Abstract

The mathematical methods required to model simple stochastic processes are reviewed briefly. These methods are used to determine the probability-density function (PDF) for noise-induced energy perturbations of isolated (solitary) optical pulses in fiber communication systems. The analytical formula is consistent with the numerical solution of the energy-moment equation. System failures are caused by large energy perturbations. For such perturbations the actual PDF differs significantly from the (ideal-ized) Gauss PDF that is often used to predict system performance.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic, San Diego, 1997), pp. 373–460.
  2. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998).
  3. K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda, and T. Misuochi, “A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,” OFC 2003, paper ThE2.
  4. J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.
  5. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
    [CrossRef] [PubMed]
  6. C. J. McKinstrie, “Gordon-Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200, 165–177 (2001) and references therein.
    [CrossRef]
  7. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communication systems using linear amplifiers,” Opt. Lett. 23, 1351–1353 (1990).
    [CrossRef]
  8. C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27, 1887–1889 (2002)and references therein.
    [CrossRef]
  9. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged dersciption of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
    [CrossRef]
  10. W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22, 226–230 (1977).
    [CrossRef]
  11. D. Anderson, “Variational approach to pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  12. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
    [CrossRef] [PubMed]
  13. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
    [CrossRef]
  14. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
    [CrossRef]
  15. J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
    [CrossRef]
  16. P. A. Humblet and M. Azizog̃lu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991).
    [CrossRef]
  17. J. S. Lee and C. S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. 12, 1224–1229 (1994).
    [CrossRef]
  18. T. Yoshino and G. P. Agrawal, “Photoelectron statistics of solitons corrupted by amplified spontaneous emission,” Phys. Rev. A 51, 1662–1668 (1995).
    [CrossRef] [PubMed]
  19. B. A. Malomed and N. Flytzanis, “Fluctuational distribution function of solitons in the nonlinear Schrödinger system,” Phys. Rev. E 48, R5–R8 (1993).
    [CrossRef]
  20. G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 25601R (2001).
    [CrossRef]
  21. R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002).
    [CrossRef]
  22. R. O. Moore, G. Biondini, and W. L. Kath, “Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems,” Opt. Lett. 28, 105–107 (2003).
    [CrossRef] [PubMed]
  23. C. J. McKinstrie and P. J. Winzer, “How to apply importance-sampling techniques to simulations of optical systems,” http://arxiv.org/physics/0309002.
  24. H. Kim and A. H. Gnauck, “Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,” IEEE Photon. Technol. Lett. 15, 320–322 (2003).
    [CrossRef]
  25. C. W. Gardiner, Handbook of Stochastic Methods, 2nd Ed. (Springer, Berlin, 2002).
  26. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  27. J. G. Proakis, Digital Communications, 3rd Ed. (McGraw-Hill, New York, 1995).
  28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), result 29.3.81.
  29. W. Horsthemke and R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1983), Sections 5.5 and 6.7.
  30. R. Graham, “Hopf bifurcation with fluctuating control parameter,” Phys. Rev. A 25, 3234–3258 (1982).
    [CrossRef]
  31. R. Graham and A. Schenzle, “Carleman imbedding of multiplicative stochastic processes,” Phys. Rev. A 25, 1731–1754 (1982).
    [CrossRef]
  32. I. S. Gradsteyn and I. M. Rhyzhik, Table of Integrals, Series and Products, 5th Ed. (Academic, San Diego, 1994), result 6.633.2.
  33. P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. 15, 840–842 (2003) and references therein.
    [CrossRef]
  34. J. D. Hoffman, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1992).

2003 (3)

H. Kim and A. H. Gnauck, “Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,” IEEE Photon. Technol. Lett. 15, 320–322 (2003).
[CrossRef]

P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. 15, 840–842 (2003) and references therein.
[CrossRef]

R. O. Moore, G. Biondini, and W. L. Kath, “Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems,” Opt. Lett. 28, 105–107 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (2)

C. J. McKinstrie, “Gordon-Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200, 165–177 (2001) and references therein.
[CrossRef]

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 25601R (2001).
[CrossRef]

1995 (1)

T. Yoshino and G. P. Agrawal, “Photoelectron statistics of solitons corrupted by amplified spontaneous emission,” Phys. Rev. A 51, 1662–1668 (1995).
[CrossRef] [PubMed]

1994 (1)

J. S. Lee and C. S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. 12, 1224–1229 (1994).
[CrossRef]

1993 (1)

B. A. Malomed and N. Flytzanis, “Fluctuational distribution function of solitons in the nonlinear Schrödinger system,” Phys. Rev. E 48, R5–R8 (1993).
[CrossRef]

1991 (2)

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[CrossRef]

P. A. Humblet and M. Azizog̃lu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

1990 (4)

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
[CrossRef]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
[CrossRef]

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communication systems using linear amplifiers,” Opt. Lett. 23, 1351–1353 (1990).
[CrossRef]

1986 (1)

1983 (1)

D. Anderson, “Variational approach to pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

1982 (2)

R. Graham, “Hopf bifurcation with fluctuating control parameter,” Phys. Rev. A 25, 3234–3258 (1982).
[CrossRef]

R. Graham and A. Schenzle, “Carleman imbedding of multiplicative stochastic processes,” Phys. Rev. A 25, 1731–1754 (1982).
[CrossRef]

1977 (1)

W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22, 226–230 (1977).
[CrossRef]

1971 (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged dersciption of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Abe, J.

K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda, and T. Misuochi, “A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,” OFC 2003, paper ThE2.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), result 29.3.81.

Agrawal, G. P.

T. Yoshino and G. P. Agrawal, “Photoelectron statistics of solitons corrupted by amplified spontaneous emission,” Phys. Rev. A 51, 1662–1668 (1995).
[CrossRef] [PubMed]

Anderson, D.

D. Anderson, “Variational approach to pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Azizog~lu, M.

P. A. Humblet and M. Azizog̃lu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

Bergano, N. S.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Biondini, G.

Cai, J. X.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Cai, Y.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Chandrasekhar, S.

P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. 15, 840–842 (2003) and references therein.
[CrossRef]

Davidson, C. R.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Domagala, G.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Falkovich, G. E.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 25601R (2001).
[CrossRef]

Firth, W. J.

W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22, 226–230 (1977).
[CrossRef]

Flytzanis, N.

B. A. Malomed and N. Flytzanis, “Fluctuational distribution function of solitons in the nonlinear Schrödinger system,” Phys. Rev. E 48, R5–R8 (1993).
[CrossRef]

Foursa, D. G.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Gardiner, C. W.

C. W. Gardiner, Handbook of Stochastic Methods, 2nd Ed. (Springer, Berlin, 2002).

Gnauck, A. H.

H. Kim and A. H. Gnauck, “Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,” IEEE Photon. Technol. Lett. 15, 320–322 (2003).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gordon, J. P.

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communication systems using linear amplifiers,” Opt. Lett. 23, 1351–1353 (1990).
[CrossRef]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic, San Diego, 1997), pp. 373–460.

Gradsteyn, I. S.

I. S. Gradsteyn and I. M. Rhyzhik, Table of Integrals, Series and Products, 5th Ed. (Academic, San Diego, 1994), result 6.633.2.

Graham, R.

R. Graham, “Hopf bifurcation with fluctuating control parameter,” Phys. Rev. A 25, 3234–3258 (1982).
[CrossRef]

R. Graham and A. Schenzle, “Carleman imbedding of multiplicative stochastic processes,” Phys. Rev. A 25, 1731–1754 (1982).
[CrossRef]

Grigoryan, V. S.

Haus, H. A.

Hoffman, J. D.

J. D. Hoffman, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1992).

Holzlöhner, R.

Horsthemke, W.

W. Horsthemke and R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1983), Sections 5.5 and 6.7.

Humblet, P. A.

P. A. Humblet and M. Azizog̃lu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

Iannone, E.

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

Ishida, K.

K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda, and T. Misuochi, “A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,” OFC 2003, paper ThE2.

Kath, W. L.

Kaup, D. J.

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

Kim, H.

P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. 15, 840–842 (2003) and references therein.
[CrossRef]

H. Kim and A. H. Gnauck, “Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,” IEEE Photon. Technol. Lett. 15, 320–322 (2003).
[CrossRef]

Kinjo, K.

K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda, and T. Misuochi, “A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,” OFC 2003, paper ThE2.

Kobayashi, T.

K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda, and T. Misuochi, “A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,” OFC 2003, paper ThE2.

Kolokolov, I.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 25601R (2001).
[CrossRef]

Kuroda, S.

K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda, and T. Misuochi, “A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,” OFC 2003, paper ThE2.

Lai, Y.

Lakoba, T. I.

Lebedev, V.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 25601R (2001).
[CrossRef]

Lee, J. S.

J. S. Lee and C. S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. 12, 1224–1229 (1994).
[CrossRef]

Lefever, R.

W. Horsthemke and R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1983), Sections 5.5 and 6.7.

Li, H.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Liu, L.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Malomed, B. A.

B. A. Malomed and N. Flytzanis, “Fluctuational distribution function of solitons in the nonlinear Schrödinger system,” Phys. Rev. E 48, R5–R8 (1993).
[CrossRef]

Mamyshev, P. V.

L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic, San Diego, 1997), pp. 373–460.

Marcuse, D.

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
[CrossRef]

Matera, F.

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

McKinstrie, C. J.

C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27, 1887–1889 (2002)and references therein.
[CrossRef]

C. J. McKinstrie, “Gordon-Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200, 165–177 (2001) and references therein.
[CrossRef]

C. J. McKinstrie and P. J. Winzer, “How to apply importance-sampling techniques to simulations of optical systems,” http://arxiv.org/physics/0309002.

Mecozzi, A.

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

Menyuk, C. R.

Misuochi, T.

K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda, and T. Misuochi, “A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,” OFC 2003, paper ThE2.

Mollenauer, L. F.

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communication systems using linear amplifiers,” Opt. Lett. 23, 1351–1353 (1990).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic, San Diego, 1997), pp. 373–460.

Moore, R. O.

Nissov, M.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Patterson, W. W.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged dersciption of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Pilipetskii, A. N.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Proakis, J. G.

J. G. Proakis, Digital Communications, 3rd Ed. (McGraw-Hill, New York, 1995).

Rhyzhik, I. M.

I. S. Gradsteyn and I. M. Rhyzhik, Table of Integrals, Series and Products, 5th Ed. (Academic, San Diego, 1994), result 6.633.2.

Schenzle, A.

R. Graham and A. Schenzle, “Carleman imbedding of multiplicative stochastic processes,” Phys. Rev. A 25, 1731–1754 (1982).
[CrossRef]

Settembre, M.

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

Shim, C. S.

J. S. Lee and C. S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. 12, 1224–1229 (1994).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), result 29.3.81.

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged dersciption of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Turitsyn, S. K.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 25601R (2001).
[CrossRef]

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged dersciption of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Winzer, P. J.

P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. 15, 840–842 (2003) and references therein.
[CrossRef]

C. J. McKinstrie and P. J. Winzer, “How to apply importance-sampling techniques to simulations of optical systems,” http://arxiv.org/physics/0309002.

Xie, C.

Yoshino, T.

T. Yoshino and G. P. Agrawal, “Photoelectron statistics of solitons corrupted by amplified spontaneous emission,” Phys. Rev. A 51, 1662–1668 (1995).
[CrossRef] [PubMed]

IEEE Photon. Technol. Lett. (2)

H. Kim and A. H. Gnauck, “Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,” IEEE Photon. Technol. Lett. 15, 320–322 (2003).
[CrossRef]

P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. 15, 840–842 (2003) and references therein.
[CrossRef]

J. Lightwave Technol. (5)

R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002).
[CrossRef]

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[CrossRef]

P. A. Humblet and M. Azizog̃lu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

J. S. Lee and C. S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. 12, 1224–1229 (1994).
[CrossRef]

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

Opt. Commun. (2)

C. J. McKinstrie, “Gordon-Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200, 165–177 (2001) and references therein.
[CrossRef]

W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22, 226–230 (1977).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (5)

D. Anderson, “Variational approach to pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

R. Graham, “Hopf bifurcation with fluctuating control parameter,” Phys. Rev. A 25, 3234–3258 (1982).
[CrossRef]

R. Graham and A. Schenzle, “Carleman imbedding of multiplicative stochastic processes,” Phys. Rev. A 25, 1731–1754 (1982).
[CrossRef]

T. Yoshino and G. P. Agrawal, “Photoelectron statistics of solitons corrupted by amplified spontaneous emission,” Phys. Rev. A 51, 1662–1668 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (2)

B. A. Malomed and N. Flytzanis, “Fluctuational distribution function of solitons in the nonlinear Schrödinger system,” Phys. Rev. E 48, R5–R8 (1993).
[CrossRef]

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 25601R (2001).
[CrossRef]

Radiophys. Quantum Electron. (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged dersciption of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Other (12)

J. D. Hoffman, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1992).

I. S. Gradsteyn and I. M. Rhyzhik, Table of Integrals, Series and Products, 5th Ed. (Academic, San Diego, 1994), result 6.633.2.

C. J. McKinstrie and P. J. Winzer, “How to apply importance-sampling techniques to simulations of optical systems,” http://arxiv.org/physics/0309002.

C. W. Gardiner, Handbook of Stochastic Methods, 2nd Ed. (Springer, Berlin, 2002).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. G. Proakis, Digital Communications, 3rd Ed. (McGraw-Hill, New York, 1995).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), result 29.3.81.

W. Horsthemke and R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1983), Sections 5.5 and 6.7.

L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic, San Diego, 1997), pp. 373–460.

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

K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda, and T. Misuochi, “A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,” OFC 2003, paper ThE2.

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

PDFs for the normalized energy E/E 0. The solid, dot-dashed and dashed curves represent the exact PDF (86), the approximate PDF (78) and Gauss PDF (75), respectively. The differences between the exact and approximate PDFs are barely perceptible. Because the noise-induced energy perturbations depend on the pulse energy, the exact and approximate PDFs (which account for this dependence) have enhanced high-energy tails and diminished low-energy tails relative to the Gauss PDF (which does not).

Fig. 2.
Fig. 2.

PDFs for the normalized energy E/E 0. The solid curve represents the analytical solution (86), whereas the dashed curve represents the numerical solution of the FPE (79) and the dots represent the results of importance-sampled simulations based on the FDE (113).

Fig. 3.
Fig. 3.

Mean and variance of the normalized energy E/E(0) plotted as functions of distance. The curves were obtained by solving Eqs. (124) and (125) numerically.

Fig. 4.
Fig. 4.

Relative energy variance plotted as a function of distance. The solid curve was obtained by solving Eqs. (124) and (125) numerically, whereas the dashed line was obtained from an approximate analytical formula.

Equations (128)

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

X . = a ( X , z ) + b ( X , z ) r ( z ) ,
r ( z ) = 0 ,
r ( z ) r ( z ) = δ ( z z ) ,
W ( z ) = 0 z r ( z ) d z
W ( z ) = 0 ,
W ( z ) W ( z ) = min ( z , z ) .
G z ( ζ ) = lim L L 2 L 2 r ( z + ζ ) r * ( z ) d z L
G e ( z , z ) = r ( z ) r * ( z ) .
r ¯ ( k ) = L 2 L 2 r ( z ) exp ( i 2 π k z ) d z
S ( k ) = lim L r ¯ ( k ) 2 L .
S ( k ) = G ( ζ ) exp ( i 2 π k ζ ) d ζ ,
G ( ζ ) = S ( k ) exp ( i 2 π k ζ ) d k .
S ( k ) = rect ( k K ) .
G ( ζ ) = K sinc ( π K ζ )
δ X = 0 δ z { a [ X ( z ) ] + b [ X ( z ) ] r ( z ) } d z ,
0 δ z a [ X ( z ) ] d z a 0 δ z ,
0 δ z b [ X ( z ) ] r ( z ) d z 0 δ z [ b 0 + b 0 b 0 0 z r ( z ) d z ] r ( z ) d z ,
δ X a 0 δ z + b 0 0 δ z r ( z ) d z + b 0 b 0 0 δ z 0 z r ( z ) r ( z ) d z d z .
δ X a 0 δ z + b 0 b 0 0 δ z 0 z r ( z ) r ( z ) d z d z .
δ X = 0 δ z { a [ X ( z ) ] + b [ X ( z ) ] r ( z + l ) } d z + O ( l ) .
δ X a 0 δ z .
δ X 2 b 0 2 0 δ z 0 δ z r ( z ) r ( z ) d z d z ,
= b 0 2 δ z .
δ X ( a 0 + b 0 b 0 2 ) δ z ,
δ X 2 b 0 2 δ z .
δ X = a δ z + b δ W .
δ Y f X δ X + f XX δ X 2 2 ,
Y . = f X X . + f XX b 2 2 .
δ X = ( a + b b X 2 ) δ z + b δ W ,
δ Y = f X ( a + b b X 2 ) δ z + f X b δ W + f XX b 2 δ z 2 .
δ Y = [ ( a g Y ) + ( b g Y ) ( b g Y ) Y 2 ] δ z + ( b g Y ) δ W .
Y . = ( a + b r ) g Y ,
= f X X . .
P ( x , δ z ) = T ( δ x , δ z | x δ x , 0 ) P ( x δ x , 0 ) d ( δ x ) ,
P + δ z z P P T ( δ x , δ z ) d ( δ x )
x [ P δ x T ( δ x , δ z ) d ( δ x ) ]
+ xx [ P δ x 2 T ( δ x , δ z ) d ( δ x ) 2 ] ,
T ( δ x , δ z ) = exp [ ( δ x a δ z ) 2 2 b 2 δ z ] ( 2 π b 2 δ z ) 1 2 .
z P = x ( a P ) + xx ( b 2 P ) 2 .
z P = x [ ( a + b x b 2 ) P ] + xx ( b 2 P ) 2 ,
z P = x ( a P ) + x [ b x ( b P ) ] 2 .
f ( X ) r = 0
X ( 0 ) X ( δ z ) + b [ X ( δ z ) ] δ z 0 r ( z ) d z ,
f [ X ( 0 ) ] r ( 0 ) f [ X ( δ z ) ] r ( 0 ) + f [ X ( δ z ) ] b [ X ( δ z ) ] δ z 0 r ( z ) r ( 0 ) d z .
f ( X ) r = f ( X ) b ( X ) 2 .
r x ( z ) = a ( X ) ,
r x ( z ) = a ( X ) + b ( X ) b ( X ) 2 .
δ r x ( z ) δ r x ( z ) = b 2 ( X ) δ ( z z ) .
z A = [ g ( z ) α ] A 2 i β tt 2 A 2 + i γ A 2 A + R ( z , t ) ,
R ( z , t ) = 0 ,
R ( z , t ) R ( z , t ) = 0 ,
R * ( z , t ) R ( z , t ) = S ( z ) δ ( z z ) δ ( t t ) ,
δ A = 0 δ z R ( z , t ) d z + O ( δ z 3 2 ) .
δ E = T 2 T 2 δ A ( t ) 2 d t .
δ E = T 2 T 2 0 δ z 0 δ z R * ( z , t ) R ( z , t ) d z d z d t .
δ E = S δ z F T ,
E = T 2 T 2 A ( t ) 2 d t .
d z E = [ g ( z ) α ] E + S ( z ) F T + T 2 T 2 [ A * ( z , t ) R ( z , t ) + A ( z , t ) R * ( z , t ) ] d t .
d z E = a ( E , z ) + b ( E , z ) r ( z ) ,
r e ( z ) = a ( E ) ,
δ r e ( z ) δ r e ( z ) = b 2 ( E ) δ ( z z ) .
A * ( z , t ) R ( z , t ) = 0 .
r e ( z ) = ( g α ) E + S F T ,
δ r e ( z ) δ r e ( z ) = 2 S E δ ( z z ) .
d z E = ( g α ) E + S F T + ( 2 S E ) 1 2 r ( z ) ,
z P = ( α g ) e ( e P ) S F T e P + S ee 2 ( e P ) .
d z E = [ g ( z ) α ] E + T 2 T 2 [ A * ( z , t ) R ( z , t ) + A ( z , t ) R * ( z , t ) ] d t .
r e ( z ) = a ( E ) + b ( E ) b ( E ) 2 ,
δ r e ( z ) δ r e ( z ) = b 2 ( E ) δ ( z z ) ,
A * ( z , t ) R ( z , t ) = S F 2 .
r e ( z ) = ( g α ) E + S F T ,
δ r e ( z ) δ r e ( z ) = 2 S E δ ( z z ) .
d z E = ( g α ) E + S ( F T 1 2 ) + ( 2 S E ) 1 2 r ( z ) ,
z P = ( α g ) e ( e P ) S F T e P + S ee 2 ( e P ) .
d ζ X = μ + ( 2 X ) 1 2 r ( ζ ) ,
d ζ X μ + 2 1 2 r ( ζ ) .
P ( x , ζ ) exp [ ( x m n ) 2 2 v n ] ( 2 π v n ) 1 2 .
d ζ Y = μ ¯ Y + r ( ζ ) 2 1 2 ,
d ζ Y μ ¯ + r ( ζ ) 2 1 2 .
P ( x , ζ ) exp [ ( x 1 2 m n ) 2 2 v n ] ( 8 π x v n ) 1 2 .
ζ P = μ x P + xx 2 ( x P )
x n ( ζ ) = 0 x n P ( x , ζ ) d ζ .
d ζ x = μ ,
d ζ x 2 = 2 ( 1 + μ ) x ,
P ¯ ( s , ζ ) = exp [ s ( 1 + s ζ ) ] ( 1 + s ζ ) μ .
x n ( ζ ) = ( 1 ) n lim s 0 d n P ¯ ( s , ζ ) d s n .
exp ( k s ) s μ ( x k ) ( μ 1 ) 2 I μ 1 [ 2 ( k x ) 1 2 ] ,
P ( x , ζ ) = [ x ( μ 1 ) 2 ζ ] exp [ ( 1 + x ) ζ ] I μ 1 ( 2 x 1 2 ζ ) .
P ( x , t ) x μ 2 exp [ ( x 1 2 1 ) 2 ζ ] ( 4 π x 3 2 ζ ) 1 2 .
ζ P = ( μ 2 ) x 0 P + x 0 x 0 2 ( x 0 P ) .
P ( x , 0 | x 0 , 0 ) = δ ( x x 0 ) .
0 P ( x , ζ | x 0 , 0 ) d x = 1 = 0 P ( x , ζ | x 0 , 0 ) d x 0 .
P ( x , ζ | x 0 , 0 ) = 0 exp ( λ ζ ) p ( x , λ ) p ˜ ( x 0 , λ ) d λ ,
x d xx 2 P ( μ 2 ) d x p = λ p ,
d xx 2 ( x p ˜ ) + ( μ 2 ) d x p ˜ = λ p ˜ .
0 P ( x , λ ) p ˜ ( x , λ ) d x = δ ( λ λ ) .
p ( x , λ ) = q ( x ) p ˜ ( x , λ ) ,
d x ( x q ) μ q = 0 .
0 p ( x , λ ) p ˜ ( x , λ ) d λ = δ ( x x ) .
f ( x ) = 0 δ ( x x ) f ( x ) d x ,
= 0 c ( λ ) p ( x , λ ) d λ , where c ( λ ) = 0 f ( x ) p ˜ ( x , λ ) d x ,
= 0 c ˜ ( λ ) p ˜ ( x , λ ) d λ , where c ˜ ( λ ) = 0 f ( x ) p ( x , λ ) d x .
p ( x , λ ) = x ( μ 1 ) 2 J μ 1 [ 2 ( λ x ) 1 2 ] ,
p ˜ ( x , λ ) = x ( 1 μ ) 2 J μ 1 [ 2 ( λ x ) 1 2 ] ,
0 exp ( γ 2 u 2 ) J n ( α u ) J n ( β u ) u d u = exp [ ( α 2 + β 2 ) 4 γ 2 ] I n ( α β 2 γ 2 ) 2 γ 2 ,
P ( x , ζ | x 0 , 0 ) = ( x x 0 ) ( μ 1 ) 2 exp [ ( x 0 + x ) ζ ] I μ 1 [ 2 ( x 0 x ) 1 2 ζ ] ζ .
μ P + x ( x P ) = 0 ,
X n + 1 = X n + a ( X n ) δ z + b ( X n ) δ W n + 1 ,
δ W m δ W n = δ z δ mn ,
X n + 1 p = X n + a ( X n ) δ z + b ( X n ) δ W n + 1 ,
X n + 1 c = X n + [ a ( X n ) + a ( X n + 1 p ) ] δ z 2
+ [ b ( X n ) + b ( X n + 1 p ) ] δ W n + 1 2 .
X n + 1 c X n + a n δ z + ( b n + b n b n δ W n + 1 2 ) δ W n + 1 .
δ X n + 1 ( a n + b n b n 2 ) δ z ,
δ X n + 1 2 b n 2 δ z .
X n + 1 = X n + ( a n + b n b n 2 ) δ z + b n δ W n + 1 .
d z X = μ σ x ( z ) + ν ( z ) X + [ 2 σ x ( z ) X ] 1 2 r ( z ) ,
σ x ( z ) = n sp ω g ( z ) E ( 0 )
d z Y = μ σ y ( z ) + [ 2 σ y ( z ) Y ] 1 2 r ( z ) ,
σ y ( z ) = [ n sp ω E ( 0 ) ] g ( z ) e 0 z ν ( z ) d z .
ζ ( z ) = 0 z σ y ( z ) d z .
d ζ Y = μ + ( 2 Y ) 1 2 r ( ζ ) .
E a = E ( 0 ) 0 l e 0 z ν ( z ) dz dz l .
ζ ( l ) = σ a 0 l e 0 l ν ( z ) dz dz 0 l e 0 z ν ( z ) dz dz l .
ζ ( z ) ρ σ a z ,
g ( z ) α p α l exp [ α p mod ( z , l ) ] exp ( α p l ) 1 ,
d z m x = ν ( z ) m x + μ σ x ( z ) ,
d z v x = 2 ν ( z ) v x + 2 σ x ( z ) m x ,

Metrics