Abstract

A closed-form theory is presented to characterize the noise accumulation in a transmission system with grating dispersion compensation, and its implications for the system performance are discussed. It is shown that the spectral distribution of the noise depends on the mapping used to compensate for the dispersion. Optimum system performance is obtained for compensation maps that are symmetric with respect to the middle point of each compensated span.

© 1998 Optical Society of America

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References

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  1. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
    [CrossRef]
  2. I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in the cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996).
    [CrossRef]
  3. I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 861–866 (1996).
    [CrossRef]
  4. N. J. Smith, N. J. Doran, F. M. Knox, and W. Forysiak, “Energy-scaling characteristics of solitons in strongly dispersion-managed fibers,” Opt. Lett. 21, 1981–1983 (1996).
    [CrossRef] [PubMed]
  5. S. K. Turitsyn, “Theory of averaged pulse propagation in high bit rate optical transmission systems with strong dispersion management,” JETP Lett. 65, 845–850 (1997).
    [CrossRef]
  6. Y. Kodama, “Nonlinear chirped RZ and NRZ pulses in optical transmission lines,” in New Trends in Optical Soliton Transmission Systems,” A. Hasegawa, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1998), paper 3-A-3.
  7. T. Georges and F. Favre, “Transmission systems based on dispersion managed solitons: theory and experiment,” in New Trends in Optical Soliton Transmission Systems,” A. Hasegawa, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1998), paper 2-A-2.
  8. J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
    [CrossRef]
  9. A. Naka, T. Matsuda, and S. Saito, “Optical RZ signal straight line transmission with dispersion compensation over 5220 km at 20 Gbit/s and 2160 km at 2×20 Gbit/s,” Electron. Lett. 32, 1694–1696 (1996).
    [CrossRef]
  10. D. Le Guen, F. Favre, M. L. Moulinard, M. Henry, G. Michaud, L. Mace, F. Devaux, B. Charbonnier, and T. Georges, “200 Gbit/s 100 km span soliton WDM transmission over 1000 km of standard fiber with dispersion compensation and pre-chirping,” in Optical Fiber Communication Conference (OFC) Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, DC, 1997), postdeadline paper PD17.
  11. N. J. Smith and N. J. Doran, “Modulational instabilities in fibers with periodic dispersion management,” Opt. Lett. 21, 570–572 (1996).
    [CrossRef] [PubMed]
  12. A. Hasegawa and Y. Kodama, “Guiding center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
    [CrossRef] [PubMed]
  13. E. Parzen, Stochastic Processes (Holden-Day, San Francisco, Calif., 1971).
  14. H. A. Haus, “Quantum noise in a solitonlike repeater system,” J. Opt. Soc. Am. B 8, 1122–1126 (1991).
    [CrossRef]
  15. M. Midrio, “Statistical properties of noisy propagation in normally dispersive nonlinear fibers,” J. Opt. Soc. Am. B 14, 2910–2914 (1997).
    [CrossRef]
  16. J. C. Bronsky and J. N. Kutz, “Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management,” Opt. Lett. 21, 937–939 (1996).
    [CrossRef]
  17. M. Midrio, “Analytical performance evaluation of nonreturn-to-zero transmission systems operating in normally dispersive nonlinear fibers,” J. Lightwave Technol. 15, 2038–2051 (1997).
    [CrossRef]

1998

1997

S. K. Turitsyn, “Theory of averaged pulse propagation in high bit rate optical transmission systems with strong dispersion management,” JETP Lett. 65, 845–850 (1997).
[CrossRef]

M. Midrio, “Analytical performance evaluation of nonreturn-to-zero transmission systems operating in normally dispersive nonlinear fibers,” J. Lightwave Technol. 15, 2038–2051 (1997).
[CrossRef]

M. Midrio, “Statistical properties of noisy propagation in normally dispersive nonlinear fibers,” J. Opt. Soc. Am. B 14, 2910–2914 (1997).
[CrossRef]

1996

1991

A. Hasegawa and Y. Kodama, “Guiding center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[CrossRef] [PubMed]

H. A. Haus, “Quantum noise in a solitonlike repeater system,” J. Opt. Soc. Am. B 8, 1122–1126 (1991).
[CrossRef]

Bennion, I.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Blow, K. J.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Bronsky, J. C.

Doran, N. J.

Evangelides, S. G.

Forysiak, W.

Gabitov, I.

Gordon, J. P.

Hasegawa, A.

A. Hasegawa and Y. Kodama, “Guiding center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[CrossRef] [PubMed]

Haus, H. A.

Holmes, P.

Knox, F. M.

N. J. Smith, N. J. Doran, F. M. Knox, and W. Forysiak, “Energy-scaling characteristics of solitons in strongly dispersion-managed fibers,” Opt. Lett. 21, 1981–1983 (1996).
[CrossRef] [PubMed]

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Kodama, Y.

A. Hasegawa and Y. Kodama, “Guiding center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[CrossRef] [PubMed]

Kutz, J. N.

Matsuda, T.

A. Naka, T. Matsuda, and S. Saito, “Optical RZ signal straight line transmission with dispersion compensation over 5220 km at 20 Gbit/s and 2160 km at 2×20 Gbit/s,” Electron. Lett. 32, 1694–1696 (1996).
[CrossRef]

Midrio, M.

M. Midrio, “Statistical properties of noisy propagation in normally dispersive nonlinear fibers,” J. Opt. Soc. Am. B 14, 2910–2914 (1997).
[CrossRef]

M. Midrio, “Analytical performance evaluation of nonreturn-to-zero transmission systems operating in normally dispersive nonlinear fibers,” J. Lightwave Technol. 15, 2038–2051 (1997).
[CrossRef]

Naka, A.

A. Naka, T. Matsuda, and S. Saito, “Optical RZ signal straight line transmission with dispersion compensation over 5220 km at 20 Gbit/s and 2160 km at 2×20 Gbit/s,” Electron. Lett. 32, 1694–1696 (1996).
[CrossRef]

Saito, S.

A. Naka, T. Matsuda, and S. Saito, “Optical RZ signal straight line transmission with dispersion compensation over 5220 km at 20 Gbit/s and 2160 km at 2×20 Gbit/s,” Electron. Lett. 32, 1694–1696 (1996).
[CrossRef]

Smith, N. J.

Turitsyn, S. K.

S. K. Turitsyn, “Theory of averaged pulse propagation in high bit rate optical transmission systems with strong dispersion management,” JETP Lett. 65, 845–850 (1997).
[CrossRef]

I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in the cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996).
[CrossRef]

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 861–866 (1996).
[CrossRef]

Electron. Lett.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

A. Naka, T. Matsuda, and S. Saito, “Optical RZ signal straight line transmission with dispersion compensation over 5220 km at 20 Gbit/s and 2160 km at 2×20 Gbit/s,” Electron. Lett. 32, 1694–1696 (1996).
[CrossRef]

J. Lightwave Technol.

M. Midrio, “Analytical performance evaluation of nonreturn-to-zero transmission systems operating in normally dispersive nonlinear fibers,” J. Lightwave Technol. 15, 2038–2051 (1997).
[CrossRef]

J. Opt. Soc. Am. B

JETP Lett.

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 861–866 (1996).
[CrossRef]

S. K. Turitsyn, “Theory of averaged pulse propagation in high bit rate optical transmission systems with strong dispersion management,” JETP Lett. 65, 845–850 (1997).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

A. Hasegawa and Y. Kodama, “Guiding center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[CrossRef] [PubMed]

Other

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, Calif., 1971).

D. Le Guen, F. Favre, M. L. Moulinard, M. Henry, G. Michaud, L. Mace, F. Devaux, B. Charbonnier, and T. Georges, “200 Gbit/s 100 km span soliton WDM transmission over 1000 km of standard fiber with dispersion compensation and pre-chirping,” in Optical Fiber Communication Conference (OFC) Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, DC, 1997), postdeadline paper PD17.

Y. Kodama, “Nonlinear chirped RZ and NRZ pulses in optical transmission lines,” in New Trends in Optical Soliton Transmission Systems,” A. Hasegawa, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1998), paper 3-A-3.

T. Georges and F. Favre, “Transmission systems based on dispersion managed solitons: theory and experiment,” in New Trends in Optical Soliton Transmission Systems,” A. Hasegawa, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1998), paper 2-A-2.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the receiver stage.

Fig. 2
Fig. 2

Schematic diagram of an amplified span in an asymmetrically compensated transmission system.

Fig. 3
Fig. 3

Equivalent form for an asymmetrically compensated transmission system.

Fig. 4
Fig. 4

Noise spectra in the presence of average normal dispersion. Panels (a), (c), (e), and (g) are for noise components in phase with the useful signal, whereas the remaining panels are for out-of-phase components. Moreover, panels (a) and (b) are for a system with constant dispersion, panels (c) and (d) are for a system with symmetric compensation, panels (e) and (f ) are for a system with no prechirping, and panels (g) and (h) are for a system with no postchirping.

Fig. 5
Fig. 5

Noise spectra in the presence of average anomalous dispersion. Panels (a), (c), (e), and (g) are for noise components in phase with the useful signal, whereas the remaining panels are for out-of-phase components. Moreover, panels (a) and (b) are for a system with constant dispersion, panels (c) and (d) are for a system with symmetric compensation, panels (e) and (f ) are for a system with no prechirping, and panels (g) and (h) are for a system with no postchirping.  

Fig. 6
Fig. 6

Standard deviation for the decision variable.

Fig. 7
Fig. 7

Ratio between the variance of the decision variable in the presence of nonlinearity and dispersion, for the case of a linear system.

Equations (44)

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i uz-12 β(z)  2ut2+γ|u|2u=0,0zzA,
β(z)=D1δ(z)+β2+D2δ(z-zA),
γ=γNL 1-exp(-ΓzA)ΓzA,0<z<zA,
i uz-12 β(z)  2ut2+γ|u|2u=0,0zNzA,
u(t, z=0)=P0+η(t),|η(t)|2=ση2P0,
Wη=η^R(Ω)η^I(Ω) η^R(Ω)η^I(Ω)*=Wη2π 1001,|Ω|<πB.
u(t, z)=[P0+η(t, z)]expiγNL 1-exp(-ΓzA)ΓzA P0 z
i ηˆz+12 β(z)Ω2ηˆ+γNL 1-exp(-ΓzA)ΓzA P0(ηˆ+η^*)
=0.
β(z)=β+δβ(z),
β=1zA 0zAβ(ξ)dξ=D1+D2zA+β2,
δβ=0,
μ^p(z, Ω)=ηˆ(z, Ω)expiΩ20zδβ(ξ)dξexp-iπp zzA,
i μ^p z+πpzA μ^p+βΩ22 μ^p
+γNL exp(-2ΓzA)-12ΓzA P0[μ^p+μ^p * cp(Ω2)]=0,
cp(Ω2)=1zA 0zA expi2πp zzA+0zδβ(ξ)dξ.
 z μ^Rμ^I=Ψ(Ω2)μ^Rμ^I=0A2A10 μ^Rμ^I,
A1=βΩ22+γNL P0 1-exp(-ΓzA)ΓzA [1+c0(Ω2)],
A2=βΩ22+γNL P0 1-exp(-ΓzA)ΓzA [1-c0(Ω2)].
μ^R(Ω, zA)μ^I(Ω, zA)=Mμ^R(Ω, 0)μ^I(Ω, 0),
M=exp[Ψ(Ω2)zA]=cos(χzA)-sin(χzA)/KK sin(χzA)cos(χzA),
μ^R(Ω, lzA)μ^I(Ω, lzA)=Mlμ^R(Ω, 0)μ^I(Ω, 0)
=cos(lχzA)-sin(lχzA)/KK sin(lχzA)cos(lχzA)×μ^R(Ω, 0)μ^I(Ω, 0),
Wμ(lzA)=WRWRIWIRWI=μ^R(Ω, lzA)μ^I(Ω, lzA) μ^R(Ω, lzA)μ^I(Ω, lzA)* .
WR=Wη2 |cos(lχzA)|2+|sin(lχ zA)|2|K|,
WRI=WIR*=Wη2 (K)* sin*(lχzA)-sin(lχ zA)K cos*(lχzA),
WI=Wη2 [|cos(lχ zA)|2+|K||sin(lχ zA)|2].
W=Wη2 l=1NWμ(lzA)= f11f12 f12*f22,
f11=N Wη2 1-1|K| cos[(N+1)χR zA]sin(NχRzA)2N sin(χR zA)+1+1|K| cosh[(N+1)χI zA]sinh(NχI zA)2N sinh(χI zA),
f12=N Wη2 (K)*-1K×sin[(N+1)χR zA]sin(NχR zA)2N sin(χR zA)-i(K)*+1K×sinh[(N+1)χI zA]sinh(NχI zA)2N sinh(χI zA),
f22=N Wη2 (1-|K|) cos[(N+1)χR zA]sin(NχR zA)2N sin(χRzA)+(1+|K|) cosh[(N+1)χI zA]sinh(NχI zA)2N sinh(χI zA),
f11=N Wη2 G11,f22=N Wη2 G22,
c0=sin[Ω2(β2-β) za/2]Ω2(β2-β) za/2
Re{μ^A}Im{μ^A}=Π(δ D1)Re{μ^in}Im{μ^in},
Π(δD1)=cosδ D1Ω22-sinδ D1Ω22sinδ D1Ω22cosδ D1Ω22.
Re{μ^out}Im{μ^out}=Π(δ D2)MΠ(δ D1)Re{μ^in}Im{μ^in},
R=Π(δ D2)WΠ(δ D2)*,
RR=cos2Ω2δ D22f11+sin2Ω2δ D22f22-sin(Ω2δ D2)Re{ f12},
RI=sin2Ω2δ D22 f11+cos2Ω2δ D22f22+sin(Ω2δ D2)Re { f12}.
BERerfcmy(T0)σy(T0)2,y(t)=dτ|u(τ)|2h(t-τ),
σy2=4 P0RR( f )|H( f )|2d f,
f(z)=exp-iΩ20zδβ(ξ)dξ
f(z)=exp[-iΩ2g(z)]0z<zA/2exp[iΩ2g(zA/2-z)]zA/2z<zA.
cp(Ω2)=1zA 0zA/2 cosΩ2g(z)+2πp zzAdz,

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