Abstract

Through a series of extensive system simulations we show that all of the previously not understood discrepancies between the Gaussian noise (GN) model and simulations can be attributed to the omission of an important, recently reported, fourth-order noise (FON) term, that accounts for the statistical dependencies within the spectrum of the interfering channel. We examine the importance of the FON term as well as the dependence of NLIN on modulation format with respect to link-length and number of spans. A computationally efficient method for evaluating the FON contribution, as well as the overall NLIN power is provided.

© 2014 Optical Society of America

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References

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  1. R.-J. Essiambre, G. Kramer, P.J. Winzer, G.J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010).
    [Crossref]
  2. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
    [Crossref]
  3. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).
    [Crossref]
  4. A. Carena, G. Bosco, V. Curri, P. Poggiolini, and F. Forghieri, “Impact of the transmitted signal initial dispersion transient on the accuracy of the GN-model of non-linear propagation,” 39th European Conference and Exhibition on Optical Communication (ECOC 2013), Paper Th.1.D.4.
  5. P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31, 1273–1282 (2013).
    [Crossref]
  6. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express,  21, pp. 25685–25699 (2013).
    [Crossref] [PubMed]
  7. A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri, “On the accuracy of the GN-model and on analytical correction terms to improve it,” arXiv preprint 1401.6946v1 (2014).
  8. A. Mecozzi and R.-J. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012).
    [Crossref]
  9. M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps,” J. Lightwave Technol. 31, 3839–3852 (2013).
    [Crossref]
  10. R. Dar, M. Shtaif, and M. Feder, “New bounds on the capacity of the nonlinear fiber-optic channel,” Opt. Lett. 39, 398–401 (2014).
    [Crossref]
  11. 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]
  12. M. Shtaif, “Analytical description of cross-phase modulation in dispersive optical fibers,” Opt. Lett. 23, 1191–1193 (1998).
    [Crossref]
  13. R. E. Caflisch, “Monte Carlo and quasi-Monte Carlo methods,” Acta Numer. 7, 1–49 (1998).
    [Crossref]

2014 (1)

2013 (3)

2012 (2)

2011 (1)

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[Crossref]

2010 (1)

1998 (2)

1991 (1)

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]

Bosco, G.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).
[Crossref]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[Crossref]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, and F. Forghieri, “Impact of the transmitted signal initial dispersion transient on the accuracy of the GN-model of non-linear propagation,” 39th European Conference and Exhibition on Optical Communication (ECOC 2013), Paper Th.1.D.4.

A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri, “On the accuracy of the GN-model and on analytical correction terms to improve it,” arXiv preprint 1401.6946v1 (2014).

Caflisch, R. E.

R. E. Caflisch, “Monte Carlo and quasi-Monte Carlo methods,” Acta Numer. 7, 1–49 (1998).
[Crossref]

Carena, A.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).
[Crossref]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[Crossref]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, and F. Forghieri, “Impact of the transmitted signal initial dispersion transient on the accuracy of the GN-model of non-linear propagation,” 39th European Conference and Exhibition on Optical Communication (ECOC 2013), Paper Th.1.D.4.

A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri, “On the accuracy of the GN-model and on analytical correction terms to improve it,” arXiv preprint 1401.6946v1 (2014).

Curri, V.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).
[Crossref]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[Crossref]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, and F. Forghieri, “Impact of the transmitted signal initial dispersion transient on the accuracy of the GN-model of non-linear propagation,” 39th European Conference and Exhibition on Optical Communication (ECOC 2013), Paper Th.1.D.4.

A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri, “On the accuracy of the GN-model and on analytical correction terms to improve it,” arXiv preprint 1401.6946v1 (2014).

Dar, R.

Essiambre, R.-J.

Feder, M.

Forestieri, E.

Forghieri, F.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).
[Crossref]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[Crossref]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, and F. Forghieri, “Impact of the transmitted signal initial dispersion transient on the accuracy of the GN-model of non-linear propagation,” 39th European Conference and Exhibition on Optical Communication (ECOC 2013), Paper Th.1.D.4.

A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri, “On the accuracy of the GN-model and on analytical correction terms to improve it,” arXiv preprint 1401.6946v1 (2014).

Foschini, G.J.

Goebel, B.

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]

Jiang, Y.

A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri, “On the accuracy of the GN-model and on analytical correction terms to improve it,” arXiv preprint 1401.6946v1 (2014).

Johannisson, P.

P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31, 1273–1282 (2013).
[Crossref]

Karlsson, M.

P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31, 1273–1282 (2013).
[Crossref]

Kramer, G.

Mecozzi, A.

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]

Poggiolini, P.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).
[Crossref]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[Crossref]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, and F. Forghieri, “Impact of the transmitted signal initial dispersion transient on the accuracy of the GN-model of non-linear propagation,” 39th European Conference and Exhibition on Optical Communication (ECOC 2013), Paper Th.1.D.4.

A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri, “On the accuracy of the GN-model and on analytical correction terms to improve it,” arXiv preprint 1401.6946v1 (2014).

Prati, G.

Secondini, M.

Shtaif, M.

Winzer, P.J.

Acta Numer. (1)

R. E. Caflisch, “Monte Carlo and quasi-Monte Carlo methods,” Acta Numer. 7, 1–49 (1998).
[Crossref]

IEEE Photon. Technol. Lett. (1)

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[Crossref]

J. Lightwave Technol. (6)

Opt. Express (1)

Opt. Lett. (2)

Other (2)

A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri, “On the accuracy of the GN-model and on analytical correction terms to improve it,” arXiv preprint 1401.6946v1 (2014).

A. Carena, G. Bosco, V. Curri, P. Poggiolini, and F. Forghieri, “Impact of the transmitted signal initial dispersion transient on the accuracy of the GN-model of non-linear propagation,” 39th European Conference and Exhibition on Optical Communication (ECOC 2013), Paper Th.1.D.4.

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Figures (5)

Fig. 1
Fig. 1 The NLIN power versus the average power per-channel in a 5 × 100km system for QPSK and 16-QAM modulation. The solid lines show the theoretical results given by Eq. (1) and the dots represent simulations. The dashed red line corresponds to the SON contribution P3χ1, which is identical to the result of the GN model. (a) Distributed amplification. (b) Lumped amplification.
Fig. 2
Fig. 2 Constellation diagrams for QPSK (top panels) and 16-QAM (bottom panels) in the cases of distributed amplification (a and b), 25 km spans (c and d), 50 km (e and f) and 100 km (g and h). In all cases a path-averaged power of −10dBm was used. The phase noise nature of NLIN is evident in the case of 16-QAM modulation, but its relative significance reduces when the span-length is large. The 100 km span case is closer to the circular noise distributions observed in [3].
Fig. 3
Fig. 3 Accumulation of the NLIN power (normalized to the received power) with the number of spans. Figure a corresponds to the case of distributed amplification whereas figures b,c and d correspond to the cases of 25 km, 50 km, and 100 km span-lengths, respectively. The solid lines show the theoretical results given by Eq. (1) and the dots represent simulations. The red dashed curve represents the SON, or equivalently the GN model result.
Fig. 4
Fig. 4 The importance of incomplete collisions is reflected in the ratio between the FON and SON coefficients χ2/χ1. When incomplete collisions dominate χ2/χ1 ≪ 1, and when their contribution is small (as occurs in single span, or distributed gain systems) χ2/χ1 ∼ 1. In (a) The total link-length is held fixed at 500 km. In (b) the span length is kept constant. In both Figs. (a) and (b), the dashed curve corresponds to distributed amplification.
Fig. 5
Fig. 5 The effect of pre-dispersion. Accumulation of the NLIN power (normalized to the received power) with the number of spans. Figures a and b correspond to distributed amplification and span-length of 100 km, respectively, where pre-dispersion of 8500 ps/nm was applied to the injected pulses. The solid lines show the theoretical results given by Eq. (1) and the dots represent simulations. The red dashed curve represents the SON, or equivalently the GN model result.

Equations (1)

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σ NLIN 2 = P 3 χ 1 SON \ GN + P 3 χ 2 ( | b | 4 | b | 2 2 2 ) FON ,

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