Abstract

We show that the timing jitter in dispersion-managed light-wave systems can be reduced considerably by replacing the erbium-doped fiber amplifiers with parametric amplifiers in which four-wave mixing is used to generate a phase-conjugated signal. We derive analytic expressions for the timing jitter by using the moment method for both the soliton and the nonsoliton systems and show that in both cases parametric amplifiers reduce the timing jitter enough that a 160-Gbit/s system is not jitter limited to distances as large as 4000 km. We include the contribution to timing jitter from the frequency shifts induced by intrapulse Raman scattering because this contribution dominates at such high bit rates. The effects of third-order dispersion are also included in the theoretical analysis.

© 2003 Optical Society of America

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References

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  1. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, 2002).
  2. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, New York, 1998), Chap. 5.
  3. G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, San Diego, Calif., 2001).
  4. S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol. 14, 243-248 (1996).
    [CrossRef]
  5. S. Chi and S. Wen, “Recovery of the soliton self-frequency shift by optical phase conjugation,” Opt. Lett. 19, 1705-1707 (1994).
    [CrossRef] [PubMed]
  6. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, San Diego, Calif., 2001).
  7. R. J. Essiambre and G. P. Agrawal, “Timing jitter of ultrashort solitons in high-speed communication systems. II. Control of jitter by periodic optical phase conjugation,” J. Opt. Soc. Am. B 14, 323-330 (1997).
    [CrossRef]
  8. R. J. Essiambre and G. P. Agrawal, “Timing jitter of ultrashort solitons in high-speed communication systems. I. General formulation and application to dispersion-decreasing fibers,” J. Opt. Soc. Am. B 14, 314-322 (1997).
    [CrossRef]
  9. J. Santhanam and G. P. Agrawal, “Raman-induced timing jitter in dispersion-managed communication systems,” IEEE J. Sel. Top. Quantum Electron. 8, 632-639 (2002).
    [CrossRef]
  10. M. E. Marhic, F. S. Yang, M. C. Ho, and L. G. Kazovsky, “High-nonlinearity fiber parametric amplifiers with periodic dispersion compensation,” J. Lightwave Technol. 17, 210-215 (1999).
    [CrossRef]
  11. C. J. McKinstrie, S. Radic, and A. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
    [CrossRef]
  12. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
    [CrossRef]
  13. J. Santhanam, C. J. McKinstrie, T. I. Lakoba, and G. P. Agrawal, “Effects of precompensation and postcompensation on timing jitter in dispersion-managed systems,” Opt. Lett. 26, 1131-1133 (2001).
    [CrossRef]
  14. V. S. Grigoryan, C. R. Menyuk, and R. M. Mu, “Calculation of timing and amplitude jitter in dispersion-managed optical fiber communications using linearization,” J. Lightwave Technol. 17, 1347-1356 (1999).
    [CrossRef]
  15. C. J. McKinstrie, J. Santhanam, and G. P. Agrawal, “Gordon-Haus timing jitter in dispersion-managed systems with lumped amplification: an analytical approach,” J. Opt. Soc. Am. B 19, 640-649 (2002).
    [CrossRef]
  16. S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
    [CrossRef]

2002 (4)

C. J. McKinstrie, S. Radic, and A. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

J. Santhanam and G. P. Agrawal, “Raman-induced timing jitter in dispersion-managed communication systems,” IEEE J. Sel. Top. Quantum Electron. 8, 632-639 (2002).
[CrossRef]

C. J. McKinstrie, J. Santhanam, and G. P. Agrawal, “Gordon-Haus timing jitter in dispersion-managed systems with lumped amplification: an analytical approach,” J. Opt. Soc. Am. B 19, 640-649 (2002).
[CrossRef]

2001 (1)

1999 (2)

1998 (1)

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

1997 (2)

1996 (1)

S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol. 14, 243-248 (1996).
[CrossRef]

1994 (1)

Agrawal, G. P.

Andrekson, P. A.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

Chi, S.

Chraplyvy, A.

C. J. McKinstrie, S. Radic, and A. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

Essiambre, R. J.

Gabitov, I.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Grigoryan, V. S.

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

Hedekvist, P. O.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

Ho, M. C.

Kazovsky, L. G.

Laedke, E. W.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Lakoba, T. I.

Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

Marhic, M. E.

McKinstrie, C. J.

Menyuk, C. R.

Mezentsev, V. K.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Mu, R. M.

Musher, S. L.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Radic, S.

C. J. McKinstrie, S. Radic, and A. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

Santhanam, J.

Schafer, T.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Shapiro, E. G.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Shirasaki, M.

S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol. 14, 243-248 (1996).
[CrossRef]

Spatschek, K. H.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Turitsyn, S. K.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Watanabe, S.

S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol. 14, 243-248 (1996).
[CrossRef]

Wen, S.

Westlund, M.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

Yang, F. S.

IEEE J. Sel. Top. Quantum Electron. (3)

J. Santhanam and G. P. Agrawal, “Raman-induced timing jitter in dispersion-managed communication systems,” IEEE J. Sel. Top. Quantum Electron. 8, 632-639 (2002).
[CrossRef]

C. J. McKinstrie, S. Radic, and A. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
[CrossRef]

J. Lightwave Technol. (3)

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

Opt. Commun. (1)

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schafer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Opt. Lett. (2)

Other (4)

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, San Diego, Calif., 2001).

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, 2002).

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, New York, 1998), Chap. 5.

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, San Diego, Calif., 2001).

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Figures (5)

Fig. 1
Fig. 1

Timing jitter for a 160-Gbit/s DM soliton system with 40-km amplifier spacing. The solid and dashed curves show, respectively, timing jitter with and without the Raman contribution for both EDFAs and parametric amplifiers. The map parameters are given in the text and result in an average dispersion of βav=-0.1 ps2/km. The dotted curve shows the acceptable timing jitter value.

Fig. 2
Fig. 2

Same as Fig. 1 except that the system was designed by use of DDFs and amplifiers are placed every 45 km. The GVD decreases exponentially over 45 km starting from its initial value of 1 ps/(km nm).

Fig. 3
Fig. 3

Comparison of numerical (stars) and analytic (solid curve) results for TR=0 (no Raman jitter) for a 160-Gbit/s DM soliton system designed with 40-km amplifier spacing and βav=-0.1 ps2/km. The dispersion map is the same as in Fig. 1.

Fig. 4
Fig. 4

Same as Fig. 3 except the Raman contribution to timing jitter is included by use of TR=3 fs. The stars and solid curves show the numerical and analytic results, respectively.

Fig. 5
Fig. 5

Timing jitter for a quasi-linear CRZ system with 40-km amplifier spacing and βav=-0.005 ps2/km. The solid and dashed curves show, respectively, timing jitter with and without the Raman contribution for both EDFAs and parametric amplifiers. The dotted line shows the acceptable timing jitter value.

Equations (60)

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i Az-β222At2-i β363At3+γ|A|2A
=-iα2 A+TRγA |A|2t,
i A*z+β222A*t2-i β363A*t3-γ|A|2A*
=-iα2 A*-TRγA*|A|2t.
A(z, t)=a exp[-(1+iC)(t-T)2/2τ2+iϕ-iΩ(t-T)],
E=-|A|2dt,
T=1E-t|A|2dt,
Ω=i2E-A*At-A A*tdt.
dEdz=-αE+i=iN(giE+δEi)δ(z-zi),
dΩdz=-TRγE2πτ3+i=1NδΩiδ(z-zi),
dTdz=β2Ω+β3(1+C2)12τ2+β3Ω26+i=1NδTiδ(z-zi),
δAi*(t)δAj(t)=Sδijδ(t-t),
δEi2=2SEi,δΩiδEi=2SCiπτi,
δΩi2=S(1+Ci2)Eiτi2,δEiδTi=0,
δTi2=Sτi2Ei,δΩiδTi=SCiEi,
E(LA)=E(0)+δE1,
Ω(LA)=-Ω(0)-bRE(0)+δΩ1,
T(LA)=T(0)-b2Ω(0)-b2RE(0)+b3+b3ΩΩ2(0)-bEΩE(0)Ω(0)+b3EE2(0)+δT1,
b2=0LAβ2(z)dz,b3=0LAβ3(1+C2)12τ2dz,
bR=-0LAq(z)dz,
b2R=-0LAdzβ2(z)0zdzq(z),
b3Ω=160LAβ3dz,
bEΩ=-160LAdzβ3(z)0zq(z)dz,
b3E=-160LAdzβ3(z)0zq(z)dz2,
q(z)=TRγ(z)p(z)2πτ3(z),
E(2LA)=E(0)+δE1+δE2,
Ω(2LA)=Ω(0)-bRδE1+δΩ1+δΩ2,
T(2LA)=T(0)-b2bRE(0)+2b3-b2δΩ1-b2RδE1+δT1+δT2.
T(NLA)=(N/2)[2b3-b2bRE0-b2RS],
σPA2=σGH2+[b22bR2(N2-4)/12+b2R2]N/2δE2+b3SN(N-2)2(N-4)/96,
σFA2=σGH2+R1(δE)2+R2δEδΩ+R3S/E,
R1=N(N-1)[bR2b22(N3-10N2+29N-9)/120+b2bRb2R(19N2-65N+48)/96+b2R2(2N-1)/6],
R2=N(N-1)b2[b2bR(N-2)(3N-1)/12+b2R(2N-1)/3],
R3=N(N-1)b3[b3(N-1)(N-2)/6+4b2(N-1)/3τ0].
σGH2=N(b22/2)δΩ2+NδT2,
σGH2=(b22/6)N(N-1)(2N-1)δΩ2+b2N(N-1)δΩδT+NδT2.
A=a sech(t-T/τ)exp(iϕ-iΩt)
δEi2=2SEi,δΩiδEi=0,
δΩi2=2S3Eiτi2,δEiδTi=0,
δTi2=π2Sτi26Ei,δΩiδTi=0.
dEdz=-αE+i(giE+δEi)δ(z-zi),
dΩdz=-4TRγE15τ3+iδΩiδ(z-zi),
dTdz=β2Ω+β318τ2+β3Ω26+iδTiδ(z-zi).
b2=β2(0)Leff,b3=β3(0)Leff/(18τ2),
bR=-4γTRLeff/(15τ03),
b2R=-2γTRβ2(0)Leff2 /(15τ03),
Leff=[1-exp(-αLA)]/α.
σFA2=σGH2+R1(δE)2+R3S/E,
σGH2=(b22/6)N(N-1)(2N-1)δΩ2+NδT2.
C1=-(C0+b2/τmin2),
τ1=τmin1+C12,τ2=τ0.
σPA2=σGH2+N/2[b22bR2N(N2-4)/12+b2R2N]δE2+b3SN(N-2)(N-4)/96E,
σFA2=σGH2+R1(δE)2+2Sπτmin (1+C02)-1/2[R2C0-R2b2(1+2C02)/τmin2]+R3S/E.
σGH2=N(S/E)τmin2[1+b22/τmin4+C02+C0b2/τmin2],
σGH2=N(S/E)τmin2[1+(C0+b2N/τmin2)2],
R2=N(N-1)b2[b2bR(N-1)2/3+b2R(2N-1)/3],
R2=N(N-1)b2[b2bR(12N3-40N2-53)/10+b2R(9N2-3N-13)]/18,
R3=N(N-1)b3{b3(N-1)(N-2)/6+(4b2/3τmin)(1+C02)-1/2×[C0/2-(2N-1)(1+2C02)(b2/3τmin2)]}.
σGH2=N(S/E)τmin2[1+3b2/4τmin2],
σGH2=N(S/E)τmin2.

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