Abstract

It is shown that in the limit when the amplifier spacing in an all-optical amplifier–repeater system is much smaller than a soliton period, an equivalent nonlinear Schrödinger equation can be developed. It contains noise sources that account for the noise introduced by the attenuation and the amplifiers. The limit on soliton propagation caused by amplifier noise, as formulated by Gordon and Haus [ Opt. Lett. 11, 665 ( 1986)], becomes rigorous for this system.

© 1991 Optical Society of America

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References

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  1. A. Hasegawa, “Amplification and reshaping of optical solitons in glass fiber—IV: use of the stimulated Raman process,” Opt. Lett. 8, 650 (1983).
    [Crossref] [PubMed]
  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 (1986).
    [Crossref]
  3. L. F. Mollenauer and K. Smith, “Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain,” Opt. Lett. 13, 675 (1988).
    [Crossref] [PubMed]
  4. L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long distance soliton propagation using lumped amplifiers and dispersion-shifted fiber,” IEEE J. Lightwave Technol. 9, 194 (1991).
    [Crossref]
  5. H. Kubota and M. Nakazawa, “Long-distance optical soliton transmission with lumped amplifiers,” IEEE J. Quantum Electron. 26, 692 (1990).
    [Crossref]
  6. K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron. Lett. 26, 539 (1990).
    [Crossref]
  7. M. Nakazawa, K. Suzuki, and Y. Kimura, “3.2–5 Gb/s, 100 error-free soliton transmissions with erbium amplifiers and repeaters,” IEEE Photonics Tech. Lett. 26, 216 (1990); K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron Lett. 26, 551 (1990).
    [Crossref]
  8. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665 (1986).
    [Crossref] [PubMed]
  9. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386 (1990).
    [Crossref]
  10. Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844 (1989); “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854 (1989).
    [Crossref] [PubMed]
  11. H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. IEEE 58, 1599 (1970).
    [Crossref]
  12. H. A. Haus and Y. Yamamoto, “Quantum circuit theory of phase-sensitive linear systems,” IEEE J. Quantum Electron. QE-23, 212 (1987).
    [Crossref]
  13. Y. Lai and H. A. Haus, “Quantum theory of self-induced transparency solitons: a linearization approach,” Phys. Rev. A 42, 2925 (1990).
    [Crossref] [PubMed]

1991 (1)

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long distance soliton propagation using lumped amplifiers and dispersion-shifted fiber,” IEEE J. Lightwave Technol. 9, 194 (1991).
[Crossref]

1990 (5)

H. Kubota and M. Nakazawa, “Long-distance optical soliton transmission with lumped amplifiers,” IEEE J. Quantum Electron. 26, 692 (1990).
[Crossref]

K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron. Lett. 26, 539 (1990).
[Crossref]

M. Nakazawa, K. Suzuki, and Y. Kimura, “3.2–5 Gb/s, 100 error-free soliton transmissions with erbium amplifiers and repeaters,” IEEE Photonics Tech. Lett. 26, 216 (1990); K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron Lett. 26, 551 (1990).
[Crossref]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386 (1990).
[Crossref]

Y. Lai and H. A. Haus, “Quantum theory of self-induced transparency solitons: a linearization approach,” Phys. Rev. A 42, 2925 (1990).
[Crossref] [PubMed]

1989 (1)

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844 (1989); “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854 (1989).
[Crossref] [PubMed]

1988 (1)

1987 (1)

H. A. Haus and Y. Yamamoto, “Quantum circuit theory of phase-sensitive linear systems,” IEEE J. Quantum Electron. QE-23, 212 (1987).
[Crossref]

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 (1986).
[Crossref]

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

1983 (1)

1970 (1)

H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. IEEE 58, 1599 (1970).
[Crossref]

Evangelides, S. G.

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long distance soliton propagation using lumped amplifiers and dispersion-shifted fiber,” IEEE J. Lightwave Technol. 9, 194 (1991).
[Crossref]

Gordon, J. P.

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 (1986).
[Crossref]

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

Hasegawa, A.

Haus, H. A.

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long distance soliton propagation using lumped amplifiers and dispersion-shifted fiber,” IEEE J. Lightwave Technol. 9, 194 (1991).
[Crossref]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386 (1990).
[Crossref]

Y. Lai and H. A. Haus, “Quantum theory of self-induced transparency solitons: a linearization approach,” Phys. Rev. A 42, 2925 (1990).
[Crossref] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844 (1989); “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854 (1989).
[Crossref] [PubMed]

H. A. Haus and Y. Yamamoto, “Quantum circuit theory of phase-sensitive linear systems,” IEEE J. Quantum Electron. QE-23, 212 (1987).
[Crossref]

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

H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. IEEE 58, 1599 (1970).
[Crossref]

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 (1986).
[Crossref]

Kimura, Y.

K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron. Lett. 26, 539 (1990).
[Crossref]

M. Nakazawa, K. Suzuki, and Y. Kimura, “3.2–5 Gb/s, 100 error-free soliton transmissions with erbium amplifiers and repeaters,” IEEE Photonics Tech. Lett. 26, 216 (1990); K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron Lett. 26, 551 (1990).
[Crossref]

Kubota, H.

H. Kubota and M. Nakazawa, “Long-distance optical soliton transmission with lumped amplifiers,” IEEE J. Quantum Electron. 26, 692 (1990).
[Crossref]

Lai, Y.

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386 (1990).
[Crossref]

Y. Lai and H. A. Haus, “Quantum theory of self-induced transparency solitons: a linearization approach,” Phys. Rev. A 42, 2925 (1990).
[Crossref] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844 (1989); “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854 (1989).
[Crossref] [PubMed]

Mollenauer, L. F.

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long distance soliton propagation using lumped amplifiers and dispersion-shifted fiber,” IEEE J. Lightwave Technol. 9, 194 (1991).
[Crossref]

L. F. Mollenauer and K. Smith, “Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain,” Opt. Lett. 13, 675 (1988).
[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 (1986).
[Crossref]

Nakazawa, M.

K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron. Lett. 26, 539 (1990).
[Crossref]

M. Nakazawa, K. Suzuki, and Y. Kimura, “3.2–5 Gb/s, 100 error-free soliton transmissions with erbium amplifiers and repeaters,” IEEE Photonics Tech. Lett. 26, 216 (1990); K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron Lett. 26, 551 (1990).
[Crossref]

H. Kubota and M. Nakazawa, “Long-distance optical soliton transmission with lumped amplifiers,” IEEE J. Quantum Electron. 26, 692 (1990).
[Crossref]

Smith, K.

Suzuki, K.

K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron. Lett. 26, 539 (1990).
[Crossref]

M. Nakazawa, K. Suzuki, and Y. Kimura, “3.2–5 Gb/s, 100 error-free soliton transmissions with erbium amplifiers and repeaters,” IEEE Photonics Tech. Lett. 26, 216 (1990); K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron Lett. 26, 551 (1990).
[Crossref]

Yamada, E.

K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron. Lett. 26, 539 (1990).
[Crossref]

Yamamoto, Y.

H. A. Haus and Y. Yamamoto, “Quantum circuit theory of phase-sensitive linear systems,” IEEE J. Quantum Electron. QE-23, 212 (1987).
[Crossref]

Electron. Lett. (1)

K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron. Lett. 26, 539 (1990).
[Crossref]

IEEE J. Lightwave Technol. (1)

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long distance soliton propagation using lumped amplifiers and dispersion-shifted fiber,” IEEE J. Lightwave Technol. 9, 194 (1991).
[Crossref]

IEEE J. Quantum Electron. (3)

H. Kubota and M. Nakazawa, “Long-distance optical soliton transmission with lumped amplifiers,” IEEE J. Quantum Electron. 26, 692 (1990).
[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 (1986).
[Crossref]

H. A. Haus and Y. Yamamoto, “Quantum circuit theory of phase-sensitive linear systems,” IEEE J. Quantum Electron. QE-23, 212 (1987).
[Crossref]

IEEE Photonics Tech. Lett. (1)

M. Nakazawa, K. Suzuki, and Y. Kimura, “3.2–5 Gb/s, 100 error-free soliton transmissions with erbium amplifiers and repeaters,” IEEE Photonics Tech. Lett. 26, 216 (1990); K. Suzuki, M. Nakazawa, E. Yamada, and Y. Kimura, “5 Gbit/s, 250 km error-free soliton transmission with Er3+-doped fibre amplifiers and repeaters,” Electron Lett. 26, 551 (1990).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Lett. (3)

Phys. Rev. A (2)

Y. Lai and H. A. Haus, “Quantum theory of self-induced transparency solitons: a linearization approach,” Phys. Rev. A 42, 2925 (1990).
[Crossref] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844 (1989); “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854 (1989).
[Crossref] [PubMed]

Proc. IEEE (1)

H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. IEEE 58, 1599 (1970).
[Crossref]

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Figures (1)

Fig. 1
Fig. 1

Noise penalty factor for nonuniform gain.

Tables (2)

Tables Icon

Table 1 Expansion Functionsa

Tables Icon

Table 2 Adjoint Functions

Equations (50)

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- i u z = 1 2 2 u t 2 + c u 2 u + i Γ u .
c = ω 0 2 n 2 / v g A eff β .
u z + Γ u = 0
u = u 0 ( t ) exp ( - Γ z ) .
- i u 0 ( t ) z exp ( - Γ z ) = [ 1 2 2 t 2 u 0 + exp ( - 2 Γ z ) c u 0 2 u 0 ] exp ( - Γ z ) .
- i Δ u 0 ( t ) Δ z = 1 2 2 t 2 u 0 + 1 l 0 l d z exp ( - 2 Γ z ) c u 0 2 u 0 = 1 2 2 u 0 t 2 + 1 - exp ( - 2 Γ l ) 2 Γ l c u 0 2 u 0 .
- i u 0 z = 1 2 2 t 2 u 0 + r 2 c u 0 2 u 0 ,
r 2 = [ 1 - exp ( - 2 Γ l ) ] / 2 Γ l .
u 0 ( z , t ) = n 0 r c 2 sech [ n 0 r 2 c 2 ( t - T - p 0 z ) ] × exp { i [ p 0 ( t - T ) - p 0 2 2 z + n 0 2 r 4 c 2 8 z + θ 0 ] } .
- i z u 0 = 1 2 2 t 2 u 0 + r 2 c u 0 2 u 0 + n ( z , t ) .
n ˜ * ( z , Ω ) n ˜ ( z , Ω ) ¯ = 1 2 π 1 Ω 2 / Δ ω g 2 + 1 θ ( G - 1 ) l × δ ( z - z ) δ ( Ω - Ω ) ,
n * ( z , t ) n ( z , t ) ¯ = [ θ ( G - 1 ) / l ] δ ( z - z ) δ ( t - t ) .
[ u ^ ( z , t ) , u ^ ( z , t ) ] = δ ( t - t ) .
u ^ = u 0 ( z , t ) + δ u ^ ( z , t ) ,
[ δ u ^ ( z , t ) , δ u ^ ( z , t ) ] = δ ( t - t ) .
- i u 0 z = 1 2 2 u 0 t 2 + r 2 c u 0 2 u 0 .
- i δ u ^ z = 1 2 2 δ u ^ t 2 + 2 r 2 c u 0 2 δ u ^ + r 2 c u 0 2 δ u ^ + n ^ ( z , t ) .
d δ u ^ ( z , t ) d z = - Γ δ u ^ ( z , t ) + β ^ ( z , t ) ,
[ β ^ ( z , t ) , β ^ ( z , t ) ] = 2 Γ δ ( z - z ) δ ( t - t ) .
δ u ^ ( l , t ) = exp ( - Γ l ) 0 l exp ( Γ z ) β ^ ( z , t ) d z + δ u ^ ( 0 , t ) exp ( - Γ l ) = N ^ ( t ) + δ u ^ ( 0 , t ) exp ( - Γ l ) ,
N ^ ( t ) exp ( - Γ l ) 0 l exp ( Γ z ) β ^ ( z , t ) d z .
[ N ^ ( t ) , N ^ ( t ) ] = exp ( - 2 Γ l ) 0 l d z 0 l d z exp [ Γ ( z + z ) ] × [ β ^ ( z ) , β ^ ( z ) ] = 2 Γ exp ( - 2 Γ l ) 0 l d z exp ( 2 Γ z ) = [ 1 - exp ( - 2 Γ l ) ] δ ( t - t ) .
n ^ l ( z , t ) G N ^ / Δ z
[ n ^ l ( z , t ) , n ^ l ( z , t ) ] = G [ 1 - exp ( - 2 Γ l ) ] l δ ( z - z ) δ ( t - t ) = G - 1 l δ ( z - z ) δ ( t - t ) .
[ n ^ g ( z , t ) , n ^ g ( z , t ) ] = - [ ( G - 1 ) / l ] δ ( z - z ) δ ( t - t ) ,
δ u ^ = ( δ u ^ 1 + i δ u ^ 2 ) exp ( i Φ )
Φ = ( n 0 2 r 4 c 2 / 8 ) z .
δ u ^ 2 z = 1 2 2 δ u ^ 1 t 2 + 3 r 2 c u 0 2 δ u ^ 1 + n ^ 1 ,
- δ u ^ 1 z = 1 2 2 δ u ^ 2 t 2 + r 2 c u 0 2 δ u ^ 2 + n ^ 2 .
n ^ = ( n ^ 1 + i n ^ 2 ) exp ( i Φ ) .
δ u ^ 1 = δ n ^ ( z ) f n ( t ) + δ T ^ ( z ) f T ( t ) + δ u ^ c 1 ,
δ u 2 = δ p ^ ( z ) f p ( t ) + δ θ ^ ( z ) f θ ( t ) + δ u ^ c 2 .
δ n ^ ( z ) = δ n ( 0 ) ,
δ θ ^ ( z ) = δ θ ( 0 ) + ( n 0 r 4 c 2 / 4 ) z δ n ( 0 ) ,
δ p ^ ( z ) = δ p ( 0 ) ,
δ T ^ ( z ) = δ T ^ ( 0 ) + z δ p ^ ( 0 ) .
f _ i f i d t = 1 ,             i = n , θ , p , T .
δ p ^ ( z ) = δ u ^ 2 ( z , t ) f _ p d t .
n ^ 2 2 = - 1 4 n ^ exp ( i Φ ) - n ^ exp ( - i Φ ) 2 = 1 4 ( n ^ n ^ + n ^ n ^ ) .
n ^ 2 ( z , t ) n ^ 2 ( z , t ) = 1 2 δ ( t - t ) δ ( z - z ) G - 1 l .
n ^ p ( z ) n ^ p ( z ) = 1 2 δ ( z - z ) G - 1 l f _ p 2 d t .
f _ p 2 d t = n 0 r 4 c 2 2 - tanh 2 x sech 2 x d x = n 0 r 4 c 2 3 .
n ^ p ( z ) n ^ p ( z ) = δ ( z - z ) n 0 r 4 c 2 6 G - 1 l .
n ^ T ( z ) n ^ T ( z ) = 1 2 δ ( z - z ) G - 1 l f _ T 2 d t = δ ( z - z ) 4 n 0 3 r 4 c 2 G - 1 l - x 2 sech 2 x d x = δ ( z - z ) 6.6 n 0 3 r 4 c 2 G - 1 l .
d δ p ^ ( z ) d z = n ^ p ( z ) ,
d δ T ^ ( z ) d z = δ p ^ ( z ) + n ^ T ( z ) .
δ T ( L ) 2 = 0 L d z 0 L d z δ p ^ ( z ) δ p ^ ( z ) + 0 L d z 0 L d z n ^ T ( z ) n ^ T ( z ) + δ T 2 ( 0 ) .
δ p ^ ( z ) δ p ^ ( z ) = 0 z d z 0 z d z n ^ p ( z ) n ^ p ( z ) + δ p ^ ( 0 ) 2 = n 0 r 4 c 2 6 ( G - 1 ) z l + δ p ^ ( 0 ) 2 ,             z < z ;
δ T ^ ( L ) 2 = n 0 r 4 c 2 18 G - 1 l L 3 + δ p ^ ( 0 ) 2 L 2 + 6.6 n 0 3 r 4 c 2 ( G - 1 ) L l + δ T ( 0 ) 2 .
f = r 2 ( G - 1 ) 2 Γ l = ( G - 1 ) 2 G ( ln G ) 2 .

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