ ( ω ) .
u ˜ p ( z b , ω ) = ( ω + i ω - i ) 2 u ˜ p ( z a , ω ) .
f ( z a , t ) = i ( / 4 ) u s ( z a , t ) .
u p ( z a , t ) = i ( u s 2 - 1 ) u s ( z a , t ) .
f ( z , t ) = i 8 exp ( i z a 2 ) d ω sech ( π 2 ω ) exp ( - i ω t - i ω 2 2 ζ ) ,
f ( z , t ) = ( π i 32 ζ ) 1 / 2 exp ( i z a 2 ) sech ( π t 2 ζ ) exp ( i t 2 2 ζ ) .
2 Re ( u s * u p ) = ( π 8 ζ ) 1 / 2 ( 1 - 2 γ 2 ) sech ( t ) cos ( ζ 2 - π 4 ) .
f ˜ ( 0 ) = - 1 2 π d t { Re [ u p ( 0 , t ) ] + i ( 2 γ 2 - 1 ) Im [ u p ( 0 , t ) ] } .
- i u 1 z = 1 2 2 u 1 t 2 + u 1 * u 1 2 + i Γ ( z ) 2 u 1 ,
- i u z = 1 2 2 u t 2 + G ( z ) u * u 2
- i u z = 1 2 2 u t 2 + [ 1 + A ( z ) ] u * u 2 .
- i u p z = 1 2 2 u p t 2 + 2 u s 2 u p + u s 2 u p * + A ( z ) ( u s 2 - 1 ) u s ,
- i f z = 1 2 2 f t 2 + 1 4 A ( z ) u s .
f ( z , t ) = i 8 0 z d s A ( x ) exp ( i s 2 ) d ω sech ( π 2 ω ) × exp [ - i ω t - i ω 2 2 ( z - s ) ] ,
f k ( z , t ) = 1 4 d ω sech ( π 2 ω ) exp ( - ι ω t ) × ( A k ω 2 + 1 + 2 k { exp [ i ( 1 2 + k ) z ] - exp ( - i ω 2 2 z ) } + A k * ω 2 + 1 - 2 k × { exp [ i ( 1 2 - k ) z ] - exp ( - i ω 2 2 z ) } ) .
u p d = ( π 2 z ) 1 / 2 sech ( π t 2 z ) p 2 Re ( A k ) - i p k Im ( A k ) k 2 - p 2 × exp [ i t 2 2 z + i 2 tan - 1 ( z t ) - i π 4 ] ,
u p r = i π k A k * 2 Ω sech ( π Ω 2 ) × exp [ i Ω t - i Ω 2 2 z + i 2 tan - 1 ( 1 Ω ) ]
Γ = Ω u p r 2 = 1 4 Ω [ π k A k sech ( π Ω 2 ) ] 2 .
f ( 0 , t ) = - 1 2 d ω sech ( π 2 ω ) exp ( - i ω t ) × [ Re ( A k ) ( ω 2 + 1 ) - i Im ( A k ) 2 k 4 k 2 - ( ω 2 + 1 ) 2 ] .
f ( 0 , t ) = n C n ( t ) 2 n sech ( t ) .
2 t 2 sech n ( t ) = n 2 sech n ( t ) - n ( n + 1 ) sech n + 2 ( t ) ,
f ( 0 , t ) = n = 0 ( - 1 ) n 2 n ( n + 1 ) Re ( A k ) + i k Im ( A k ) k 2 - 4 n 2 ( n + 1 ) 2 × exp [ - ( 2 n + 1 ) t ] + π A k 4 α sec ( π 2 α ) exp ( - α t ) + i π A k * 4 Ω sech ( π 2 Ω ) exp ( i Ω t )
A n = 1 ( - 1 ) n + 1 [ 1 + k 2 4 n 2 ( n + 1 ) 2 - k 2 ] exp [ - ( 2 n + 1 ) t ] 4 n ( n + 1 ) .
n = 1 ( - x ) n + 1 n ( n + 1 ) = ( 1 + x ) ln ( 1 + x ) - x ,
f ( 0 , t ) = A 4 { n = 1 ( - 1 ) n + 1 k 2 exp [ - ( 2 n + 1 ) t ] n ( n + 1 ) [ 4 n 2 ( n + 1 ) 2 - k 2 ] + 2 cosh ( t ) ln [ 1 + exp ( - 2 t ) ] - exp ( - t ) + π 2 α sec ( π 2 α ) exp ( - α t ) + i π 2 Ω sech ( π 2 Ω ) exp ( i Ω t ) } .
u p ( 0 , t ) = A { k 2 n = 1 ( - 1 ) n 2 n ( n + 1 ) + ( 2 n + 1 ) γ + γ 2 2 n ( n + 1 ) [ 4 n 2 ( n + 1 ) 2 - k 2 ] × exp [ - ( 2 n + 1 ) t ] - 1 2 exp ( - t ) sech 2 ( t ) - π 4 α [ k + γ ( α + γ ) ] sec ( π 2 α ) exp ( - α t ) + i π 8 Ω sech ( π 2 Ω ) [ ( Ω + i γ ) 2 exp ( i Ω t ) - sech 2 ( t ) exp ( - i Ω t ) ] } ,

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