Abstract

We study the properties of nonlinear interference noise (NLIN) in fiber-optic communications systems with large accumulated dispersion. Our focus is on settling the discrepancy between the results of the Gaussian noise (GN) model (according to which NLIN is additive Gaussian) and a recently published time-domain analysis, which attributes drastically different properties to the NLIN. Upon reviewing the two approaches we identify several unjustified assumptions that are key in the derivation of the GN model, and that are responsible for the discrepancy. We derive the true NLIN power and verify that the NLIN is not additive Gaussian, but rather it depends strongly on the data transmitted in the channel of interest. In addition we validate the time-domain model numerically and demonstrate the strong dependence of the NLIN on the interfering channels’ modulation format.

© 2013 Optical Society of America

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  1. A.  Mecozzi, C. B.  Clausen, M.  Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394(2000).
    [CrossRef]
  2. F.  Forghieri, R.W.  Tkack, A. R.  Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” Ch. 8 in Optical Fiber Telecommunications IIIA, P.  Kaminow, T. L.  Koch, eds. (Academic Press, 1997).
  3. E.  Ip, J. M.  Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008).
    [CrossRef]
  4. R. I.  Killey, P. M.  Watts, V.  Mikhailov, M.  Glick, P.  Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett., 17, 714–716(2005).
    [CrossRef]
  5. A.  Splett, C.  Kurtzke, K.  Petermann, “Ultimate transmission capacity of amplified fiber communication systems taking into account fiber nonlinearities,” in 19th European Conference on Optical Communication (ECOC), ECOC Technical Digest, (1993), Paper MoC2.4.
  6. P.  Poggiolini, A.  Carena, V.  Curri, G.  Bosco, F.  Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett., 23, 742–744, (2011).
    [CrossRef]
  7. A.  Carena, V.  Curri, G.  Bosco, P.  Poggiolini, F.  Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).
    [CrossRef]
  8. P.  Poggiolini, G.  Bosco, A.  Carena, V.  Curri, F.  Forghieri, “A detailed analytical derivation of the GN model of non-linear interference in coherent optical transmission systems,” July2, 2012[Online]. Available: arXiv:1209.0394v12 [physics.optics].
  9. P.  Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. of Light-wave Technol. 30, 3857–3879, (2012).
    [CrossRef]
  10. A.  Carena, G.  Bosco, V.  Curri, P.  Poggiolini, M.  Tapia Taiba, F.  Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC, 2010, Paper P4.07.
  11. E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.
  12. P.  Johannisson, M.  Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” IEEE J. of Lightwave Technol. 31, 1273–1282 (2013).
    [CrossRef]
  13. A.  Bononi, P.  Serena, “An alternative derivation of Johannissons regular perturbation model,” July19, 2012[Online]. Available: arXiv:1207.4729v1 [physics.optics].
  14. G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
    [CrossRef]
  15. A.  Bononi, N.  Rossi, P.  Serena, “Nonlinear threshold decrease with distance in 112 Gb/s PDM-QPSK coherent systems,” Proceeding of the European Conf. on Opt. Comm. (ECOC), Paper We.2.C.4, Amsterdam (2012)
  16. S. J.  Savory, “Approximations for the nonlinear self-channel interference of channels with rectangular spectra,” IEEE Photon. Technol. Lett. 25, 961–964, (2013)
    [CrossRef]
  17. A.  Mecozzi, R.  Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012).
    [CrossRef]
  18. X.  Liu, A. R.  Chraplyvy, P. J.  Winzer, R. W.  Tkach, S.  Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013).
    [CrossRef]
  19. R.  Dar, M.  Shtaif, M.  Feder, “New bounds on the capacity of fiber-optics communications,” arXiv:1305.1762v1 [physics.optics].
  20. R.  Dar, M.  Shtaif, M.  Feder, “Improved bounds on the nonlinear fiber-channel capacity,” in Proc. ECOC, 2013, paper P.4.16.
  21. S.  Kumar, D.  Yang, “Second-order theory for self-phase modulation andcross-phase modulation in optical fibers,” J. of Lightwave Technol. 23, 2073–2080, (2005)
    [CrossRef]
  22. We have assumed a fixed position independent dispersion parameter β″. The generalization to position dependent dispersion is straightforward, but we avoid it as it considerably complicates the notation.
  23. Although terms involving X0,k,mwith m≠ k also contribute to phase noise, they are much smaller than the terms proportional to X0,m,m(particularly in the limit of large dispersion) and we omit them from the analysis presented in this paper.
  24. X.  Zhou, “Hardware efficient carrier recovery algorithms for single-carrier QAM systems,” in Advanced Photonics Congress, OSA Technical Digest (online)(Optical Society of America, 2012), paper SpTu3A.1.
    [CrossRef]
  25. A.  Tolmachev, I.  Tselniker, M.  Meltsin, I.  Sigron, M.  Nazarathy, “Efficient multiplier-free fpga demonstration of polar-domain multi-symbol-delay-detector (MSDD) for high performance phase recovery of 16-QAM,” Proceedings of Optical Fiber Comm. Conf. OFC 2012, Paper OMC8
  26. A.  Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13(1994).
    [CrossRef]
  27. R.-J.  Essiambre, G.  Kramer, P.J.  Winzer, G.J.  Foschini, B.  Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701, (2010).
    [CrossRef]
  28. S.  Asmussen, P.W.  Glynn, Stochastic Simulation, Algorithms and Analysis (Springer, 2007).
  29. The numerical curves in Figs. 3 and 4 reproduce the plots reported in [20], but with larger statsitics.

2013 (3)

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

X.  Liu, A. R.  Chraplyvy, P. J.  Winzer, R. W.  Tkach, S.  Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013).
[CrossRef]

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

2012 (4)

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

P.  Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. of Light-wave Technol. 30, 3857–3879, (2012).
[CrossRef]

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

A.  Mecozzi, R.  Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012).
[CrossRef]

2011 (1)

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

2010 (1)

2008 (1)

E.  Ip, J. M.  Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008).
[CrossRef]

2005 (2)

R. I.  Killey, P. M.  Watts, V.  Mikhailov, M.  Glick, P.  Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett., 17, 714–716(2005).
[CrossRef]

S.  Kumar, D.  Yang, “Second-order theory for self-phase modulation andcross-phase modulation in optical fibers,” J. of Lightwave Technol. 23, 2073–2080, (2005)
[CrossRef]

2000 (1)

A.  Mecozzi, C. B.  Clausen, M.  Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394(2000).
[CrossRef]

1994 (1)

A.  Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13(1994).
[CrossRef]

Asmussen, S.

S.  Asmussen, P.W.  Glynn, Stochastic Simulation, Algorithms and Analysis (Springer, 2007).

Bayvel, P.

R. I.  Killey, P. M.  Watts, V.  Mikhailov, M.  Glick, P.  Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett., 17, 714–716(2005).
[CrossRef]

Bononi, A.

A.  Bononi, N.  Rossi, P.  Serena, “Nonlinear threshold decrease with distance in 112 Gb/s PDM-QPSK coherent systems,” Proceeding of the European Conf. on Opt. Comm. (ECOC), Paper We.2.C.4, Amsterdam (2012)

A.  Bononi, P.  Serena, “An alternative derivation of Johannissons regular perturbation model,” July19, 2012[Online]. Available: arXiv:1207.4729v1 [physics.optics].

Bosco, G.

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

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

P.  Poggiolini, A.  Carena, V.  Curri, G.  Bosco, 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, M.  Tapia Taiba, F.  Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC, 2010, Paper P4.07.

P.  Poggiolini, G.  Bosco, A.  Carena, V.  Curri, F.  Forghieri, “A detailed analytical derivation of the GN model of non-linear interference in coherent optical transmission systems,” July2, 2012[Online]. Available: arXiv:1209.0394v12 [physics.optics].

E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.

Carena, A.

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

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

P.  Poggiolini, A.  Carena, V.  Curri, G.  Bosco, 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, M.  Tapia Taiba, F.  Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC, 2010, Paper P4.07.

P.  Poggiolini, G.  Bosco, A.  Carena, V.  Curri, F.  Forghieri, “A detailed analytical derivation of the GN model of non-linear interference in coherent optical transmission systems,” July2, 2012[Online]. Available: arXiv:1209.0394v12 [physics.optics].

E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.

Chandrasekhar, S.

X.  Liu, A. R.  Chraplyvy, P. J.  Winzer, R. W.  Tkach, S.  Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013).
[CrossRef]

Chraplyvy, A. R.

X.  Liu, A. R.  Chraplyvy, P. J.  Winzer, R. W.  Tkach, S.  Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013).
[CrossRef]

F.  Forghieri, R.W.  Tkack, A. R.  Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” Ch. 8 in Optical Fiber Telecommunications IIIA, P.  Kaminow, T. L.  Koch, eds. (Academic Press, 1997).

Cigliutti, R.

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.

Clausen, C. B.

A.  Mecozzi, C. B.  Clausen, M.  Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394(2000).
[CrossRef]

Curri, V.

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

A.  Carena, V.  Curri, G.  Bosco, P.  Poggiolini, 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, 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, M.  Tapia Taiba, F.  Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC, 2010, Paper P4.07.

P.  Poggiolini, G.  Bosco, A.  Carena, V.  Curri, F.  Forghieri, “A detailed analytical derivation of the GN model of non-linear interference in coherent optical transmission systems,” July2, 2012[Online]. Available: arXiv:1209.0394v12 [physics.optics].

E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.

Dar, R.

R.  Dar, M.  Shtaif, M.  Feder, “Improved bounds on the nonlinear fiber-channel capacity,” in Proc. ECOC, 2013, paper P.4.16.

R.  Dar, M.  Shtaif, M.  Feder, “New bounds on the capacity of fiber-optics communications,” arXiv:1305.1762v1 [physics.optics].

Essiambre, R.

Essiambre, R.-J.

Feder, M.

R.  Dar, M.  Shtaif, M.  Feder, “Improved bounds on the nonlinear fiber-channel capacity,” in Proc. ECOC, 2013, paper P.4.16.

R.  Dar, M.  Shtaif, M.  Feder, “New bounds on the capacity of fiber-optics communications,” arXiv:1305.1762v1 [physics.optics].

Forghieri, F.

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

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

P.  Poggiolini, A.  Carena, V.  Curri, G.  Bosco, 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, M.  Tapia Taiba, F.  Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC, 2010, Paper P4.07.

F.  Forghieri, R.W.  Tkack, A. R.  Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” Ch. 8 in Optical Fiber Telecommunications IIIA, P.  Kaminow, T. L.  Koch, eds. (Academic Press, 1997).

P.  Poggiolini, G.  Bosco, A.  Carena, V.  Curri, F.  Forghieri, “A detailed analytical derivation of the GN model of non-linear interference in coherent optical transmission systems,” July2, 2012[Online]. Available: arXiv:1209.0394v12 [physics.optics].

E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.

Foschini, G.J.

Glick, M.

R. I.  Killey, P. M.  Watts, V.  Mikhailov, M.  Glick, P.  Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett., 17, 714–716(2005).
[CrossRef]

Glynn, P.W.

S.  Asmussen, P.W.  Glynn, Stochastic Simulation, Algorithms and Analysis (Springer, 2007).

Goebel, B.

Ip, E.

E.  Ip, J. M.  Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008).
[CrossRef]

Jiang, Y.

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

Johannisson, P.

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

Kahn, J. M.

E.  Ip, J. M.  Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008).
[CrossRef]

Karlsson, M.

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

Killey, R. I.

R. I.  Killey, P. M.  Watts, V.  Mikhailov, M.  Glick, P.  Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett., 17, 714–716(2005).
[CrossRef]

Kramer, G.

Kumar, S.

S.  Kumar, D.  Yang, “Second-order theory for self-phase modulation andcross-phase modulation in optical fibers,” J. of Lightwave Technol. 23, 2073–2080, (2005)
[CrossRef]

Kurtzke, C.

A.  Splett, C.  Kurtzke, K.  Petermann, “Ultimate transmission capacity of amplified fiber communication systems taking into account fiber nonlinearities,” in 19th European Conference on Optical Communication (ECOC), ECOC Technical Digest, (1993), Paper MoC2.4.

Liu, X.

X.  Liu, A. R.  Chraplyvy, P. J.  Winzer, R. W.  Tkach, S.  Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013).
[CrossRef]

Mecozzi, A.

A.  Mecozzi, R.  Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012).
[CrossRef]

A.  Mecozzi, C. B.  Clausen, M.  Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394(2000).
[CrossRef]

Meltsin, M.

A.  Tolmachev, I.  Tselniker, M.  Meltsin, I.  Sigron, M.  Nazarathy, “Efficient multiplier-free fpga demonstration of polar-domain multi-symbol-delay-detector (MSDD) for high performance phase recovery of 16-QAM,” Proceedings of Optical Fiber Comm. Conf. OFC 2012, Paper OMC8

Mikhailov, V.

R. I.  Killey, P. M.  Watts, V.  Mikhailov, M.  Glick, P.  Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett., 17, 714–716(2005).
[CrossRef]

Nazarathy, M.

A.  Tolmachev, I.  Tselniker, M.  Meltsin, I.  Sigron, M.  Nazarathy, “Efficient multiplier-free fpga demonstration of polar-domain multi-symbol-delay-detector (MSDD) for high performance phase recovery of 16-QAM,” Proceedings of Optical Fiber Comm. Conf. OFC 2012, Paper OMC8

Nespola, A.

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.

Papoulis, A.

A.  Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13(1994).
[CrossRef]

Petermann, K.

A.  Splett, C.  Kurtzke, K.  Petermann, “Ultimate transmission capacity of amplified fiber communication systems taking into account fiber nonlinearities,” in 19th European Conference on Optical Communication (ECOC), ECOC Technical Digest, (1993), Paper MoC2.4.

Poggiolini, P.

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

A.  Carena, V.  Curri, G.  Bosco, P.  Poggiolini, 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, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. of Light-wave Technol. 30, 3857–3879, (2012).
[CrossRef]

P.  Poggiolini, A.  Carena, V.  Curri, G.  Bosco, 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, M.  Tapia Taiba, F.  Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC, 2010, Paper P4.07.

P.  Poggiolini, G.  Bosco, A.  Carena, V.  Curri, F.  Forghieri, “A detailed analytical derivation of the GN model of non-linear interference in coherent optical transmission systems,” July2, 2012[Online]. Available: arXiv:1209.0394v12 [physics.optics].

E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.

Rossi, N.

A.  Bononi, N.  Rossi, P.  Serena, “Nonlinear threshold decrease with distance in 112 Gb/s PDM-QPSK coherent systems,” Proceeding of the European Conf. on Opt. Comm. (ECOC), Paper We.2.C.4, Amsterdam (2012)

Sasaki, T.

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

Savory, S. J.

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

Serena, P.

A.  Bononi, N.  Rossi, P.  Serena, “Nonlinear threshold decrease with distance in 112 Gb/s PDM-QPSK coherent systems,” Proceeding of the European Conf. on Opt. Comm. (ECOC), Paper We.2.C.4, Amsterdam (2012)

A.  Bononi, P.  Serena, “An alternative derivation of Johannissons regular perturbation model,” July19, 2012[Online]. Available: arXiv:1207.4729v1 [physics.optics].

Shtaif, M.

A.  Mecozzi, C. B.  Clausen, M.  Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394(2000).
[CrossRef]

R.  Dar, M.  Shtaif, M.  Feder, “Improved bounds on the nonlinear fiber-channel capacity,” in Proc. ECOC, 2013, paper P.4.16.

R.  Dar, M.  Shtaif, M.  Feder, “New bounds on the capacity of fiber-optics communications,” arXiv:1305.1762v1 [physics.optics].

Sigron, I.

A.  Tolmachev, I.  Tselniker, M.  Meltsin, I.  Sigron, M.  Nazarathy, “Efficient multiplier-free fpga demonstration of polar-domain multi-symbol-delay-detector (MSDD) for high performance phase recovery of 16-QAM,” Proceedings of Optical Fiber Comm. Conf. OFC 2012, Paper OMC8

Splett, A.

A.  Splett, C.  Kurtzke, K.  Petermann, “Ultimate transmission capacity of amplified fiber communication systems taking into account fiber nonlinearities,” in 19th European Conference on Optical Communication (ECOC), ECOC Technical Digest, (1993), Paper MoC2.4.

Tapia Taiba, M.

A.  Carena, G.  Bosco, V.  Curri, P.  Poggiolini, M.  Tapia Taiba, F.  Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC, 2010, Paper P4.07.

Tkach, R. W.

X.  Liu, A. R.  Chraplyvy, P. J.  Winzer, R. W.  Tkach, S.  Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013).
[CrossRef]

Tkack, R.W.

F.  Forghieri, R.W.  Tkack, A. R.  Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” Ch. 8 in Optical Fiber Telecommunications IIIA, P.  Kaminow, T. L.  Koch, eds. (Academic Press, 1997).

Tolmachev, A.

A.  Tolmachev, I.  Tselniker, M.  Meltsin, I.  Sigron, M.  Nazarathy, “Efficient multiplier-free fpga demonstration of polar-domain multi-symbol-delay-detector (MSDD) for high performance phase recovery of 16-QAM,” Proceedings of Optical Fiber Comm. Conf. OFC 2012, Paper OMC8

Torrengo, E.

E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.

Tselniker, I.

A.  Tolmachev, I.  Tselniker, M.  Meltsin, I.  Sigron, M.  Nazarathy, “Efficient multiplier-free fpga demonstration of polar-domain multi-symbol-delay-detector (MSDD) for high performance phase recovery of 16-QAM,” Proceedings of Optical Fiber Comm. Conf. OFC 2012, Paper OMC8

Watts, P. M.

R. I.  Killey, P. M.  Watts, V.  Mikhailov, M.  Glick, P.  Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett., 17, 714–716(2005).
[CrossRef]

Winzer, P. J.

X.  Liu, A. R.  Chraplyvy, P. J.  Winzer, R. W.  Tkach, S.  Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013).
[CrossRef]

Winzer, P.J.

Yamamoto, Y.

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

Yang, D.

S.  Kumar, D.  Yang, “Second-order theory for self-phase modulation andcross-phase modulation in optical fibers,” J. of Lightwave Technol. 23, 2073–2080, (2005)
[CrossRef]

Zeolla, D.

E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.

Zhou, X.

X.  Zhou, “Hardware efficient carrier recovery algorithms for single-carrier QAM systems,” in Advanced Photonics Congress, OSA Technical Digest (online)(Optical Society of America, 2012), paper SpTu3A.1.
[CrossRef]

IEEE Photon. Technol. Lett. (3)

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

G.  Bosco, R.  Cigliutti, A.  Nespola, A.  Carena, V.  Curri, F.  Forghieri, Y.  Yamamoto, T.  Sasaki, Y.  Jiang, P.  Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012)
[CrossRef]

A.  Mecozzi, C. B.  Clausen, M.  Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394(2000).
[CrossRef]

IEEE J. of Lightwave Technol. (1)

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

IEEE Photon. Technol. Lett. (1)

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

IEEE Photon. Technol. Lett. (1)

R. I.  Killey, P. M.  Watts, V.  Mikhailov, M.  Glick, P.  Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett., 17, 714–716(2005).
[CrossRef]

J. Lightwave Technol. (1)

E.  Ip, J. M.  Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008).
[CrossRef]

J. Lightwave Technol. (3)

J. of Light-wave Technol. (1)

P.  Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. of Light-wave Technol. 30, 3857–3879, (2012).
[CrossRef]

J. of Lightwave Technol. (1)

S.  Kumar, D.  Yang, “Second-order theory for self-phase modulation andcross-phase modulation in optical fibers,” J. of Lightwave Technol. 23, 2073–2080, (2005)
[CrossRef]

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

A.  Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13(1994).
[CrossRef]

Nature Photonics (1)

X.  Liu, A. R.  Chraplyvy, P. J.  Winzer, R. W.  Tkach, S.  Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013).
[CrossRef]

Other (15)

R.  Dar, M.  Shtaif, M.  Feder, “New bounds on the capacity of fiber-optics communications,” arXiv:1305.1762v1 [physics.optics].

R.  Dar, M.  Shtaif, M.  Feder, “Improved bounds on the nonlinear fiber-channel capacity,” in Proc. ECOC, 2013, paper P.4.16.

A.  Carena, G.  Bosco, V.  Curri, P.  Poggiolini, M.  Tapia Taiba, F.  Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC, 2010, Paper P4.07.

E.  Torrengo, R.  Cigliutti, G.  Bosco, A.  Carena, V.  Curri, P.  Poggiolini, A.  Nespola, D.  Zeolla, F.  Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.

A.  Bononi, P.  Serena, “An alternative derivation of Johannissons regular perturbation model,” July19, 2012[Online]. Available: arXiv:1207.4729v1 [physics.optics].

P.  Poggiolini, G.  Bosco, A.  Carena, V.  Curri, F.  Forghieri, “A detailed analytical derivation of the GN model of non-linear interference in coherent optical transmission systems,” July2, 2012[Online]. Available: arXiv:1209.0394v12 [physics.optics].

A.  Bononi, N.  Rossi, P.  Serena, “Nonlinear threshold decrease with distance in 112 Gb/s PDM-QPSK coherent systems,” Proceeding of the European Conf. on Opt. Comm. (ECOC), Paper We.2.C.4, Amsterdam (2012)

We have assumed a fixed position independent dispersion parameter β″. The generalization to position dependent dispersion is straightforward, but we avoid it as it considerably complicates the notation.

Although terms involving X0,k,mwith m≠ k also contribute to phase noise, they are much smaller than the terms proportional to X0,m,m(particularly in the limit of large dispersion) and we omit them from the analysis presented in this paper.

X.  Zhou, “Hardware efficient carrier recovery algorithms for single-carrier QAM systems,” in Advanced Photonics Congress, OSA Technical Digest (online)(Optical Society of America, 2012), paper SpTu3A.1.
[CrossRef]

A.  Tolmachev, I.  Tselniker, M.  Meltsin, I.  Sigron, M.  Nazarathy, “Efficient multiplier-free fpga demonstration of polar-domain multi-symbol-delay-detector (MSDD) for high performance phase recovery of 16-QAM,” Proceedings of Optical Fiber Comm. Conf. OFC 2012, Paper OMC8

S.  Asmussen, P.W.  Glynn, Stochastic Simulation, Algorithms and Analysis (Springer, 2007).

The numerical curves in Figs. 3 and 4 reproduce the plots reported in [20], but with larger statsitics.

A.  Splett, C.  Kurtzke, K.  Petermann, “Ultimate transmission capacity of amplified fiber communication systems taking into account fiber nonlinearities,” in 19th European Conference on Optical Communication (ECOC), ECOC Technical Digest, (1993), Paper MoC2.4.

F.  Forghieri, R.W.  Tkack, A. R.  Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” Ch. 8 in Optical Fiber Telecommunications IIIA, P.  Kaminow, T. L.  Koch, eds. (Academic Press, 1997).

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Figures (5)

Fig. 1
Fig. 1

Received constellations (after compensating for the average nonlinear phase-rotation) of the channel of interest after 500 km of fiber. The per-channel power was −2dBm. The channel of interest is QPSK-modulated in the left column and 16-QAM modulated in the right column. In the top panel (Figs. (a) and (b)) the modulation of the interfering channels is QPSK. In the middle panel (Figs. (c) and (d)) the modulation of the interfering channels is 16-QAM, and in the bottom panel (Figs. (e) and (f)) the modulation of the interfering channel is Gaussian. The dominance of phase noise is evident in the middle and bottom panels, whereas in the top panel phase-noise is negligible.

Fig. 2
Fig. 2

The electric field intensities of a single channel operating with Nyquist sinc-shaped pulses at a baud-rate of 100 GHz after being dispersed by 8500 ps/nm/km (equivalent to 500 km in standard fiber). The solid (black), dashed (blue), and dash-dotted (red) curves correspond to QPSK, 16-QAM and Gaussian modulation, respectively. In spite of the apparent similarity between the dispersed waveforms as demonstrated in this figure, the NLIN strongly depends on the modulation format.

Fig. 3
Fig. 3

The phase-noise autocorrelation function R θ (l) of the phase-noise Eq. (15)(dashed-red) and as obtained from the simulations (solid blue) for −6dBm per-channel average power [29 ].

Fig. 4
Fig. 4

The complete NLIN variance Δ a 0 2 normalized to the average symbol energy (top solid curve), the phase noise Δ θ 2as obtained from the simulations (center solid curve) and the variance of the residual noise Δ a r 2 normalized to the average symbol energy [29]. The dashed curve (red) shows the analytical result for Δ θ 2, Eq. (13). It is within 20% of the numerically obtained Δ θ 2.

Fig. 5
Fig. 5

(a) The SON coefficient χ 1(blue diamonds) and the FON coefficient χ 2(red circles) as a function of the spacing between channels. The coefficients in the figure are normalized by γ 2 L 2/ T 3and hence they are unit-less. The green squares show χ 1χ 2. (b) The NLIN power versus the average power per-channel for QPSK, 16-QAM and Gaussian modulation. The symbols show the results of a split-step-simulation performed with the same parameters as in Fig. 1. The solid lines represent Eq. (29), whereas the dashed green line shows the prediction of the GN model Eq. (28) . It is correct only with Gaussian modulation, but severely overestimates the actual noise in other formats.

Equations (30)

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u ( 0 ) ( z , t ) = k a k g ( 0 ) ( z , t k T ) + k b k e i Ω t + i β Ω 2 2 z g ( 0 ) ( z , t k T β Ω z ) ,
u ( 1 ) ( z , t ) z = i 2 β t 2 u ( 1 ) ( z , t ) + i γ f ( z ) | u ( 0 ) ( z , t ) | 2 u ( 0 ) ( z , t ) ,
u ( 1 ) ( L , t ) = i γ 0 L d z U ( L z ) f ( z ) | u ( 0 ) ( z , t ) | 2 u ( 0 ) ( z , t ) .
Δ a 0 = u ( 1 ) ( L , t ) g ( 0 ) * ( L , t ) d t = i γ 0 L d z f ( z ) d t g ( 0 ) * ( z , t ) | u ( 0 ) ( z , t ) | 2 u ( 0 ) ( z , t ) ,
Δ a 0 = i γ h , k , m ( a h a k * a m S h , k , m + 2 a h b k * b m X h , k , m ) .
S h , k , m = 0 L d z f ( z ) d t g ( 0 ) * ( z , t ) g ( 0 ) ( z , t h T ) g ( 0 ) * ( z , t k T ) g ( 0 ) ( z , t m T ) ,
X h , k , m = 0 L d z f ( z ) d t g ( 0 ) * ( z , t ) g ( 0 ) ( z , t h T ) × g ( 0 ) * ( z , t k T β Ω z ) g ( 0 ) ( z , t m T β Ω z ) ,
Δ a 0 p = i a 0 ( 2 γ m | b m | 2 X 0 , m , m ) = i a 0 θ ,
θ = 2 γ | b 0 | 2 m X 0 , m , m , and θ 2 = 4 γ 2 m , m | b m | 2 | b m | 2 X 0 , m , m X 0 , m , m
Δ θ 2 = θ 2 θ 2 = 4 γ 2 ( | b 0 | 4 | b 0 | 2 2 ) m X 0 , m , m 2 .
g ( 0 ) ( z , t ) i 2 π β z exp ( it 2 2 β z ) g ˜ ( 0 , t β z ) .
X 0 , m , m = z 0 L d z f ( z ) d ν 4 π 2 β z | g ˜ ( 0 , ν ) | 2 | g ˜ ( 0 , ν Ω m T β z ) | 2 .
X 0 , m , m = d ν 2 π | g ˜ ( 0 , ν ) | 2 d z z m f ( z m ) 2 π β z 2 | g ˜ ( 0 , ν Ω m T β z ) | 2 { 1 β Ω 0 m | β Ω | L T 0 otherwise .
Δ θ 2 = ( | b 0 | 4 | b 0 | 2 2 ) 4 γ 2 L | β Ω | T .
R θ ( l ) = 4 γ 2 m n | b m | 2 | b n + l | 2 X 0 , m , m X 0 , n , n * θ 2 = Δ θ 2 [ 1 | l | T | β Ω | L ] + ,
R θ ( l ) = s Δ θ 2 ( Ω s ) [ 1 | l | T | β Ω s | L ] + ,
u ( 0 ) ( z , t ) = 1 M T [ n ν n ( z ) e i 2 π M T n t + e i Ω t n ξ n ( z ) e i 2 π M T n t ] .
Δ u ( t ) = 2 l m n , m n ρ l m n ν l ξ m ξ n * ,
ρ l m n = γ ( M T ) 3 / 2 e i 2 π M T ( l + m n ) t × 1 e i ( 2 π M T ) 2 β N L s ( m n ) ( l q M n ) 1 e i ( 2 π M T ) 2 β L s ( m n ) ( l q M n ) 1 e α L s e i ( 2 π M T ) 2 β L s ( m n ) ( l q M n ) α i ( 2 π M T ) 2 β ( m n ) ( l q M n ) ,
| Δ u ( t ) | 2 = 4 l m n l m n ρ l m n ρ l m n * ν l ν l * ξ m ξ n * ξ m * ξ n ,
| Δ u | 2 = 4 l m n , m n | ρ l m n | 2 | ν l | 2 | ξ m | 2 | ξ n | 2 ,
ξ n = 1 M T 0 M T x ( t ) e i 2 π M T n t d t = g ˜ ( ω n ) 1 M T k = 0 M 1 b k e i 2 π M k n ,
ξ n ξ n * = | g ˜ ( ω n ) | 2 T | b 0 | 2 δ n n .
ξ m ξ n * ξ m * ξ n = | b 0 | 2 2 T 2 | g ˜ ( ω m ) | 2 | g ˜ ( ω n ) | 2 ( δ m n δ m n + δ m m δ n n ) + | b 0 | 4 2 | b 0 | 2 2 M T 2 𝒫 m n m n
𝒫 m n m n = g ˜ ( ω m ) g ˜ * ( ω n ) g ˜ * ( ω m ) g ˜ ( ω n ) × ( δ n n + m m M + δ n n + m m + δ n n + m m + M ) .
| Δ u | 2 = | a 0 | 2 | b 0 | 2 2 χ 1 + | a 0 | 2 ( | b 0 | 4 2 | b 0 | 2 2 ) χ 2
χ 1 = 4 T 3 l m n , m n | g ˜ ( ω l ) | 2 | g ˜ ( ω m ) | 2 | g ˜ ( ω n ) | 2 | ρ l m n | 2 ,
χ 2 = 4 M T 3 l m n m n | g ˜ ( ω l ) | 2 𝒫 m n m n ρ l m n ρ l m n *
| Δ u | 2 GN = P 3 s χ 1 ( Ω s )
| Δ u | 2 Full = P 3 s [ χ 1 ( Ω s ) χ 2 ( Ω s ) ] + P 3 ( | b 0 | 4 | b 0 | 2 2 1 ) s χ 2 ( Ω s ) ,

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