Abstract

The GN-model has been proposed as an approximate but sufficiently accurate tool for predicting uncompensated optical coherent transmission system performance, in realistic scenarios. For this specific use, the GN-model has enjoyed substantial validation, both simulative and experimental. Recently, however, it has been pointed out that its predictions, when used to obtain a detailed picture of non-linear interference (NLI) noise accumulation along a link, may be affected by a substantial NLI overestimation error, especially in the first spans of the link. In this paper we analyze in detail the GN-model errors. We discuss recently proposed formulas for correcting such errors and show that they neglect several contributions to NLI, so that they may substantially underestimate NLI in specific situations, especially over low-dispersion fibers. We derive a complete set of formulas accounting for all single, cross, and multi-channel effects, This set constitutes what we have called the enhanced GN-model (EGN-model). We extensively validate the EGN model by comparison with accurate simulations in several different system scenarios. The overall EGN model accuracy is found to be very good when assessing detailed span-by-span NLI accumulation and excellent when estimating realistic system maximum reach. The computational complexity vs. accuracy trade-offs of the various versions of the GN and EGN models are extensively discussed.

© 2014 Optical Society of America

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  1. A. Splett, C. Kurzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” in Proc. ECOC 1993 (1993), paper MoC2.4.
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  4. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008).
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    [CrossRef]
  7. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012).
    [CrossRef]
  8. P. Johannisson, “Analytical modeling of nonlinear propagation in a strongly dispersive optical communication system,” arXiv:1205.2193, [physics.optics] (2012).
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    [CrossRef]
  12. P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol.31(8), 1273–1282 (2013).
    [CrossRef]
  13. P. Serena and A. Bononi, “An alternative approach to the Gaussian noise model and its system implications,” J. Lightwave Technol.31(22), 3489–3499 (2013).
    [CrossRef]
  14. P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN model of fiber non-linear propagation and its applications,” J. Lightwave Technol.32(4), 694–721 (2014).
    [CrossRef]
  15. S. Kilmurray, T. Fehenberger, P. Bayvel, and R. I. Killey, “Comparison of the nonlinear transmission performance of quasi-Nyquist WDM and reduced guard interval OFDM,” Opt. Express20(4), 4198–4205 (2012).
    [CrossRef] [PubMed]
  16. J. Pan, P. Isautier, M. Filer, S. Tibuleac, and S. E. Ralph, “Gaussian noise model aided in-band crosstalk analysis in ROADM-enabled DWDM networks,” in Proc. of OFC (2014), paper Th1I.1.
    [CrossRef]
  17. E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” Opt. Express19(26), B790–B798 (2011).
    [CrossRef] [PubMed]
  18. J.-X. Cai, O. V. Sinkin, H. Zhang, H. G. Batshon, M. Mazurczyk, D. G. Foursa, A. Pilipetskii, and G. Mohs, “Nonlinearity compensation benefit in high capacity ultra-long haul transmission systems,” in Proc. of ECOC (2013), paper We.4.D.2.
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  19. A. J. Stark, Y.-T. Hsueh, T. F. Detwiler, M. M. Filer, S. Tibuleac, and S. E. Ralph, “System performance prediction with the Gaussian noise model in 100G PDM-QPSK coherent optical networks,” J. Lightwave Technol.31(21), 3352–3360 (2013).
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    [CrossRef]
  21. J.-X. Cai, H. G. Batshon, H. Zhang, M. Mazurczyk, O. V. Sinkin, D. G. Foursa, and A. N. Pilipetskii, “Transmission performance of coded modulation formats in a wide range of spectral efficiencies,” in Proc. of OFC (2014), paper M2C.3.
    [CrossRef]
  22. 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,” in Proc. of ECOC (2013), paper Th.1.D.4.
    [CrossRef]
  23. P. Serena and A. Bononi, “On the accuracy of the Gaussian nonlinear model for dispersion-unmanaged coherent links,” in Proc. of ECOC (2013), paper Th.1.D.3.
    [CrossRef]
  24. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express21(22), 25685–25699 (2013).
    [CrossRef] [PubMed]
  25. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Accumulation of nonlinear interference noise in multi-span fiber-optic systems,” arXiv:1310.6137, [physics.optics] (2013).
  26. A. Carena, G. Bosco, V. Curri, P. Poggiolini, Y. Jiang, and F. Forghieri, “A simple and effective closed-form GN model correction formula accounting for signal non-Gaussian distribution,” arXiv:1402.3528, [physics.optics] (2014).
  27. 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:1401.6946, [physics.optics] (2014).
  28. S. J. Savory, “Approximations for the nonlinear self-channel interference of channels with rectangular spectra,” IEEE Photon. Technol. Lett.25(10), 961–964 (2013).
    [CrossRef]
  29. P. Johannisson and E. Agrell, “Modeling of nonlinear signal distortion in fiber-optical networks,” arXiv:1309.4000, [physics.optics] (2013).
  30. A. Bononi, O. Beucher, and P. Serena, “Single- and cross-channel nonlinear interference in the Gaussian Noise model with rectangular spectra,” Opt. Express21(26), 32254–32268 (2013).
    [CrossRef] [PubMed]
  31. M. Secondini and E. Forestieri, “Analytical fiber-optic channel model in the presence of cross-phase modulations,” IEEE Photon. Technol. Lett.24(22), 2016–2019 (2012).
    [CrossRef]

2014 (2)

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN model of fiber non-linear propagation and its applications,” J. Lightwave Technol.32(4), 694–721 (2014).
[CrossRef]

2013 (6)

2012 (4)

2011 (2)

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” Opt. Express19(26), B790–B798 (2011).
[CrossRef] [PubMed]

2010 (1)

2008 (1)

2003 (1)

H. Louchet, A. Hodzic, and K. Petermann, “Analytical model for the performance evaluation of DWDM transmission systems,” IEEE Photon. Technol. Lett.15(9), 1219–1221 (2003).
[CrossRef]

2002 (1)

Bauwelinck, J.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

Bayvel, P.

Beucher, O.

Bononi, A.

Bosco, G.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN model of fiber non-linear propagation and its applications,” J. Lightwave Technol.32(4), 694–721 (2014).
[CrossRef]

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012).
[CrossRef]

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” Opt. Express19(26), B790–B798 (2011).
[CrossRef] [PubMed]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

Carena, A.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN model of fiber non-linear propagation and its applications,” J. Lightwave Technol.32(4), 694–721 (2014).
[CrossRef]

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012).
[CrossRef]

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” Opt. Express19(26), B790–B798 (2011).
[CrossRef] [PubMed]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

Chen, X.

Cho, P.

Cigliutti, R.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” Opt. Express19(26), B790–B798 (2011).
[CrossRef] [PubMed]

Curri, V.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN model of fiber non-linear propagation and its applications,” J. Lightwave Technol.32(4), 694–721 (2014).
[CrossRef]

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012).
[CrossRef]

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” Opt. Express19(26), B790–B798 (2011).
[CrossRef] [PubMed]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

Dar, R.

Detwiler, T. F.

Feder, M.

Fehenberger, T.

Filer, M. M.

Forestieri, E.

M. Secondini and E. Forestieri, “Analytical fiber-optic channel model in the presence of cross-phase modulations,” IEEE Photon. Technol. Lett.24(22), 2016–2019 (2012).
[CrossRef]

Forghieri, F.

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN model of fiber non-linear propagation and its applications,” J. Lightwave Technol.32(4), 694–721 (2014).
[CrossRef]

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012).
[CrossRef]

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” Opt. Express19(26), B790–B798 (2011).
[CrossRef] [PubMed]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

Hirano, M.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

Hodzic, A.

H. Louchet, A. Hodzic, and K. Petermann, “Analytical model for the performance evaluation of DWDM transmission systems,” IEEE Photon. Technol. Lett.15(9), 1219–1221 (2003).
[CrossRef]

Hsueh, Y.-T.

Jiang, Y.

Johannisson, P.

P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol.31(8), 1273–1282 (2013).
[CrossRef]

Karagodsky, V.

Karlsson, M.

P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol.31(8), 1273–1282 (2013).
[CrossRef]

Khurgin, J.

Killey, R. I.

Kilmurray, S.

Louchet, H.

H. Louchet, A. Hodzic, and K. Petermann, “Analytical model for the performance evaluation of DWDM transmission systems,” IEEE Photon. Technol. Lett.15(9), 1219–1221 (2003).
[CrossRef]

Mecozzi, A.

Meiman, Y.

Nazarathy, M.

Nespola, A.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” Opt. Express19(26), B790–B798 (2011).
[CrossRef] [PubMed]

Noe, R.

Petermann, K.

H. Louchet, A. Hodzic, and K. Petermann, “Analytical model for the performance evaluation of DWDM transmission systems,” IEEE Photon. Technol. Lett.15(9), 1219–1221 (2003).
[CrossRef]

Poggiolini, P.

Ralph, S. E.

Sasaki, T.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

Savory, S. J.

S. J. Savory, “Approximations for the nonlinear self-channel interference of channels with rectangular spectra,” IEEE Photon. Technol. Lett.25(10), 961–964 (2013).
[CrossRef]

Secondini, M.

M. Secondini and E. Forestieri, “Analytical fiber-optic channel model in the presence of cross-phase modulations,” IEEE Photon. Technol. Lett.24(22), 2016–2019 (2012).
[CrossRef]

Serena, P.

Shieh, W.

Shpantzer, I.

Shtaif, M.

Stark, A. J.

Straullu, S.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

Tang, J.

Tibuleac, S.

Torrengo, E.

Verheyen, K.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

Weidenfeld, R.

Yamamoto, Y.

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

Zeolla, D.

IEEE Photon. Technol. Lett. (5)

H. Louchet, A. Hodzic, and K. Petermann, “Analytical model for the performance evaluation of DWDM transmission systems,” IEEE Photon. Technol. Lett.15(9), 1219–1221 (2003).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

A. Nespola, S. Straullu, A. Carena, G. Bosco, R. Cigliutti, V. Curri, P. Poggiolini, M. Hirano, Y. Yamamoto, T. Sasaki, J. Bauwelinck, K. Verheyen, and F. Forghieri, “GN-model validation over seven fiber types in uncompensated PM-16QAM Nyquist-WDM links,” IEEE Photon. Technol. Lett.26(2), 206–209 (2014).
[CrossRef]

M. Secondini and E. Forestieri, “Analytical fiber-optic channel model in the presence of cross-phase modulations,” IEEE Photon. Technol. Lett.24(22), 2016–2019 (2012).
[CrossRef]

S. J. Savory, “Approximations for the nonlinear self-channel interference of channels with rectangular spectra,” IEEE Photon. Technol. Lett.25(10), 961–964 (2013).
[CrossRef]

J. Lightwave Technol. (7)

Opt. Express (6)

Other (13)

P. Johannisson and E. Agrell, “Modeling of nonlinear signal distortion in fiber-optical networks,” arXiv:1309.4000, [physics.optics] (2013).

A. Splett, C. Kurzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” in Proc. ECOC 1993 (1993), paper MoC2.4.

J.-X. Cai, H. G. Batshon, H. Zhang, M. Mazurczyk, O. V. Sinkin, D. G. Foursa, and A. N. Pilipetskii, “Transmission performance of coded modulation formats in a wide range of spectral efficiencies,” in Proc. of OFC (2014), paper M2C.3.
[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,” in Proc. of ECOC (2013), paper Th.1.D.4.
[CrossRef]

P. Serena and A. Bononi, “On the accuracy of the Gaussian nonlinear model for dispersion-unmanaged coherent links,” in Proc. of ECOC (2013), paper Th.1.D.3.
[CrossRef]

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Accumulation of nonlinear interference noise in multi-span fiber-optic systems,” arXiv:1310.6137, [physics.optics] (2013).

A. Carena, G. Bosco, V. Curri, P. Poggiolini, Y. Jiang, and F. Forghieri, “A simple and effective closed-form GN model correction formula accounting for signal non-Gaussian distribution,” arXiv:1402.3528, [physics.optics] (2014).

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:1401.6946, [physics.optics] (2014).

J. Pan, P. Isautier, M. Filer, S. Tibuleac, and S. E. Ralph, “Gaussian noise model aided in-band crosstalk analysis in ROADM-enabled DWDM networks,” in Proc. of OFC (2014), paper Th1I.1.
[CrossRef]

J.-X. Cai, O. V. Sinkin, H. Zhang, H. G. Batshon, M. Mazurczyk, D. G. Foursa, A. Pilipetskii, and G. Mohs, “Nonlinearity compensation benefit in high capacity ultra-long haul transmission systems,” in Proc. of ECOC (2013), paper We.4.D.2.
[CrossRef]

P. Johannisson, “Analytical modeling of nonlinear propagation in a strongly dispersive optical communication system,” arXiv:1205.2193, [physics.optics] (2012).

A. Bononi and P. Serena, “An alternative derivation of Johannisson’s regular perturbation model,” arXiv:1207.4729, [physics.optics] (2012).

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “A detailed analytical derivation of the GN model of non-linear interference in coherent optical transmission systems,” arXiv:1209.0394, [physics.optics] (2012).

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Figures (9)

Fig. 1
Fig. 1

Plot of normalized Self-Channel Interference (SCI), η SCI , vs. number of spans in the link, assuming a single PM-QPSK channel over (from top to bottom) SMF, NZDSF and LS, with span length 100 km. Red dashed line: simulation. Blue solid line: GN-model. Green solid line: EGN-model (Eq. (5)).

Fig. 2
Fig. 2

Integration regions to obtain the power spectrum of XCI, G XCI GN (f) , at f=0 (i.e., at the center of CUT), due to a single adjacent INT channel, assuming that its center frequency is slightly higher than the symbol rate. The XPM approximation [24] of Eq. (14) considers the X1 regions only. The full XCI formula of Eq. (18) accounts for all X1-X4 regions.

Fig. 3
Fig. 3

Plot of normalized non-linearity coefficient η vs. number of spans in the link, assuming three PM-QPSK channels over (from top to bottom) SMF, NZDSF and LS, with span length 100 km. The CUT is the center channel. The spacing is 1.05 times the symbol rate. Red dashed line: simulation, with single-channel non-linearity (SCI) removed. Blue solid line: GN-model without SCI. Magenta solid line: the XPM approximation η XPM of [24] (Eq. (22) of this paper). Green solid line: η XCI estimated through the EGN-model (Eq. (20)).

Fig. 4
Fig. 4

Integration regions in the [ f 1 , f 2 ] plane needed to obtain the power spectrum of NLI for f = 0, due to two adjacent INT channels with spacing slightly higher than the symbol rate. The full XCI formula of Eq. (20) accounts for all X1-X4 regions. The XPM approximation [24] (Eq. (22) here) considers the X1 regions only. SCI is the center region S. MCI is the red/pink regions. The M0 region has only the GN-model term, the red M1 ones have both the GN-model term and non-Gaussianity correction terms.

Fig. 5
Fig. 5

Integration regions in the [ f 1 , f 2 ] plane needed to obtain the power spectrum of NLI for f= R s /2 , due to two adjacent INT channels with spacing slightly higher than the symbol rate. Notice that all regions change shape vs. Figure 4. Also, the maximum FWM efficiency now falls on the translated red-dashed axes, which do not coincide with the [ f 1 , f 2 ] axes. The lower M0 and M1 MCI regions are now close to such maxima.

Fig. 6
Fig. 6

Plot of normalized non-linearity coefficient η vs. number of spans in the link, assuming three PM-QPSK channels over (from top to bottom) SMF, NZDSF and LS, with span length 100 km. The CUT is the center channel. The spacing is 1.05 times the symbol rate. Red dashed line: simulation, with single-channel non-linearity (SCI) removed. Blue solid line: GN-model without SCI. Magenta solid line: the XPM approximation η XPM of [24] (Eq. (22) of this paper). Green solid line: η XMCI (i.e., XCI + MCI) estimated through the EGN-model (Eq. (28)).

Fig. 7
Fig. 7

Integration regions in the [ f 1 , f 2 ] plane needed to obtain the power spectrum of NLI for f = 0, for a nine-channel WDM system with four left and four right INT channels adjacent to the CUT, with spacing slightly higher than the symbol rate. SCI is the center region. XCI and MCI regions are color-coded (see legend). The white-filled regions (all of type M0) have only the GN-model term, all others have both the GN-model term and one or more non-Gaussianity correction terms. Note that XPM amounts to the X1 regions only.

Fig. 8
Fig. 8

Plot of normalized non-linearity coefficient η vs. number of spans in the link, assuming nine PM-QPSK channels over (from top to bottom) SMF, NZDSF and LS, with span length 100 km. The CUT is the center channel. The spacing is 1.05 times the symbol rate. Red dashed line: simulation, with single-channel non-linearity (SCI) removed. Blue solid line: GN-model without SCI. Magenta solid line: the XPM approximation η XPM of [24] (Eq. (22)). Green solid line: η XMCI (i.e., XCI + MCI) estimated through the EGN-model (Eq. (28)).

Fig. 9
Fig. 9

Plot of maximum system reach for 15-channel PM-QPSK and PM-16QAM systems at 32 GBaud, vs. channel spacing, over four different fiber types: PSCF, SMF, NZDSF and LS. The span length is 120 km for PM-QPSK and 85 km for PM-16QAM. Small filled circles: analytical predictions. Square hollow markers: simulations. Lines were added to connect analytical points as a visual aid. Dashed line: GN-model. Solid line: EGN-model.

Tables (1)

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Table 1 Values of Φ a and Ψ a

Equations (45)

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G NLI EGN ( f )= G NLI GN ( f )+ G NLI corr ( f )
G NLI EGN ( f )= G SCI EGN ( f )+ G XCI EGN ( f )+ G MCI EGN ( f )
S CUT (t)= n ( a x,n x ^ + a y,n y ^ ) s CUT ( tn T s )
P CUT =E{ | a x | 2 + | a y | 2 }, P INT =E{ | b x | 2 + | b y | 2 }
G SCI EGN (f)= P SCI 3 [ κ 1 (f)+ Φ a κ 2 (f)+ Ψ a κ 3 (f) ]
Φ a = Ε{ | a | 4 } Ε 2 { | a | 2 } 2, Ψ a = Ε{ | a | 6 } Ε 3 { | a | 2 } 9 Ε{ | a | 4 } Ε 2 { | a | 2 } +12
κ 1 (f)= 16 27 R s 3 R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 | s CUT ( f 1 ) | 2 | s CUT ( f 2 ) | 2 | s CUT ( f 1 + f 2 f) | 2 | μ( f 1 , f 2 ,f ) | 2
κ 2 (f)= 80 81 R s 2 R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 R s /2 + R s /2 d f 2 | s CUT ( f 1 ) | 2 s CUT ( f 2 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f) s CUT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f ) + 16 81 R s 2 R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 R s /2 + R s /2 d f 2 | s CUT ( f 1 + f 2 f) | 2 s CUT ( f 1 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f 2 ) s CUT ( f 2 )μ( f 1 , f 2 ,f ) μ ( f 1 + f 2 f 2 , f 2 ,f )
κ 3 (f)= 16 81 R s R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 s CUT ( f 1 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f) s CUT ( f 1 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f )
μ( f 1 , f 2 ,f )=ζ( f 1 , f 2 ,f )ν( f 1 , f 2 ,f )
ζ( f 1 , f 2 ,f )=γ 1 e 2α L s e j4 π 2 β 2 ( f 1 f)( f 2 f) L s 2αj4 π 2 β 2 ( f 1 f)( f 2 f)
ν( f 1 , f 2 ,f )= sin( 2 β 2 π 2 ( f 1 f)( f 2 f) N s L s ) sin( 2 β 2 π 2 ( f 1 f)( f 2 f) L s ) e j2 β 2 π 2 ( f 1 f)( f 2 f)( N s 1) L s
η SCI = P CUT 3 R s /2 R s /2 G SCI EGN ( f )df
G XPM (f)= P CUT P INT 2 [ κ 11 (f)+ Φ b κ 12 (f) ]
Φ b = Ε{ | b | 4 } Ε 2 { | b | 2 } 2
κ 11 (f)= 32 27 R s 3 R s /2 + R s /2 d f 1 f c R s /2 f c + R s /2 d f 2 | s CUT ( f 1 ) | 2 | s INT ( f 2 ) | 2 | s INT ( f 1 + f 2 f) | 2 | μ( f 1 , f 2 ,f ) | 2
κ 12 (f)= 80 81 R s 2 R s /2 + R s /2 d f 1 f c R s /2 f c + R s /2 d f 2 f c R s /2 f c + R s /2 d f 2 | s CUT ( f 1 ) | 2 s INT ( f 2 ) s INT ( f 2 ) s INT ( f 1 + f 2 f) s INT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f )
G XCI EGN (f)= P CUT P INT 2 [ κ 11 (f)+ Φ b κ 12 (f) ]+ P CUT 2 P INT [ κ 21 (f)+ Φ a κ 22 (f) ]+ P CUT 2 P INT [ κ 31 (f)+ Φ a κ 32 (f) ]+ P INT 3 [ κ 41 (f)+ Φ b κ 42 (f)+ Ψ b κ 43 (f) ]
Φ a = Ε{ | a | 4 } Ε 2 { | a | 2 } 2, Φ b = Ε{ | b | 4 } Ε 2 { | b | 2 } 2, Ψ b = Ε{ | b | 6 } Ε 3 { | b | 2 } 9 Ε{ | b | 4 } Ε 2 { | b | 2 } +12
η XCI = P ch 3 R s /2 R s /2 G XCI EGN ( f ) df
P ch = P INT = P CUT
η XPM = P ch 3 R s /2 R s /2 G XPM ( f )df
G MCI EGN (f)= P CUT P INT,1 P INT,-1 κ M0 (f)+ P INT,1 2 P INT,-1 [ κ M1,1 (f)+ Φ b κ M1,2 (f) ]
κ M0 (f)=2 16 27 R s 3 f c R s /2 f c + R s /2 d f 1 f c R s /2 f c + R s /2 d f 2 | s INT,1 ( f 1 ) | 2 | s INT,-1 ( f 2 ) | 2 | s CUT ( f 1 + f 2 f) | 2 | μ( f 1 , f 2 ,f ) | 2
κ M1,1 (f)=4 16 27 R s 3 f c R s /2 f c + R s /2 d f 1 f c R s /2 f c + R s /2 d f 2 | s INT,-1 ( f 1 ) | 2 | s INT,1 ( f 2 ) | 2 | s INT,1 ( f 1 + f 2 f) | 2 | μ( f 1 , f 2 ,f ) | 2
κ M1,2 (f)=2 80 81 R s 2 f c R s /2 f c + R s /2 d f 1 f c R s /2 f c + R s /2 d f 2 f c R s /2 f c + R s /2 d f 2 | s INT,-1 ( f 1 ) | 2 s INT,1 ( f 2 ) s INT,1 ( f 2 ) s INT,1 ( f 1 + f 2 f) s INT,1 ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f )
η MCI = P ch 3 R s /2 R s /2 G MCI EGN (f)df
η XMCI = η XCI + η MCI
κ 21 (f)= 32 27 R s 3 f c R s /2 f c + R s /2 d f 1 R s /2 + R s /2 d f 2 | s CUT ( f 2 ) | 2 | s CUT ( f 1 + f 2 f) | 2 | s INT ( f 1 ) | 2 | μ( f 1 , f 2 ,f ) | 2
κ 22 (f)= 80 81 R s 2 f c R s /2 f c + R s /2 d f 1 R s /2 + R s /2 d f 2 R s /2 + R s /2 d f 2 | s INT ( f 1 ) | 2 s CUT ( f 2 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f) s CUT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f )
κ 31 (f)= 16 27 R s 3 R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 | s CUT ( f 1 ) | 2 | s CUT ( f 2 ) | 2 | s INT ( f 1 + f 2 f) | 2 | μ( f 1 , f 2 ,f ) | 2
κ 32 (f)= 16 81 R s 2 R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 R s /2 + R s /2 d f 2 | s INT ( f 1 + f 2 f) | 2 s CUT ( f 1 ) s CUT ( f 2 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f 2 )μ( f 1 , f 2 ,f ) μ ( f 1 + f 2 f 2 , f 2 ,f )
κ 41 (f)= 16 27 R s 3 f c R s /2 f c + R s /2 d f 1 f c R s /2 f c + R s /2 d f 2 | s INT ( f 1 ) | 2 | s INT ( f 2 ) | 2 | s INT ( f 1 + f 2 f) | 2 | μ( f 1 , f 2 ,f ) | 2
κ 42 (f)= 80 81 R s 2 f c R s /2 f c + R s /2 d f 1 f c R s /2 f c + R s /2 d f 2 f c R s /2 f c + R s /2 d f 2 | s INT ( f 1 ) | 2 s INT ( f 2 ) s INT ( f 2 ) s INT ( f 1 + f 2 f ) s INT ( f 1 + f 2 f )μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f ) + 16 81 R s 2 f c R s /2 f c + R s /2 d f 1 f c R s /2 f c + R s /2 d f 2 f c R s /2 f c + R s /2 d f 2 | s INT ( f 1 + f 2 f ) | 2 s INT ( f 1 ) s INT ( f 2 ) s INT ( f 1 + f 2 f 2 ) s INT ( f 2 )μ( f 1 , f 2 ,f ) μ ( f 1 + f 2 f 2 , f 2 ,f )
κ 43 (f)= 16 81 R s f c R s /2 f c + R s /2 d f 1 f c R s /2 f c + R s /2 d f 2 f c R s /2 f c + R s /2 d f 1 f c R s /2 f c + R s /2 d f 2 s INT ( f 1 ) s INT ( f 2 ) s INT ( f 1 + f 2 f) s INT ( f 1 ) s INT ( f 2 ) s INT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f )
G MCI EGN (f)= G MCI GN (f)+ G MCI corr
P CUT = P INT,i = P ch i= ( N ch 1 ) /2 ,,1,1,, ( N ch 1 ) /2
G MCI corr = Φ b P ch 3 ( κ M1,2 (f)+ κ M2,2 (f)+ κ M3,2 (f) )
κ M1,2 (f)=2 80 81 R s 2 f c R s /2 f c + R s /2 d f 1 n f c R s /2 n f c + R s /2 d f 2 n f c R s /2 n f c + R s /2 d f 2 | s INT -1 ( f 1 ) | 2 s INT n ( f 2 ) s INT n ( f 2 ) s INT n ( f 1 + f 2 f) s INT n ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f ) ( withn=1,2,, ( N ch 1 ) /2 )
κ M2,2 (f)=2 80 81 R s 2 f c R s /2 f c + R s /2 d f 1 n f c R s /2 n f c + R s /2 d f 2 n f c R s /2 n f c + R s /2 d f 2 | s INT 1 ( f 1 ) | 2 s INT n ( f 2 ) s INT n ( f 2 ) s INT n ( f 1 + f 2 f) s INT n ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f ) ( withn=2,3,, ( N ch 1 ) /2 )
κ M3,2 (f)=2 16 81 R s 2 m f c R s /2 m f c + R s /2 d f 1 m f c R s /2 m f c + R s /2 d f 2 m f c R s /2 m f c + R s /2 d f 2 | s INT n ( f 1 + f 2 f) | 2 s INT m ( f 1 ) s INT m ( f 2 ) s INT m ( f 2 ) s INT m ( f 1 + f 2 f 2 )μ( f 1 , f 2 ,f ) μ ( f 1 + f 2 f 2 , f 2 ,f ) ( with{ n=2,3,, ( N ch 1 ) /2 m={ n/ 2,n iseven ( n±1 ) / 2,n isodd )
κ 2 (f)= 80 81 R s 2 R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 R s /2 + R s /2 d f 2 | s CUT ( f 1 ) | 2 s CUT ( f 2 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f) s CUT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f ) + 16 81 R s 2 R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 R s /2 + R s /2 d f 2 | s CUT ( f 1 + f 2 f) | 2 s CUT ( f 1 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f 2 ) s CUT ( f 2 )μ( f 1 , f 2 ,f ) μ ( f 1 + f 2 f 2 , f 2 ,f )
κ 21 (f)= 80 81 R s 2 R s /2 + R s /2 d f 1 | s CUT ( f 1 ) | 2 R s /2 + R s /2 d f 2 s CUT ( f 2 ) s CUT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) R s /2 + R s /2 d f 2 s CUT ( f 2 ) s CUT ( f 1 + f 2 f) μ ( f 1 , f 2 ,f ) = 80 81 R s 2 R s /2 + R s /2 d f 1 | s CUT ( f 1 ) | 2 | R s /2 + R s /2 d f 2 s CUT ( f 2 ) s CUT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) | 2
κ 22 (f)= 16 81 R s 2 R s /2 + R s /2 d f 3 R s /2 + R s /2 d f 2 R s /2 + R s /2 d f 2 | s CUT ( f 3 ) | 2 s CUT ( f 3 f 2 +f) s CUT ( f 2 ) s CUT ( f 3 f 2 +f) s CUT ( f 2 )μ( f 3 f 2 +f, f 2 ,f ) μ ( f 3 f 2 +f, f 2 ,f ) = 16 81 R s 2 R s /2 + R s /2 d f 3 | s CUT ( f 3 ) | 2 R s /2 + R s /2 d f 2 s CUT ( f 2 ) s CUT ( f 3 f 2 +f)μ( f 3 f 2 +f, f 2 ,f ) R s /2 + R s /2 d f 2 s CUT ( f 2 ) s CUT ( f 3 f 2 +f) μ ( f 3 f 2 +f, f 2 ,f ) = 16 81 R s 2 R s /2 + R s /2 d f 3 | s CUT ( f 3 ) | 2 | R s /2 + R s /2 d f 2 s CUT ( f 2 ) s CUT ( f 3 f 2 +f)μ( f 3 f 2 +f, f 2 ,f ) | 2
κ 3 (f)= 16 81 R s R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 s CUT ( f 1 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f) s CUT ( f 1 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) μ ( f 1 , f 2 ,f ) = 16 81 R s R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 s CUT ( f 1 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 s CUT ( f 1 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f) μ ( f 1 , f 2 ,f ) = 16 81 R s | R s /2 + R s /2 d f 1 R s /2 + R s /2 d f 2 s CUT ( f 1 ) s CUT ( f 2 ) s CUT ( f 1 + f 2 f)μ( f 1 , f 2 ,f ) | 2

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