Abstract

For a typical nonreturn-to-zero pulse propagating in existing normally dispersive fibers, we provide a uniform description of the optical transmission. There are three distinct regimes of the governing nonlinear Schrödinger equation: the fully nonlinear dispersive regime, which covers a small region of the pulse, and two limiting asymptotic regimes, namely, nonlinear, weakly dispersive (for the bulk of the pulse) and linear dispersive (for the tails). For prediction of pulse degradation, the asymptotic regimes admit accurate, simplified models for both nonlinear-dispersive pulse spreading and the onset of optical shocks and oscillations at the fronts.

© 1999 Optical Society of America

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References

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  1. N. S. Bergano and C. R. Davidson, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
    [CrossRef]
  2. D. Marcuse, “rms width of pulses in nonlinear dispersive fibers,” J. Lightwave Technol. 10, 17–21 (1992).
    [CrossRef]
  3. Y. Kodama and S. Wabnitz, “Analytic theory of guiding-center nonreturn-to-zero and return-to-zero signal transmission in normally dispersive fibers,” Opt. Lett. 20, 2291–2293 (1995).
    [CrossRef]
  4. M. G. Forest and K. T.-R. McLaughlin, “Onset of oscillations in nonsoliton pulses in nonlinear dispersive fibers,” J. Nonlinear Sci. 7, 43–62 (1998).
    [CrossRef]
  5. J. C. Bronski and J. N. Kutz, “Guiding-center pulse dynamics in nonreturn-to-zero communications system with mean-zero dispersion,” J. Opt. Soc. Am. B 14, 903–911 (1997).
    [CrossRef]
  6. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear-optical fibers,” J. Opt. Soc. Am. B 10, 1185–1190 (1993).
    [CrossRef]
  7. J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
    [CrossRef] [PubMed]
  8. S. Jin, C. D. Levermore, and D. W. McLaughlin, “The behavior of solutions of the NLS equation in the semi-classical limit,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, eds., NATO ASI Ser., Ser. B 320, 235–255 (1994).
    [CrossRef]
  9. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 10.
  10. J. C. Bronski and D. W. McLaughlin, “Semiclassical behavior in the NLS equation: optical shocks-focusing instabilities,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, ed., NATO ASI Ser., Ser. B 320, 21–38 (1994).
    [CrossRef]
  11. G. B. Whitham, Linear and Nonlinear Waves. Pure and Applied Mathematics. (Wiley-Interscience, New York, 1974).
  12. Y. Kodama and S. Wabnitz, “Compensation of NRZ signal distortion by initial frequency shifting,” Electron. Lett. 31, 1761–1762 (1995).
    [CrossRef]
  13. See for example, 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]

1998

M. G. Forest and K. T.-R. McLaughlin, “Onset of oscillations in nonsoliton pulses in nonlinear dispersive fibers,” J. Nonlinear Sci. 7, 43–62 (1998).
[CrossRef]

See for example, 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]

1997

1995

Y. Kodama and S. Wabnitz, “Analytic theory of guiding-center nonreturn-to-zero and return-to-zero signal transmission in normally dispersive fibers,” Opt. Lett. 20, 2291–2293 (1995).
[CrossRef]

N. S. Bergano and C. R. Davidson, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
[CrossRef]

Y. Kodama and S. Wabnitz, “Compensation of NRZ signal distortion by initial frequency shifting,” Electron. Lett. 31, 1761–1762 (1995).
[CrossRef]

1994

S. Jin, C. D. Levermore, and D. W. McLaughlin, “The behavior of solutions of the NLS equation in the semi-classical limit,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, eds., NATO ASI Ser., Ser. B 320, 235–255 (1994).
[CrossRef]

J. C. Bronski and D. W. McLaughlin, “Semiclassical behavior in the NLS equation: optical shocks-focusing instabilities,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, ed., NATO ASI Ser., Ser. B 320, 21–38 (1994).
[CrossRef]

1993

1992

D. Marcuse, “rms width of pulses in nonlinear dispersive fibers,” J. Lightwave Technol. 10, 17–21 (1992).
[CrossRef]

1989

J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
[CrossRef] [PubMed]

Anderson, D.

Bergano, N. S.

N. S. Bergano and C. R. Davidson, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
[CrossRef]

Bronski, J. C.

J. C. Bronski and J. N. Kutz, “Guiding-center pulse dynamics in nonreturn-to-zero communications system with mean-zero dispersion,” J. Opt. Soc. Am. B 14, 903–911 (1997).
[CrossRef]

J. C. Bronski and D. W. McLaughlin, “Semiclassical behavior in the NLS equation: optical shocks-focusing instabilities,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, ed., NATO ASI Ser., Ser. B 320, 21–38 (1994).
[CrossRef]

Davidson, C. R.

N. S. Bergano and C. R. Davidson, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
[CrossRef]

Desaix, M.

Evangelides, S. G.

Forest, M. G.

M. G. Forest and K. T.-R. McLaughlin, “Onset of oscillations in nonsoliton pulses in nonlinear dispersive fibers,” J. Nonlinear Sci. 7, 43–62 (1998).
[CrossRef]

Gordon, J. P.

Grischkowsky, D.

J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
[CrossRef] [PubMed]

Holmes, P.

Jin, S.

S. Jin, C. D. Levermore, and D. W. McLaughlin, “The behavior of solutions of the NLS equation in the semi-classical limit,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, eds., NATO ASI Ser., Ser. B 320, 235–255 (1994).
[CrossRef]

Karlsson, M.

Kodama, Y.

Kutz, J. N.

Levermore, C. D.

S. Jin, C. D. Levermore, and D. W. McLaughlin, “The behavior of solutions of the NLS equation in the semi-classical limit,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, eds., NATO ASI Ser., Ser. B 320, 235–255 (1994).
[CrossRef]

Lisak, M.

Marcuse, D.

D. Marcuse, “rms width of pulses in nonlinear dispersive fibers,” J. Lightwave Technol. 10, 17–21 (1992).
[CrossRef]

McLaughlin, D. W.

S. Jin, C. D. Levermore, and D. W. McLaughlin, “The behavior of solutions of the NLS equation in the semi-classical limit,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, eds., NATO ASI Ser., Ser. B 320, 235–255 (1994).
[CrossRef]

J. C. Bronski and D. W. McLaughlin, “Semiclassical behavior in the NLS equation: optical shocks-focusing instabilities,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, ed., NATO ASI Ser., Ser. B 320, 21–38 (1994).
[CrossRef]

McLaughlin, K. T.-R.

M. G. Forest and K. T.-R. McLaughlin, “Onset of oscillations in nonsoliton pulses in nonlinear dispersive fibers,” J. Nonlinear Sci. 7, 43–62 (1998).
[CrossRef]

Quiroga-Teixeiro, M. L.

Rothenberg, J. E.

J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
[CrossRef] [PubMed]

Wabnitz, S.

Electron. Lett.

Y. Kodama and S. Wabnitz, “Compensation of NRZ signal distortion by initial frequency shifting,” Electron. Lett. 31, 1761–1762 (1995).
[CrossRef]

J. Lightwave Technol.

N. S. Bergano and C. R. Davidson, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
[CrossRef]

D. Marcuse, “rms width of pulses in nonlinear dispersive fibers,” J. Lightwave Technol. 10, 17–21 (1992).
[CrossRef]

J. Nonlinear Sci.

M. G. Forest and K. T.-R. McLaughlin, “Onset of oscillations in nonsoliton pulses in nonlinear dispersive fibers,” J. Nonlinear Sci. 7, 43–62 (1998).
[CrossRef]

J. Opt. Soc. Am. B

NATO ASI Ser., Ser. B

S. Jin, C. D. Levermore, and D. W. McLaughlin, “The behavior of solutions of the NLS equation in the semi-classical limit,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, eds., NATO ASI Ser., Ser. B 320, 235–255 (1994).
[CrossRef]

J. C. Bronski and D. W. McLaughlin, “Semiclassical behavior in the NLS equation: optical shocks-focusing instabilities,” in Singular Limits of Dispersive Waves, N. M. Ercolani, I. Gabitov, D. Levermore, and D. Serre, ed., NATO ASI Ser., Ser. B 320, 21–38 (1994).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
[CrossRef] [PubMed]

Other

G. B. Whitham, Linear and Nonlinear Waves. Pure and Applied Mathematics. (Wiley-Interscience, New York, 1974).

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 10.

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Figures (6)

Fig. 1
Fig. 1

Simulation of TAT12-13 and TCP5 communications links with ZT=500 km, P0=4 mW, and t0=200 ps, so =0.5840. We note that because O(1) and the total distance is only 500 km, no wave-breaking or oscillatory phenomena are observed.

Fig. 2
Fig. 2

Simulation of the optical communications line (5 Gbits/s) proposed by Marcuse2 with ZT=10,000 km, P0=1 mW, and Tpls=200 ps, so =0.031. An initial super-Gaussian (n=10) pulse profile was assumed. Note the predicted linear shock phenomenon, which occurs in the tail of the pulse [UO(10-3)] for Zb0.19.

Fig. 3
Fig. 3

Same parameters as for Fig. 2 but with a Gaussian initial pulse (n=2). The Gaussian exhibits the predicted nonlinear shock phenomenon, which occurs for UO(1) at Zb0.6.

Fig. 4
Fig. 4

Simulation of the 10-Gbit/s optical communications line with ZT=10,000 km, P0=1 mW, and Tpls=100 ps, so =0.031 as in Figs. 2 and 3. An initial super-Gaussian (n=10) pulse profile was assumed. Note the effects of nonlinear spreading, which broadens the pulse to approximately 30 times its original width.

Fig. 5
Fig. 5

Simulation of the 10-Gbit/s optical communications line of Fig. 4 with initial quadratically chirped pulses: U(T, 0)=exp[-(T/w)2±i0.3(T/w)2/]. Note the enhanced pulse shaping and onset of oscillations over Z[0, 0.3] (0–3000 km) for both the positive (bottom) and negative (top) signs of chirp.

Fig. 6
Fig. 6

Simulation of the 10-Gbit/s optical communications line of Figs. 4 and 5 with initial quadratically chirped pulses {U(T, 0)=exp[-(T/w)2±i0.3(T/w)2/]} and including third-order dispersion and Raman effects. Note that these higher-order effects have virtually no effect on the pulse spreading and onset of oscillations.

Equations (17)

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

i Qz-k22Qt2+γ|Q|2Q=0,
i UZ-222UT2+|U|2U=0,
U(Z, T)=A(Z, T)expi S(Z, T),
-ASZ+½AST2=(2/2)ATT-A3,
AZ-½(2ATST+ASTT)=0.
βZ-3β-α4βT=0,
αZ-β-3α4αT=0.
2ATT2A31.
U(T, 0)=exp[-(T/w)n],
n2w21-n+nTwnTwn-2 exp2Twn12.
2ATT(Tl)2A3(Tl)=,
2ATT(Tr)2A3(Tr)=1/.
C<Zb<2C exp1-1n,
C=eTpls6(n-1)1γP0k1/2e2(1-1/n)-1/n
i UZ-222UT2=0.
i UZ-222UT2+|U|2U-iβ3 3UT3-βrU T(|U|2)
=0,

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