Abstract

Based on first-order perturbation theory of the soliton, the Gordon–Haus timing jitter induced by amplifier noise is found to be non-Gaussian distributed. Both frequency and timing jitter have larger tail probabilities than Gaussian distribution given by the linearized perturbation theory. The timing jitter has a larger discrepancy from Gaussian distribution than does the frequency jitter.

© 2003 Optical Society of America

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References

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  1. J. P. Gordon and H. A. Haus, Opt. Lett. 11, 865 (1986).
  2. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, New York, 1998), Chap 5.
  3. R. O. Moore, G. Biondini, and W. L. Kath, Opt. Lett. 28, 105 (2003).
    [CrossRef] [PubMed]
  4. Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763–915 (1989).
    [CrossRef]
  5. D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
    [CrossRef] [PubMed]
  6. T. Georges, Opt. Fiber Technol. 1, 97 (1995).
    [CrossRef]
  7. C. R. Menyuk, Opt. Lett. 20, 285 (1995).
    [CrossRef] [PubMed]
  8. T. Georges, Electron. Lett. 31, 1174 (1995).
    [CrossRef]
  9. T. Georges, Opt. Commun. 123, 617 (1996).
    [CrossRef]
  10. O. Leclerc, E. Desurvire, P. Brindel, and E. Maunand, Opt. Fiber Technol. 5, 301 (1999).
    [CrossRef]
  11. G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turisyn, Phys. Rev. E 63, 025601 (2001).
    [CrossRef]
  12. C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, Berlin, 1985).
  13. P. A. Humblet and M. Azizog̃lu, J. Lightwave Technol. 9, 1576 (1991).
    [CrossRef]
  14. P. Shum and H. Ghafouri-Shiraz, Opt. Laser Technol. 28, 535 (1996).
    [CrossRef]
  15. E. M. Stein and J. C. Stein, Rev. Financ. Stud. 4, 727 (1991).
    [CrossRef]
  16. R. H. Cameron and W. T. Martin, Bull. Am. Math. Soc. 51, 73 (1945).
    [CrossRef]
  17. K.-P. Ho, in Advances in Optics and Laser Research, Vol. 3, W. T. Arkin, ed. (Nova Science, Hauppauge, N.Y., 2003).
  18. K.-P. Ho, Opt. Lett. 28, 1350 (2003).
    [CrossRef] [PubMed]
  19. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), Sect. 6.561.
  20. D. J. Kaup, Phys. Rev. A 44, 4582 (1991).
    [CrossRef] [PubMed]
  21. H. A. Haus, W. S. Wong, and F. I. Khatri, J. Opt. Soc. Am. B 14, 304 (1997).
    [CrossRef]

2003 (3)

R. O. Moore, G. Biondini, and W. L. Kath, Opt. Lett. 28, 105 (2003).
[CrossRef] [PubMed]

K.-P. Ho, in Advances in Optics and Laser Research, Vol. 3, W. T. Arkin, ed. (Nova Science, Hauppauge, N.Y., 2003).

K.-P. Ho, Opt. Lett. 28, 1350 (2003).
[CrossRef] [PubMed]

2001 (1)

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turisyn, Phys. Rev. E 63, 025601 (2001).
[CrossRef]

1999 (1)

O. Leclerc, E. Desurvire, P. Brindel, and E. Maunand, Opt. Fiber Technol. 5, 301 (1999).
[CrossRef]

1998 (1)

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

1997 (1)

H. A. Haus, W. S. Wong, and F. I. Khatri, J. Opt. Soc. Am. B 14, 304 (1997).
[CrossRef]

1996 (2)

T. Georges, Opt. Commun. 123, 617 (1996).
[CrossRef]

P. Shum and H. Ghafouri-Shiraz, Opt. Laser Technol. 28, 535 (1996).
[CrossRef]

1995 (3)

T. Georges, Opt. Fiber Technol. 1, 97 (1995).
[CrossRef]

C. R. Menyuk, Opt. Lett. 20, 285 (1995).
[CrossRef] [PubMed]

T. Georges, Electron. Lett. 31, 1174 (1995).
[CrossRef]

1991 (3)

E. M. Stein and J. C. Stein, Rev. Financ. Stud. 4, 727 (1991).
[CrossRef]

P. A. Humblet and M. Azizog̃lu, J. Lightwave Technol. 9, 1576 (1991).
[CrossRef]

D. J. Kaup, Phys. Rev. A 44, 4582 (1991).
[CrossRef] [PubMed]

1990 (1)

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

1989 (1)

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

1986 (1)

J. P. Gordon and H. A. Haus, Opt. Lett. 11, 865 (1986).

1985 (1)

C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, Berlin, 1985).

1980 (1)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), Sect. 6.561.

1945 (1)

R. H. Cameron and W. T. Martin, Bull. Am. Math. Soc. 51, 73 (1945).
[CrossRef]

Azizog~lu, M.

P. A. Humblet and M. Azizog̃lu, J. Lightwave Technol. 9, 1576 (1991).
[CrossRef]

Biondini, G.

Brindel, P.

O. Leclerc, E. Desurvire, P. Brindel, and E. Maunand, Opt. Fiber Technol. 5, 301 (1999).
[CrossRef]

Cameron, R. H.

R. H. Cameron and W. T. Martin, Bull. Am. Math. Soc. 51, 73 (1945).
[CrossRef]

Desurvire, E.

O. Leclerc, E. Desurvire, P. Brindel, and E. Maunand, Opt. Fiber Technol. 5, 301 (1999).
[CrossRef]

Falkovich, G. E.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turisyn, Phys. Rev. E 63, 025601 (2001).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, Berlin, 1985).

Georges, T.

T. Georges, Opt. Commun. 123, 617 (1996).
[CrossRef]

T. Georges, Electron. Lett. 31, 1174 (1995).
[CrossRef]

T. Georges, Opt. Fiber Technol. 1, 97 (1995).
[CrossRef]

Ghafouri-Shiraz, H.

P. Shum and H. Ghafouri-Shiraz, Opt. Laser Technol. 28, 535 (1996).
[CrossRef]

Gordon, J. P.

J. P. Gordon and H. A. Haus, Opt. Lett. 11, 865 (1986).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), Sect. 6.561.

Haus, H. A.

H. A. Haus, W. S. Wong, and F. I. Khatri, J. Opt. Soc. Am. B 14, 304 (1997).
[CrossRef]

J. P. Gordon and H. A. Haus, Opt. Lett. 11, 865 (1986).

Ho, K.-P.

K.-P. Ho, in Advances in Optics and Laser Research, Vol. 3, W. T. Arkin, ed. (Nova Science, Hauppauge, N.Y., 2003).

K.-P. Ho, Opt. Lett. 28, 1350 (2003).
[CrossRef] [PubMed]

Humblet, P. A.

P. A. Humblet and M. Azizog̃lu, J. Lightwave Technol. 9, 1576 (1991).
[CrossRef]

Iannone, E.

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

Kath, W. L.

Kaup, D. J.

D. J. Kaup, Phys. Rev. A 44, 4582 (1991).
[CrossRef] [PubMed]

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

Khatri, F. I.

H. A. Haus, W. S. Wong, and F. I. Khatri, J. Opt. Soc. Am. B 14, 304 (1997).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Kolokolov, I.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turisyn, Phys. Rev. E 63, 025601 (2001).
[CrossRef]

Lebedev, V.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turisyn, Phys. Rev. E 63, 025601 (2001).
[CrossRef]

Leclerc, O.

O. Leclerc, E. Desurvire, P. Brindel, and E. Maunand, Opt. Fiber Technol. 5, 301 (1999).
[CrossRef]

Malomed, B. A.

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Martin, W. T.

R. H. Cameron and W. T. Martin, Bull. Am. Math. Soc. 51, 73 (1945).
[CrossRef]

Matera, F.

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

Maunand, E.

O. Leclerc, E. Desurvire, P. Brindel, and E. Maunand, Opt. Fiber Technol. 5, 301 (1999).
[CrossRef]

Mecozzi, A.

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

Menyuk, C. R.

Moore, R. O.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), Sect. 6.561.

Settembre, M.

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

Shum, P.

P. Shum and H. Ghafouri-Shiraz, Opt. Laser Technol. 28, 535 (1996).
[CrossRef]

Stein, E. M.

E. M. Stein and J. C. Stein, Rev. Financ. Stud. 4, 727 (1991).
[CrossRef]

Stein, J. C.

E. M. Stein and J. C. Stein, Rev. Financ. Stud. 4, 727 (1991).
[CrossRef]

Turisyn, S. K.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turisyn, Phys. Rev. E 63, 025601 (2001).
[CrossRef]

Wong, W. S.

H. A. Haus, W. S. Wong, and F. I. Khatri, J. Opt. Soc. Am. B 14, 304 (1997).
[CrossRef]

Bull. Am. Math. Soc. (1)

R. H. Cameron and W. T. Martin, Bull. Am. Math. Soc. 51, 73 (1945).
[CrossRef]

Electron. Lett. (1)

T. Georges, Electron. Lett. 31, 1174 (1995).
[CrossRef]

J. Lightwave Technol. (1)

P. A. Humblet and M. Azizog̃lu, J. Lightwave Technol. 9, 1576 (1991).
[CrossRef]

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

H. A. Haus, W. S. Wong, and F. I. Khatri, J. Opt. Soc. Am. B 14, 304 (1997).
[CrossRef]

Opt. Commun. (1)

T. Georges, Opt. Commun. 123, 617 (1996).
[CrossRef]

Opt. Fiber Technol. (2)

O. Leclerc, E. Desurvire, P. Brindel, and E. Maunand, Opt. Fiber Technol. 5, 301 (1999).
[CrossRef]

T. Georges, Opt. Fiber Technol. 1, 97 (1995).
[CrossRef]

Opt. Laser Technol. (1)

P. Shum and H. Ghafouri-Shiraz, Opt. Laser Technol. 28, 535 (1996).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (2)

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

D. J. Kaup, Phys. Rev. A 44, 4582 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (1)

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turisyn, Phys. Rev. E 63, 025601 (2001).
[CrossRef]

Rev. Financ. Stud. (1)

E. M. Stein and J. C. Stein, Rev. Financ. Stud. 4, 727 (1991).
[CrossRef]

Rev. Mod. Phys. (1)

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Other (4)

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

C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, Berlin, 1985).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), Sect. 6.561.

K.-P. Ho, in Advances in Optics and Laser Research, Vol. 3, W. T. Arkin, ed. (Nova Science, Hauppauge, N.Y., 2003).

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Figures (2)

Fig. 1
Fig. 1

Probability density functions of (a) frequency and (b) timing jitters compared with a Gaussian distribution (dashed curve).

Fig. 2
Fig. 2

Tail probabilities of (a) frequency and (b) timing jitters compared with a complementary error function (dashed curve).

Equations (22)

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

dAdζ=nAζ,
dΩdζ=nΩζ,
dTdζ=-Ω+nTζ.
EnAζ1nAζ2=Aσn2δζ1-ζ2,
EnΩζ1nΩζ2=A3σn2δζ1-ζ2,
EnTζ1nTζ2=π212A3σn2δζ1-ζ2,
Aζ=A01/2+wA22
Ωζ=0ζA01/2+wAζ12dwΩ,
σΩ2ζ=13A0σn2ζ+124σn4ζ2,
TGHζ=-0ζζ-ζ1A01/2+wAζ12dwΩ,
σT2ζ=19A0σn2ζ3+1144σn4ζ4.
ΨΩζν=G1ν2σn26,
ΨTGHζν=G2ν2σn26,
G1λ=Eexp-λ0ζA01/2+wAζ122dζ1,
G2λ=Eexp-λ0ζζ-ζ12×A01/2+wAζ122dζ1,
G1λ=exp-2A0σnλ tanhζσn2λcosh1/2ζσn2λ,
G2λ=fλζfλ01/2 expλ2σn2A0β2-λA0ζ3/3,
β2=0ζ1fλζ1ζ1ζζ-ζ22fλζ2dζ22dζ1,
fλζ1=ζ-ζ1I-1/4λσn2ζ-ζ12,
ζ1ζζ-ζ22fλζ2dζ2=ζ-ζ13/2λσn×I3/4λσn2ζ-ζ12,
β2=2λσn-7/20λσnζ2/2x1/2I3/42xI-1/4-2xdx.
G1λexp-λA0ζ,  G2λexp-13λA0ζ3,

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