Abstract

Effects of initial overlap between two solitons at different carrier wavelengths are studied theoretically and numerically as a function of the degree of overlap and the frequency separation of the two carriers. When the two solitons are fully overlapped, the carrier frequencies of the emerging two solitons are shifted toward each other by an amount proportional to the initial power and inversely proportional to the initial frequency separation with little effect on their amplitudes provided that the frequency separation is sufficiently larger than the spectral width of the solitons.

© 1991 Optical Society of America

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References

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  1. Y. Kodama, K. Nozaki, Opt. Lett. 12, 1038 (1987).
    [CrossRef] [PubMed]
  2. P. A. Andrekson, W. A. Olsson, J. R. Simpson, T. Tanbun-Ek, R. A. Logan, P. C. Becker, K. W. Wecht, Appl. Phys. Lett. 57, 1715 (1990).
    [CrossRef]
  3. A. Hasegawa, F. D. Tappert, Appl. Phys. Lett. 23, 142 (1973). Note that solitons that are periodically amplified are also described by Eq. (1) aproximately; see A. Hasegawa, Y. Kodama, Opt. Lett. 15, 1443 (1990).
    [CrossRef] [PubMed]
  4. V. E. Zakharov, A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].
  5. L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, “Wavelength-division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” submitted to IEEE J. Lightwave Technol.
  6. E. A. Overman, The Ohio State University, Columbus, Ohio (personal communication).

1990

P. A. Andrekson, W. A. Olsson, J. R. Simpson, T. Tanbun-Ek, R. A. Logan, P. C. Becker, K. W. Wecht, Appl. Phys. Lett. 57, 1715 (1990).
[CrossRef]

1987

1973

A. Hasegawa, F. D. Tappert, Appl. Phys. Lett. 23, 142 (1973). Note that solitons that are periodically amplified are also described by Eq. (1) aproximately; see A. Hasegawa, Y. Kodama, Opt. Lett. 15, 1443 (1990).
[CrossRef] [PubMed]

1971

V. E. Zakharov, A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

Andrekson, P. A.

P. A. Andrekson, W. A. Olsson, J. R. Simpson, T. Tanbun-Ek, R. A. Logan, P. C. Becker, K. W. Wecht, Appl. Phys. Lett. 57, 1715 (1990).
[CrossRef]

Becker, P. C.

P. A. Andrekson, W. A. Olsson, J. R. Simpson, T. Tanbun-Ek, R. A. Logan, P. C. Becker, K. W. Wecht, Appl. Phys. Lett. 57, 1715 (1990).
[CrossRef]

Evangelides, S. G.

L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, “Wavelength-division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” submitted to IEEE J. Lightwave Technol.

Gordon, J. P.

L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, “Wavelength-division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” submitted to IEEE J. Lightwave Technol.

Hasegawa, A.

A. Hasegawa, F. D. Tappert, Appl. Phys. Lett. 23, 142 (1973). Note that solitons that are periodically amplified are also described by Eq. (1) aproximately; see A. Hasegawa, Y. Kodama, Opt. Lett. 15, 1443 (1990).
[CrossRef] [PubMed]

Kodama, Y.

Logan, R. A.

P. A. Andrekson, W. A. Olsson, J. R. Simpson, T. Tanbun-Ek, R. A. Logan, P. C. Becker, K. W. Wecht, Appl. Phys. Lett. 57, 1715 (1990).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, “Wavelength-division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” submitted to IEEE J. Lightwave Technol.

Nozaki, K.

Olsson, W. A.

P. A. Andrekson, W. A. Olsson, J. R. Simpson, T. Tanbun-Ek, R. A. Logan, P. C. Becker, K. W. Wecht, Appl. Phys. Lett. 57, 1715 (1990).
[CrossRef]

Overman, E. A.

E. A. Overman, The Ohio State University, Columbus, Ohio (personal communication).

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

Simpson, J. R.

P. A. Andrekson, W. A. Olsson, J. R. Simpson, T. Tanbun-Ek, R. A. Logan, P. C. Becker, K. W. Wecht, Appl. Phys. Lett. 57, 1715 (1990).
[CrossRef]

Tanbun-Ek, T.

P. A. Andrekson, W. A. Olsson, J. R. Simpson, T. Tanbun-Ek, R. A. Logan, P. C. Becker, K. W. Wecht, Appl. Phys. Lett. 57, 1715 (1990).
[CrossRef]

Tappert, F. D.

A. Hasegawa, F. D. Tappert, Appl. Phys. Lett. 23, 142 (1973). Note that solitons that are periodically amplified are also described by Eq. (1) aproximately; see A. Hasegawa, Y. Kodama, Opt. Lett. 15, 1443 (1990).
[CrossRef] [PubMed]

Wecht, K. W.

P. A. Andrekson, W. A. Olsson, J. R. Simpson, T. Tanbun-Ek, R. A. Logan, P. C. Becker, K. W. Wecht, Appl. Phys. Lett. 57, 1715 (1990).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

Appl. Phys. Lett.

P. A. Andrekson, W. A. Olsson, J. R. Simpson, T. Tanbun-Ek, R. A. Logan, P. C. Becker, K. W. Wecht, Appl. Phys. Lett. 57, 1715 (1990).
[CrossRef]

A. Hasegawa, F. D. Tappert, Appl. Phys. Lett. 23, 142 (1973). Note that solitons that are periodically amplified are also described by Eq. (1) aproximately; see A. Hasegawa, Y. Kodama, Opt. Lett. 15, 1443 (1990).
[CrossRef] [PubMed]

Opt. Lett.

Zh. Eksp. Teor. Fiz.

V. E. Zakharov, A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

Other

L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, “Wavelength-division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” submitted to IEEE J. Lightwave Technol.

E. A. Overman, The Ohio State University, Columbus, Ohio (personal communication).

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Figures (2)

Fig. 1
Fig. 1

Frequency shift Δκ as a function of the degree of overlap θ0.

Fig. 2
Fig. 2

Results of numerical simulation on the initially overlapped two-soliton propagation at Z = 1.84 for different frequency spacings (a) κ = 2.68, (b) κ = 5.36, and (c) κ = 10.7. (a) The central frequency shift is so large that two solitons fail to separate; (b) solitons are separated with small frequency shift (see the small shift of the left soliton to the right); (c) solitons are separated with little frequency shift but with some deformation.

Equations (18)

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i q Z + 1 2 2 q T 2 + q 2 q = 0.
v ( T , 0 ) = S 1 ( T - θ 1 , η 1 , κ 1 , σ 1 ) + S 2 ( T - θ 2 , η 2 , κ 2 , σ 2 ) ,
S i ( T - θ i , η i , κ i , σ i ) = η i sech η i ( T - θ i ) × exp [ - i κ i ( T - θ i ) + i σ i ] .
i ψ 1 T + v ψ 2 = ζ ψ 1 , - i ψ 2 T - v * ψ 1 = ζ ψ 2 .
v ( T , 0 ) = η 1 sech η 1 ( T + θ 0 ) + η 2 sech η 2 ( T - θ 0 ) exp [ - i κ ( T - θ 0 ) + i σ 0 ] ,
ψ 1 T = O ( 1 ) .
ψ 1 = ϕ 1 - 1 κ S 2 exp ( - i κ T + ) ϕ 2 + O ( 1 κ 2 ) , ψ 2 = ϕ 2 + 1 κ S 2 * exp ( i κ T + ) ϕ 1 + O ( 1 κ 2 ) ,
i ϕ 1 T + S 1 ϕ 2 = ζ ϕ 1 + 1 κ ( - S 2 2 ϕ 1 ) + O [ exp ( ± i k T + ) κ , 1 κ 2 ] , - i ϕ 2 T - S 1 * ϕ 1 = ζ ϕ 2 + 1 κ ( - S 2 2 ϕ 2 ) + O [ exp ( ± i κ T + ) κ , 1 κ 2 ] ,
i ϕ 10 T + S 1 ϕ 20 + 1 κ S 2 2 ϕ 10 = ζ ϕ 10 + O ( 1 κ 2 ) , - i ϕ 20 T - S 1 * ϕ 10 + 1 κ S 2 2 ϕ 20 = ζ ϕ 20 + O ( 1 κ 2 ) .
ϕ i 0 = ϕ i 0 0 + 1 κ ϕ i 0 1 + O ( 1 κ 2 ) , ζ = ζ 1 0 + 1 κ ζ 1 1 + O ( 1 κ 2 ) ,
ϕ 10 0 = exp ( - 1 2 η 1 T + ) sech η 1 T + , ϕ 20 0 = i exp ( 1 2 η 1 T + ) sech η 1 T + .
i ϕ 10 1 T + S 1 ϕ 20 1 + S 2 2 ϕ 10 0 = i η 1 2 ϕ 10 1 + ζ 1 1 ϕ 10 0 , - i ϕ 20 1 T - S 1 * ϕ 10 1 + S 2 2 ϕ 20 0 = i η 1 2 ϕ 20 1 + ζ 1 1 ϕ 20 0 .
ζ 1 1 = - S 2 2 ϕ 10 0 ϕ 20 0 d T - ϕ 10 0 ϕ 20 0 d T = η 1 η 2 2 2 - sech 2 η 1 ( T + θ 0 ) sech 2 η 2 ( T - θ 0 ) d T .
Δ κ 1 = 4 η 2 3 κ .
2 π δ f = ± 4 3 1.76 2 τ s 2 1 2 π Δ f ,
δ f = 0.105 τ s 2 Δ f .
ζ 2 1 = - η 1 2 η 2 2 - sech 2 η 1 ( T + θ 0 ) sech 2 η 2 ( T - θ 0 ) d T .
q ( T , 0 ) = sech T + sech T exp ( i κ T ) .

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