Abstract

Two methods of controlling the temporal diffusion of solitons in long-distance transmission are investigated theoretically. One, pursued experimentally by Nakazawa et al. [ Electron. Lett. 27, 1270 ( 1991)] uses amplitude modulation synchronized with the bit rate. The other method, first proposed recently, uses filtering. The latter is better adapted to wavelength division multiplexing. It is shown that the so-called Gordon-Haus limit of soliton propagation can be extended significantly.

© 1992 Optical Society of America

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References

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  1. N. Nakazawa, Y. Kimura, K. Suzuki, and H. Kubota, J. Appl. Phys. 66, 2803 (1989).
    [Crossref]
  2. A. Chraplyvy, AT&T Bell Laboratories, Holmdel N.J. 07733 (personal communication, 1991).
  3. L. F. Mollenauer, B. M. Nyman, M. J. Neubelt, G. Raydon, and S. G. Evangelides, Electron. Lett. 27, 178 (1991).
    [Crossref]
  4. M. Nakazawa, K. Suzuki, H. Kubota, E. Yamada, and Y. Kimura, IEEE J. Quantum Electron. 26, 2095 (1990).
    [Crossref]
  5. L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, IEEE J. Lightwave Technol. 9, 362 (1991).
    [Crossref]
  6. J. P. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986).
    [Crossref] [PubMed]
  7. M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
    [Crossref]
  8. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, Opt. Lett. 16, 1841 (1991).We have learned that Kodama and Hasegawa have arrived independently at the same scheme [Y. Kodama and A. Hasegawa, Opt. Lett. 17, 31 (1992)].
    [Crossref] [PubMed]
  9. H. A. Haus, J. Opt. Soc. Am. B 8, 1122 (1991).
    [Crossref]
  10. L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, IEEE J. Lightwave Technol. 9, 194 (1991).
    [Crossref]
  11. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
    [Crossref]
  12. D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
    [Crossref] [PubMed]

1991 (6)

L. F. Mollenauer, B. M. Nyman, M. J. Neubelt, G. Raydon, and S. G. Evangelides, Electron. Lett. 27, 178 (1991).
[Crossref]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, IEEE J. Lightwave Technol. 9, 362 (1991).
[Crossref]

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[Crossref]

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, Opt. Lett. 16, 1841 (1991).We have learned that Kodama and Hasegawa have arrived independently at the same scheme [Y. Kodama and A. Hasegawa, Opt. Lett. 17, 31 (1992)].
[Crossref] [PubMed]

H. A. Haus, J. Opt. Soc. Am. B 8, 1122 (1991).
[Crossref]

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, IEEE J. Lightwave Technol. 9, 194 (1991).
[Crossref]

1990 (3)

H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
[Crossref]

D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[Crossref] [PubMed]

M. Nakazawa, K. Suzuki, H. Kubota, E. Yamada, and Y. Kimura, IEEE J. Quantum Electron. 26, 2095 (1990).
[Crossref]

1989 (1)

N. Nakazawa, Y. Kimura, K. Suzuki, and H. Kubota, J. Appl. Phys. 66, 2803 (1989).
[Crossref]

1986 (1)

Chraplyvy, A.

A. Chraplyvy, AT&T Bell Laboratories, Holmdel N.J. 07733 (personal communication, 1991).

Evangelides, S. G.

L. F. Mollenauer, B. M. Nyman, M. J. Neubelt, G. Raydon, and S. G. Evangelides, Electron. Lett. 27, 178 (1991).
[Crossref]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, IEEE J. Lightwave Technol. 9, 362 (1991).
[Crossref]

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, IEEE J. Lightwave Technol. 9, 194 (1991).
[Crossref]

Gordon, J. P.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, IEEE J. Lightwave Technol. 9, 362 (1991).
[Crossref]

J. P. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986).
[Crossref] [PubMed]

Haus, H. A.

Kaup, D. J.

D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[Crossref] [PubMed]

Kimura, Y.

M. Nakazawa, K. Suzuki, H. Kubota, E. Yamada, and Y. Kimura, IEEE J. Quantum Electron. 26, 2095 (1990).
[Crossref]

N. Nakazawa, Y. Kimura, K. Suzuki, and H. Kubota, J. Appl. Phys. 66, 2803 (1989).
[Crossref]

Kubota, H.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[Crossref]

M. Nakazawa, K. Suzuki, H. Kubota, E. Yamada, and Y. Kimura, IEEE J. Quantum Electron. 26, 2095 (1990).
[Crossref]

N. Nakazawa, Y. Kimura, K. Suzuki, and H. Kubota, J. Appl. Phys. 66, 2803 (1989).
[Crossref]

Lai, Y.

Mecozzi, A.

Mollenauer, L. F.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, IEEE J. Lightwave Technol. 9, 362 (1991).
[Crossref]

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, IEEE J. Lightwave Technol. 9, 194 (1991).
[Crossref]

L. F. Mollenauer, B. M. Nyman, M. J. Neubelt, G. Raydon, and S. G. Evangelides, Electron. Lett. 27, 178 (1991).
[Crossref]

Moores, J. D.

Nakazawa, M.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[Crossref]

M. Nakazawa, K. Suzuki, H. Kubota, E. Yamada, and Y. Kimura, IEEE J. Quantum Electron. 26, 2095 (1990).
[Crossref]

Nakazawa, N.

N. Nakazawa, Y. Kimura, K. Suzuki, and H. Kubota, J. Appl. Phys. 66, 2803 (1989).
[Crossref]

Neubelt, M. J.

L. F. Mollenauer, B. M. Nyman, M. J. Neubelt, G. Raydon, and S. G. Evangelides, Electron. Lett. 27, 178 (1991).
[Crossref]

Nyman, B. M.

L. F. Mollenauer, B. M. Nyman, M. J. Neubelt, G. Raydon, and S. G. Evangelides, Electron. Lett. 27, 178 (1991).
[Crossref]

Raydon, G.

L. F. Mollenauer, B. M. Nyman, M. J. Neubelt, G. Raydon, and S. G. Evangelides, Electron. Lett. 27, 178 (1991).
[Crossref]

Suzuki, K.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[Crossref]

M. Nakazawa, K. Suzuki, H. Kubota, E. Yamada, and Y. Kimura, IEEE J. Quantum Electron. 26, 2095 (1990).
[Crossref]

N. Nakazawa, Y. Kimura, K. Suzuki, and H. Kubota, J. Appl. Phys. 66, 2803 (1989).
[Crossref]

Yamada, E.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[Crossref]

M. Nakazawa, K. Suzuki, H. Kubota, E. Yamada, and Y. Kimura, IEEE J. Quantum Electron. 26, 2095 (1990).
[Crossref]

Electron. Lett. (2)

L. F. Mollenauer, B. M. Nyman, M. J. Neubelt, G. Raydon, and S. G. Evangelides, Electron. Lett. 27, 178 (1991).
[Crossref]

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[Crossref]

IEEE J. Lightwave Technol. (2)

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, IEEE J. Lightwave Technol. 9, 362 (1991).
[Crossref]

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, IEEE J. Lightwave Technol. 9, 194 (1991).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Nakazawa, K. Suzuki, H. Kubota, E. Yamada, and Y. Kimura, IEEE J. Quantum Electron. 26, 2095 (1990).
[Crossref]

J. Appl. Phys. (1)

N. Nakazawa, Y. Kimura, K. Suzuki, and H. Kubota, J. Appl. Phys. 66, 2803 (1989).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Lett. (2)

Phys. Rev. A (1)

D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[Crossref] [PubMed]

Other (1)

A. Chraplyvy, AT&T Bell Laboratories, Holmdel N.J. 07733 (personal communication, 1991).

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Figures (2)

Fig. 1
Fig. 1

Variance of the timing fluctuations versus distance (in soliton period units) for the modulation case. The values of the parameters are μτ2/D = 1.0 and ωmτ = 0.25.

Fig. 2
Fig. 2

Variance of the timing fluctuations versus distance (in soliton period units) for the filtering case. The value of the parameters is 1/Ωf2lD = 0.1.

Equations (75)

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d = 1 2 Γ ln ( 1 r 2 ) ,
r 2 = 1 exp ( 2 Γ l ) 2 Γ l
z u = j D 2 t 2 u j r 2 δ | u | 2 u + Δ g u + μ ( cos ω m t 1 ) u + 1 Ω f 2 l ( 2 t 2 + 2 j Δ ω t Δ ω 2 ) u + S ( z , t ) .
D = 1 2 β .
δ = 2 π λ n 2 ω 0 A eff ,
E R 2 ( cos ω m t 1 ) ,
μ = E R 2 l ,
z u = j D 2 t 2 u j r 2 δ | u | 2 u + Δ g u μ 2 ω m 2 t 2 u + 1 Ω f 2 l ( 2 t 2 + 2 j Δ ω t Δ ω 2 ) u + S ( z , t ) .
u o ( z , t ) = A 0 sech ( t T τ ) × exp j ( p t D p 2 z + D z τ 2 + θ ) ,
2 A 0 2 τ = n .
τ = ( 2 D r 2 δ A 0 2 ) 1 / 2 = 4 D r 2 δ n .
z u = j D 2 t 2 u 2 r 2 δ | u | 2 u + Δ g u μ 2 ω m 2 [ ( t T ) 2 + 2 ( t T ) T + T 2 ] u + 1 Ω f 2 l ( 2 t 2 2 j p t p 2 ) u + S ( z , t ) .
u ( z , t ) = [ U 0 ( t ) + Δ u ( z , t ) ] exp j ( D p 2 z + D z τ 2 + θ 0 ) ,
U 0 A 0 sech ( t T τ ) exp ( j p t )
z Δ u + j ( D p 2 + D / τ 2 ) Δ u = j ( D 2 t 2 Δ u + 2 r 2 δ | U 0 | 2 Δ u + r 2 δ U 0 2 Δ u * ) + Δ g U 0 μ 2 ω m 2 [ ( t T ) 2 + 2 ( t T ) T + T 2 ] U 0 + 1 Ω f 2 l ( 2 t 2 2 j p t p 2 ) U 0 + S ( z , t ) .
Δ u = Δ n ( z ) f n ( t ) + Δ θ f θ ( t ) + Δ p ( z ) f p ( t ) + Δ T ( z ) f T ( t ) + Δ u c ,
f n = n u 0 ( z = 0 , t ) | p = T = θ = 0 = 1 n [ 1 t τ tanh ( t / τ ) ] A 0 sech ( t / τ ) ,
f θ = θ u 0 ( z = 0 , t ) | p = T = θ = 0 = j A 0 sech ( t / τ ) ,
f T = T u 0 ( z = 0 , t ) | p = T = θ = 0 = 1 τ tanh ( t / τ ) A 0 sech ( t / τ ) ,
f p = p u 0 ( z = 0 , t ) | p = T = θ = 0 = j t A 0 sech ( t / τ ) .
Re f i * f j d t = δ i j , i , j = n , θ , p , T
f n ( t ) = 2 A 0 sech ( t / τ ) ,
f θ ( t ) = ( 2 j / n ) [ 1 ( t / τ ) tanh ( t / τ ) ] A 0 sech ( t / τ ) ,
f p ( t ) = j [ 2 / ( n τ ) tanh ( t / τ ) ] A 0 sech ( t / τ ) ,
f T ( t ) = ( 2 / n ) t A 0 sech ( t / τ ) .
z Δ n ( z ) = S n ( t ) ,
z Δ θ ( z ) = 2 D τ 2 Δ n n + S θ ( z ) ,
z Δ T ( z ) = 2 D Δ p + S T ( z ) ,
z Δ p ( z ) = S p ( z ) ,
S i ( z ) = Re S ( z , t ) f i ( t ) d t .
z Δ p ( z ) = 4 3 1 Ω f 2 l 1 τ 2 Δ p ( z ) + S p ( z ) ,
z Δ T ( z ) = 2 D Δ p τ 2 π 2 6 μ ω m 2 Δ T ( z ) + S T ( z ) ,
f T ( t ) t U 0 ( t ) d t = 2 A o 2 r 2 n τ 3 π 2 6 = τ 2 π 2 6 ,
| f p ( t ) | t U 0 ( t ) d t = 4 3 A o 2 r 2 n r = 2 3 τ 2 .
S * ( z , t ) S ( z , t ) = 2 Γ δ ( t t ) δ ( z z ) .
f = ( G 1 ) 2 G ( ln G ) 2 ,
S p ( z ) S p ( z ) = δ ( z z ) 4 3 r 2 n f Γ τ 2 .
1 / 2 π L d z f ( z ) exp ( jKz ) = f ( K ) , L / 2 π d K f ( K ) exp ( jKz ) = f ( z ) ,
| Δ p ( K ) | 2 = N p 2 π K 2 ,
N p = 4 3 r 2 n f Γ τ 2 .
Δ p ( z ) Δ p ( z ) = N p z , z < z .
| Δ T ( K ) | 2 = ( 2 D ) 2 N p / 2 π K 2 ( K 2 + 1 / z T 2 ) ,
z T 6 μ ω m 2 π 2 τ 2 .
lim K 0 | Δ T ( K ) | 2 = ( 2 D ) 2 N p 2 π ( K / z T ) 2 .
Δ T ( z ) Δ T ( z ) = ( 2 D ) 2 N p z T 2 z ; z < z .
Δ T 2 = 88 ps 2 .
| Δ p ( K ) | 2 = N p / 2 π K 2 + 1 / z p 2 ,
z p 3 Ω f 2 l τ 2 4 .
S T ( z ) S T ( z ) = N T δ ( z z ) ,
N T f π 2 3 1 r 2 n τ 2 Γ .
| Δ T ( K ) | 2 ( 2 D ) 2 N p z p 2 + N T 2 π K 2 ,
Δ T ( z ) Δ T ( z ) = [ ( 2 D ) 2 N p z p 2 + N T ] z , z < z .
n z = d t 2 U 0 ( Δ g μ 2 ω m 2 t 2 + 1 Ω f 2 l 2 t 2 ) U 0 = 0 ,
Δ g = 1 24 π 2 τ 2 μ ω m 2 + 1 3 τ 2 1 Ω f 2 l .
z Δ n = Δ n n d t 2 U 0 ( Δ g μ 2 ω m 2 t 2 + 1 Ω f 2 l 2 t 2 ) U 0 = Δ n d t 4 U 0 ( Δ g μ 2 ω m 2 t 2 + 1 Ω f 2 l 2 t 2 ) n U 0 2 n d g d n Δ n ,
z Δ n ( z ) = 2 n d g d n Δ n μ ω m 2 n Δ T 2 2 Ω f 2 l n Δ p 2 + S n ( z ) .
2 n d g d n = 4 1 24 π 2 τ 2 μ ω m 2 .
2 n d g d n = 4 1 3 τ 2 1 Ω m 2 l .
z u = j D 2 t 2 u + Δ g u + 1 Ω f 2 l z u + S ( z , t ) .
S * ( z , t ) S ( z , t ) = 2 Γ f δ ( z z ) δ ( t t ) .
d t f ( t ) exp ( j Ω t ) = f ( Ω ) , 1 2 π d Ω f ( Ω ) exp ( j Ω t ) = f ( t ) ,
S * ( z , Ω ) S ( z , Ω ) = 2 Γ f 2 π δ ( Ω Ω ) δ ( z z ) . ( 6.9 )
u ( z , Ω ) = u ( 0 , Ω ) exp { [ Δ g ( 1 Ω f 2 l j D ) Ω 2 ] z } + 0 z d z exp { [ Δ g ( 1 Ω f 2 l j D ) Ω 2 ] ( z z ) } × S ( z , Ω ) .
u * ( z , Ω ) u ( z , Ω ) = 2 Γ f exp { [ ( 2 Δ g 1 Ω f 2 l ) ( Ω 2 + Ω 2 ) + j D ( Ω 2 Ω 2 ) ] z } 1 [ ( 2 Δ g 1 Ω f 2 l ) ( Ω 2 + Ω 2 ) + j D ( Ω 2 Ω 2 ) ] 2 π δ ( Ω Ω )
u * ( z , t ) u ( z , t ) = 2 Γ f d Ω 2 π exp { 2 ( Δ g 1 Ω f 2 l Ω 2 ) z } 1 2 ( Δ g 1 Ω f 2 l Ω 2 ) .
u * ( z , t ) u ( z , t ) 2 Γ z f 1 2 π exp ( 2 Δ g z ) 2 Δ g z × ( Ω f 2 l 2 Δ g 2 Δ g z 1 ) 1 / 2 .
n c 2 Γ z · f 1 2 π exp ( 2 Δ g z ) 2 Δ g z [ 1 6 1 ( 2 Δ g z 1 ) ] 1 / 2 · t o τ .
Δ g z = π 12 1 D l Ω f 2 z z o ,
z o = π 4 τ 2 D .
Δ T ( z ) 2 = Δ T ( 0 ) 2 + Δ p ( 0 ) 2 f 1 ( z ) + 2 3 π 2 2 Γ f 1 r 2 n τ 2 4 f 2 ( z ) + ( 2 D ) 2 2 Γ f 1 r 2 n 2 9 τ 2 f 3 ( z ) ,
f 1 ( z ) = z p 2 [ 1 exp ( z / z p ) ] 2 ,
f 2 ( z ) = z ,
f 3 ( z ) = 3 z p 3 [ ( z / z p ) 3 / 2 + 2 exp ( z / z p ) ½ exp ( 2 z / z p ) ] .
f 2 ( z ) = ( z T / 2 ) [ 1 exp ( 2 z / z T ) ] .
R 3 f 3 ( L ) = 0.1372 f 1 t s t w 2 R 3 A eff / h ( 2 Γ ) n 2 D ,

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