Abstract

Using narrow solitons is a natural way to increase the data-transmission rate in soliton-based optical communications lines. However, in the present work it is demonstrated that it does not make sense to employ ultrashort solitons whose dispersion lengths Z0 are much smaller than the amplification spacing Za: In this case, the interaction between the solitons gives rise to stable bound states in which the temporal separation between the solitons only increases with a decrease of the soliton width. This is the reason why, when Z0 and Za are close, the case is potentially the most promising. Also in this case, however, resonance effects are possible. In the present work, resonance excitation of the internal vibrations of a fundamental soliton by the periodic amplification is considered analytically. Both the fundamental and the second resonances are analyzed in detail by means of a variational approximation. The most salient feature of the resonant propagation of the soliton is that it is bistable. Interaction between vibrating solitons is also considered, and it is shown that this interaction can be suppressed by vibrations with fairly small amplitudes, preventing formation of bound states. These resonant features are beneficial for use in communication lines. Emission of radiation by the vibrating soliton, which is a detrimental effect, is considered too, and it is demonstrated that it can be suppressed in two different ways: by the straightforward use of notch filters, or, in a smarter way, by the vibrations being switched into a range in which the first vibrational harmonic does not radiate.

© 1996 Optical Society of America

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1995 (4)

G. P. Agrawal and R.-J. Essiambre, J. Opt. Soc. Am. B 12, 2420 (1995).
[CrossRef]

W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
[CrossRef]

R. G. Bauer and L. A. Melnikov, Opt. Commun. 115, 190 (1995).
[CrossRef]

I. M. Uzunov, R. Muschall, M. Gölles, F. Lederer, and S. Wabnitz, Opt. Commun. 118, 577 (1995).
[CrossRef]

1994 (6)

1993 (5)

M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, Electron. Lett. 29, 729 (1993).
[CrossRef]

J. N. Elgin and S. M. J. Kelly, Opt. Lett. 18, 787 (1993).
[CrossRef] [PubMed]

B. A. Malomed, D. F. Parker, and N. F. Smyth, Phys. Rev. E 48, 1418 (1993).
[CrossRef]

K. C. Chan and H. F. Liu, Opt. Lett. 18, 1150 (1993).
[CrossRef]

H. H. Y. Cheng and H. F. Liu, Opt. Lett. 18, 1777 (1993).
[CrossRef] [PubMed]

1992 (2)

1991 (5)

Y. Kodama and A. Hasegawa, Phys. Rev. Lett. 66, 161 (1991); Opt. Lett. 15, 1443 (1991).
[CrossRef]

M. Nakazawa, E. Yoshida, and Y. Kimiura, Appl. Phys. Lett. 59, 2073 (1991); V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, Electron. Lett. 28, 1391 (1992); N. K. Smith, K. J. Blow, and I. Andonovic, J. Lightwave Technol. 10, 1329 (1992); N. Pandit, D. U. Noske, and J. R. Taylor, Opt. Lett. 17, 1515 (1992).
[CrossRef]

B. A. Malomed, Phys. Rev. A 44, 6954 (1991); Phys. Rev. E 47, 2874 (1993); Europhys. Lett. 30, 507 (1995).
[CrossRef] [PubMed]

G. P. Agrawal, Phys. Rev. A 44, 7493 (1991).
[CrossRef] [PubMed]

A. Meccozi, J. D. Moores, H. A. Haus, and Y. Lai, Opt. Lett. 16, 1841 (1991); Y. Kodama and A. Hasegawa, Opt. Lett. 17, 31 (1992).
[CrossRef] [PubMed]

1989 (1)

Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

1988 (2)

D. Anderson, M. Lisak, and T. Reichel, J. Opt. Soc. Am. B 5, 207 (1988).
[CrossRef]

D. Anderson, M. Lisak, and T. Reichel, Phys. Rev. A 38, 1618 (1988).
[CrossRef] [PubMed]

1987 (2)

Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
[CrossRef]

B. A. Malomed, Opt. Commun. 61, 192 (1987).
[CrossRef]

1986 (1)

1983 (1)

D. Anderson, Phys. Rev. A 27, 3135 (1983).
[CrossRef]

1982 (2)

R. K. Bullough, A. P. Fordy, and S. V. Manakov, Phys. Lett. A 91, 98 (1982).
[CrossRef]

A. Hasegawa and Y. Kodama, Opt. Lett. 7, 339 (1982).
[CrossRef]

1979 (1)

A. Bondeson, M. Lisak, and D. Anderson, Phys. Scri. 20, 479 (1979).
[CrossRef]

1974 (1)

J. Satsuma and N. Yajima, Prog. Theor. Phys. 55, Suppl., 284 (1974).
[CrossRef]

Agrawal, G. P.

Akiba, S.

M. Suzuki, N. Edagawa, H. Taga, H. Tanaka, S. Yamamoto, and S. Akiba, IEEE Photonics Technol. Lett. 6, 757 (1994).
[CrossRef]

Anderson, D.

Bauer, R. G.

R. G. Bauer and L. A. Melnikov, Opt. Commun. 115, 190 (1995).
[CrossRef]

Bondeson, A.

A. Bondeson, M. Lisak, and D. Anderson, Phys. Scri. 20, 479 (1979).
[CrossRef]

Bullough, R. K.

R. K. Bullough, A. P. Fordy, and S. V. Manakov, Phys. Lett. A 91, 98 (1982).
[CrossRef]

Chan, K. C.

Cheng, H. H. Y.

Desurvire, E.

E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley-Interscience, New York, 1994).

Doran, N. J.

Edagawa, N.

M. Suzuki, N. Edagawa, H. Taga, H. Tanaka, S. Yamamoto, and S. Akiba, IEEE Photonics Technol. Lett. 6, 757 (1994).
[CrossRef]

Elgin, J. N.

Essiambre, R.-J.

Evangelides, S. G.

Fordy, A. P.

R. K. Bullough, A. P. Fordy, and S. V. Manakov, Phys. Lett. A 91, 98 (1982).
[CrossRef]

Forysiak, W.

Gölles, M.

I. M. Uzunov, R. Muschall, M. Gölles, F. Lederer, and S. Wabnitz, Opt. Commun. 118, 577 (1995).
[CrossRef]

Gordon, J. P.

Grimshaw, R.

R. Grimshaw, J. He, and B. A. Malomed, Phys. Scri. (to be published).

Hasegawa, A.

Y. Kodama and A. Hasegawa, Phys. Rev. Lett. 66, 161 (1991); Opt. Lett. 15, 1443 (1991).
[CrossRef]

Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
[CrossRef]

A. Hasegawa and Y. Kodama, Opt. Lett. 7, 339 (1982).
[CrossRef]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, Oxford, 1995).

Haus, H.

Haus, H. A.

He, J.

R. Grimshaw, J. He, and B. A. Malomed, Phys. Scri. (to be published).

Kath, W. L.

W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
[CrossRef]

Kelly, S. M. J.

Kimiura, Y.

M. Nakazawa, E. Yoshida, and Y. Kimiura, Appl. Phys. Lett. 59, 2073 (1991); V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, Electron. Lett. 28, 1391 (1992); N. K. Smith, K. J. Blow, and I. Andonovic, J. Lightwave Technol. 10, 1329 (1992); N. Pandit, D. U. Noske, and J. R. Taylor, Opt. Lett. 17, 1515 (1992).
[CrossRef]

Kimura, Y.

M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, Electron. Lett. 29, 729 (1993).
[CrossRef]

Kivshar, Yu. S.

Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Knox, F. M.

Kodama, Y.

Y. Kodama and S. Wabnitz, Opt. Lett. 19, 162 (1994).
[CrossRef]

Y. Kodama and A. Hasegawa, Phys. Rev. Lett. 66, 161 (1991); Opt. Lett. 15, 1443 (1991).
[CrossRef]

Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
[CrossRef]

A. Hasegawa and Y. Kodama, Opt. Lett. 7, 339 (1982).
[CrossRef]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, Oxford, 1995).

Kubota, H.

M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, Electron. Lett. 29, 729 (1993).
[CrossRef]

Lai, Y.

Landau, L. D.

L. D. Landau and E. M. Lifshits, Mechanics, in Russian (Nauka, Moscow, 1973) [English translation (Pergamon, New York, 1976)].

Lederer, F.

I. M. Uzunov, R. Muschall, M. Gölles, F. Lederer, and S. Wabnitz, Opt. Commun. 118, 577 (1995).
[CrossRef]

Lifshits, E. M.

L. D. Landau and E. M. Lifshits, Mechanics, in Russian (Nauka, Moscow, 1973) [English translation (Pergamon, New York, 1976)].

Lisak, M.

Liu, H. F.

Malomed, B.

Malomed, B. A.

B. A. Malomed and N. F. Smyth, Phys. Rev. E 50, 1535 (1994).
[CrossRef]

B. A. Malomed, Opt. Lett. 19, 341 (1994).
[CrossRef] [PubMed]

B. A. Malomed, D. F. Parker, and N. F. Smyth, Phys. Rev. E 48, 1418 (1993).
[CrossRef]

B. A. Malomed, Phys. Rev. A 44, 6954 (1991); Phys. Rev. E 47, 2874 (1993); Europhys. Lett. 30, 507 (1995).
[CrossRef] [PubMed]

Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

B. A. Malomed, Opt. Commun. 61, 192 (1987).
[CrossRef]

R. Grimshaw, J. He, and B. A. Malomed, Phys. Scri. (to be published).

Manakov, S. V.

R. K. Bullough, A. P. Fordy, and S. V. Manakov, Phys. Lett. A 91, 98 (1982).
[CrossRef]

Meccozi, A.

Melnikov, L. A.

R. G. Bauer and L. A. Melnikov, Opt. Commun. 115, 190 (1995).
[CrossRef]

Mollenauer, L. F.

Moores, J. D.

Muschall, R.

I. M. Uzunov, R. Muschall, M. Gölles, F. Lederer, and S. Wabnitz, Opt. Commun. 118, 577 (1995).
[CrossRef]

Nakazawa, M.

M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, Electron. Lett. 29, 729 (1993).
[CrossRef]

M. Nakazawa, E. Yoshida, and Y. Kimiura, Appl. Phys. Lett. 59, 2073 (1991); V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, Electron. Lett. 28, 1391 (1992); N. K. Smith, K. J. Blow, and I. Andonovic, J. Lightwave Technol. 10, 1329 (1992); N. Pandit, D. U. Noske, and J. R. Taylor, Opt. Lett. 17, 1515 (1992).
[CrossRef]

Parker, D. F.

B. A. Malomed, D. F. Parker, and N. F. Smyth, Phys. Rev. E 48, 1418 (1993).
[CrossRef]

Quiroga-Teixeiro, M.

Reichel, T.

D. Anderson, M. Lisak, and T. Reichel, J. Opt. Soc. Am. B 5, 207 (1988).
[CrossRef]

D. Anderson, M. Lisak, and T. Reichel, Phys. Rev. A 38, 1618 (1988).
[CrossRef] [PubMed]

Satsuma, J.

J. Satsuma and N. Yajima, Prog. Theor. Phys. 55, Suppl., 284 (1974).
[CrossRef]

Smyth, N. F.

W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
[CrossRef]

B. A. Malomed and N. F. Smyth, Phys. Rev. E 50, 1535 (1994).
[CrossRef]

B. A. Malomed, D. F. Parker, and N. F. Smyth, Phys. Rev. E 48, 1418 (1993).
[CrossRef]

Suzuki, K.

M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, Electron. Lett. 29, 729 (1993).
[CrossRef]

Suzuki, M.

M. Suzuki, N. Edagawa, H. Taga, H. Tanaka, S. Yamamoto, and S. Akiba, IEEE Photonics Technol. Lett. 6, 757 (1994).
[CrossRef]

Taga, H.

M. Suzuki, N. Edagawa, H. Taga, H. Tanaka, S. Yamamoto, and S. Akiba, IEEE Photonics Technol. Lett. 6, 757 (1994).
[CrossRef]

Takaya, M.

M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, Electron. Lett. 29, 729 (1993).
[CrossRef]

Tanaka, H.

M. Suzuki, N. Edagawa, H. Taga, H. Tanaka, S. Yamamoto, and S. Akiba, IEEE Photonics Technol. Lett. 6, 757 (1994).
[CrossRef]

Uzunov, I. M.

I. M. Uzunov, R. Muschall, M. Gölles, F. Lederer, and S. Wabnitz, Opt. Commun. 118, 577 (1995).
[CrossRef]

Wabnitz, S.

I. M. Uzunov, R. Muschall, M. Gölles, F. Lederer, and S. Wabnitz, Opt. Commun. 118, 577 (1995).
[CrossRef]

Y. Kodama and S. Wabnitz, Opt. Lett. 19, 162 (1994).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, Prog. Theor. Phys. 55, Suppl., 284 (1974).
[CrossRef]

Yamada, E.

M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, Electron. Lett. 29, 729 (1993).
[CrossRef]

Yamamoto, S.

M. Suzuki, N. Edagawa, H. Taga, H. Tanaka, S. Yamamoto, and S. Akiba, IEEE Photonics Technol. Lett. 6, 757 (1994).
[CrossRef]

Yoshida, E.

M. Nakazawa, E. Yoshida, and Y. Kimiura, Appl. Phys. Lett. 59, 2073 (1991); V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, Electron. Lett. 28, 1391 (1992); N. K. Smith, K. J. Blow, and I. Andonovic, J. Lightwave Technol. 10, 1329 (1992); N. Pandit, D. U. Noske, and J. R. Taylor, Opt. Lett. 17, 1515 (1992).
[CrossRef]

Appl. Phys. Lett. (1)

M. Nakazawa, E. Yoshida, and Y. Kimiura, Appl. Phys. Lett. 59, 2073 (1991); V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, Electron. Lett. 28, 1391 (1992); N. K. Smith, K. J. Blow, and I. Andonovic, J. Lightwave Technol. 10, 1329 (1992); N. Pandit, D. U. Noske, and J. R. Taylor, Opt. Lett. 17, 1515 (1992).
[CrossRef]

Electron. Lett. (1)

M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, Electron. Lett. 29, 729 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

M. Suzuki, N. Edagawa, H. Taga, H. Tanaka, S. Yamamoto, and S. Akiba, IEEE Photonics Technol. Lett. 6, 757 (1994).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (3)

I. M. Uzunov, R. Muschall, M. Gölles, F. Lederer, and S. Wabnitz, Opt. Commun. 118, 577 (1995).
[CrossRef]

R. G. Bauer and L. A. Melnikov, Opt. Commun. 115, 190 (1995).
[CrossRef]

B. A. Malomed, Opt. Commun. 61, 192 (1987).
[CrossRef]

Opt. Lett. (11)

Phys. Lett. A (1)

R. K. Bullough, A. P. Fordy, and S. V. Manakov, Phys. Lett. A 91, 98 (1982).
[CrossRef]

Phys. Rev. A (4)

D. Anderson, Phys. Rev. A 27, 3135 (1983).
[CrossRef]

D. Anderson, M. Lisak, and T. Reichel, Phys. Rev. A 38, 1618 (1988).
[CrossRef] [PubMed]

B. A. Malomed, Phys. Rev. A 44, 6954 (1991); Phys. Rev. E 47, 2874 (1993); Europhys. Lett. 30, 507 (1995).
[CrossRef] [PubMed]

G. P. Agrawal, Phys. Rev. A 44, 7493 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (3)

B. A. Malomed, D. F. Parker, and N. F. Smyth, Phys. Rev. E 48, 1418 (1993).
[CrossRef]

B. A. Malomed and N. F. Smyth, Phys. Rev. E 50, 1535 (1994).
[CrossRef]

W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

Y. Kodama and A. Hasegawa, Phys. Rev. Lett. 66, 161 (1991); Opt. Lett. 15, 1443 (1991).
[CrossRef]

Phys. Scri. (1)

A. Bondeson, M. Lisak, and D. Anderson, Phys. Scri. 20, 479 (1979).
[CrossRef]

Prog. Theor. Phys. (1)

J. Satsuma and N. Yajima, Prog. Theor. Phys. 55, Suppl., 284 (1974).
[CrossRef]

Rev. Mod. Phys. (1)

Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Other (4)

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, Oxford, 1995).

R. Grimshaw, J. He, and B. A. Malomed, Phys. Scri. (to be published).

L. D. Landau and E. M. Lifshits, Mechanics, in Russian (Nauka, Moscow, 1973) [English translation (Pergamon, New York, 1976)].

E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley-Interscience, New York, 1994).

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Equations (41)

Equations on this page are rendered with MathJax. Learn more.

i u z + 1 2 u r r + | u | 2 u = i γ 0 ( z ) u + i γ 1 u r r ,
γ 0 ( z ) = Γ 0 + Γ 1 n = + δ ( z Z a n ) ,
γ ¯ 0 = 1 3 γ 1 W 2 ,
2 W + | μ ( τ ) | 2 d τ .
γ 0 ( z ) γ ¯ 0 + γ ˜ 0 ( z ) .
U ( z , τ ) exp [ Γ ˜ ( z ) ] u ( z , τ ) ,
Γ ˜ ( z ) γ ˜ 0 ( z ) d z .
i U z + 1 2 U r r + exp [ 2 Γ ˜ ( z ) ] | U | 2 U = i ( γ ¯ 0 U + γ 1 U r r ) .
U sol = A sech ( τ a ) exp ( i ϕ + i b τ 2 ) ,
b = 1 2 a 1 d a d z .
π ( a ) = 1 2 a 2 W exp [ ( 4 π Γ ˜ ( z ) ) ] a 1 .
d 2 a d ζ 2 = a 3 W 0 exp [ Δ ( ζ ) ] a 2 λ γ 1 W 0 2 d a d ζ ,
Δ ( ζ ) = 4 π Γ ˜ ( z )
b = 1 2 a 1 d a d z 4 π 2 γ 1 a 2 .
d W d z = 4 γ ¯ 0 W 4 3 γ 1 W 3 4 3 π 2 γ 1 W 2 a 1 b 2 .
Δ ( ζ ) = 4 π 2 Γ 1 n = 1 n 1 sin ( π 2 Z a n ζ ) ,
d 2 α d x 2 + λ γ 1 d α d x + α 3 α 2 + 6 α 3 = Δ 0 cos ( k x ) ,
α W 0 a 1 , k π 2 / Z a W 0 2 , x W 0 2 ζ .
α ( x ) = α 1 cos ( k x ) + α 0 + α 2 cos ( 2 k x ) + + ,
eff = + 3 2 α 1 2 ,
Δ 0 2 > 4 9 3 λ 3 γ 1 3 ,
α 1 ( + 3 α 1 2 ) = 1 2 Δ 0 .
3 < ( 9 4 Δ 0 ) 2 .
( α 1 ) max = Δ 0 / λ γ 1 ,
α ( x ) = 1 3 Δ 0 cos ( k x ) + β ( x ) ,
d 2 β d x 2 + λ γ 1 d β d x + β 3 β 2 + 6 β 3 = 2 Δ 0 cos ( k x ) β .
β ( x ) = β 1 cos ( k x + δ ) + β 0 + β 2 cos ( 2 k x + 2 δ ) ,
eff = + 3 2 β 1 2 .
β 1 2 = 1 6 ( ± Δ 0 2 λ 2 γ 1 2 ) .
U exp ( W 0 | τ | + i ϕ + i b τ 2 + i Ω | τ | ) .
Ω = 4 3 γ ¯ 0 γ 1 ,
τ 2 ( 2 W 0 α 1 ) 2 .
V ( T , Δ ϕ ) ( W 0 | T | ) cos [ Δ ϕ + 1 2 ( b 1 b 2 ) T 2 ] × cos [ Ω | T | + 1 2 ( b 1 + b 2 ) T 2 ] ,
T 0 = π / 2 Ω .
T 0 = π W 0 / 2 Γ 0 .
α 1 2 γ 1 2 .
U cw = U 0 exp ( i K z i ω τ ) ,
d ϕ d z = 1 2 W 0 2 K 0 .
Q = ± 2 π W 0 2
0 4 π π 0.27,
= 3 α 1 2 + 1 2 Δ 0 α 1 1 .

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