Ripple distribution for nonlinear fiber-optic channels

Mariia Sorokina; Stylianos Sygletos; Sergei Turitsyn

doi:10.1364/OE.25.002228

Ripple distribution for nonlinear fiber-optic channels

Mariia Sorokina, Stylianos Sygletos, and Sergei Turitsyn

Optics Express
Vol. 25,
Issue 3,
pp. 2228-2238
(2017)
•https://doi.org/10.1364/OE.25.002228

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Mariia Sorokina, Stylianos Sygletos, and Sergei Turitsyn, "Ripple distribution for nonlinear fiber-optic channels," Opt. Express 25, 2228-2238 (2017)

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Related Topics
Table of Contents Category
- Optical Communications
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fiber optics communications (060.2330)
- Information theoretical analysis (110.3055)
- Nonlinear optics, fibers (190.4370)

About this Article
History
- Original Manuscript: November 2, 2016
- Revised Manuscript: January 10, 2017
- Manuscript Accepted: January 15, 2017
- Published: January 27, 2017
Virtual Issues
Optics Express Nonlinearity Mitigation for Coherent Transmission Systems (2017)

Accessible

Open Access

Abstract

We demonstrate data rates above the threshold imposed by nonlinearity on conventional optical signals by applying novel probability distribution, which we call ripple distribution, adapted to the properties of the fiber channel. Our results offer a new direction for signal coding, modulation and practical nonlinear distortions compensation algorithms.

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References

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Publication

A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” Tech. Digest of European Conference on Optical Communication paper MoC2.4. (1993).
P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
[Crossref] [PubMed]
R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).
[Crossref] [PubMed]
R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]
E. E. Narimanov and P. Mitra, “The channel capacity of a fiber optics communication system: perturbation theory,” J. Lightwave Technol. 20(3), 530–537 (2002).
[Crossref]
D. J. Richardson, “Filling the light pipe,” Science 330, 327–328 (2010).
[Crossref] [PubMed]
E. Temprana, E. Myslivets, B.P.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[Crossref] [PubMed]
P. J. Winzer, “Scaling optical fiber networks: challenges and solutions,” Optics and Photonics News, 26, 28–35 (2015).
[Crossref]
C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).
[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]
I. B. Djordjevic, H. G. Batshon, L. Xu, and T. Wang, “Coded polarization-multiplexed iterative polar modulation (PM-IPM) for beyond 400 Gb/s serial optical transmission,” in Proc. Optical Fiber Communication Conference, Los Angeles, CA, Mar. 2010, p. OMK2
T. Fehenberger, A. Alvarado Segovia, G. Bocherer, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photon. Technol. Lett. 28(7) 786–789 (2016).
[Crossref]
C. Pan and F. R. Kschischang, “Probabilistic 16-QAM Shaping in WDM Systems,” Journal of Lightwave Technology (in press).
F. Buchali, G. Bocherer, W. Idler, L. Schmalen, P. Schulte, and F. Steiner, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: an experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016).
[Crossref]
E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightwave Technol. 32(16), 2862–2876 (2014).
[Crossref]
M. Sorokina, A. Ellis, and S. Turitsyn, “Optical information capacity processing,” chapter in All-Optical Signal Processing325–354 (2015).
E. Agrell, G. Durisi, and P. Johannisson, “Information-theory-friendly models for fiberoptic channels: A primer,” IEEE Information Theory Workshop (2015).
M. H. Taghavi, G. C. Papen, and P.H. Siegel, “On the multiuser capacity of WDM in a nonlinear optical fiber: coherent communication,” IEEE Trans. Inf. Theory, 52(11), 5008–5022 (2006).
[Crossref]
H. Song and M. Brandt-Pearce, “A 2-D discrete-time model of physical impairments in wavelength-division multiplexing systems,” J. Lightwave Technol. 30(5), 713–726 (2012).
[Crossref]
R. Dar, M. Shtaif, and M. Feder, “New bounds on the capacity of the nonlinear fiber-optic channel,” Optics Letters 39, 398–401 (2014).
[Crossref] [PubMed]
Z. Tao, Y. Zhao, Y. Fan, L. Dou, T. Hoshida, and J. C. Rasmussen, “Analytical intrachannel nonlinear models to predict the nonlinear noise waveform,” J. Lightwave Technol. 33(10), 2011–2018 (2015).
[Crossref]
M. Secondini and E. Forestieri, “Scope and limitations of the nonlinear Shannon limit,” online version of 24.10.2016 http://ieeexplore.ieee.org/document/7637002/ .
M. Secondini, chapter in “Roadmap of optical communications,” by E. Agrell et al. Journal of Optics 18 (6), 063002 (2016).
[Crossref]
M. Sorokina, S. Sygletos, and S. K. Turitsyn, “Shannon capacity of nonlinear communication channels,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2016), paper SM3F.4.
M. Sorokina, S. Sygletos, and S. K. Turitsyn, “Ripple distribution for nonlinear fiber-optic channels,” https://arxiv.org/abs/1610.06937
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]
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(23), 3839–3852 (2013).
[Crossref]
L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “A discrete-time model for uncompensated single-channel fiber-optical links,” IEEE Trans. Commun. 60(11), 3440–3450 (2012).
[Crossref]
R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]
R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Inter-channel nonlinear interference noise in WDM systems: modeling and mitigation,” J. Lightwave Technol. 33(5), 1044–1053 (2015).
[Crossref]
M. Shtaif, R. Dar, A. Mecozzi, and M. Feder, “Nonlinear interference noise in WDM systems and approaches for its cancelation,” in Optical Communication (ECOC 2014), 39th European Conference and Exhibition on Optical Communications, paper We1.3.1.
K. S. Turitsyn, S.A. Derevyanko, I.V. Yurkevich, and S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Letters 91, 203901 (2003).
[Crossref]
M. Secondini, E. Forestieri, and C. R. Menyuk, “A combined regular-logarithmic perturbation method for signal-noise interaction in amplified optical systems”, J. Lightwave Technol. 27(16), 3358–3369 (2009).
[Crossref]
K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers”, J. Lightwave Technol. 15(12), 2232–2241 (1997).
[Crossref]
M. Schetzen, “The Volterra and Wiener theories of nonlinear systems”. (Krieger, 2006).
A. Amari, P. Ciblat, and Y. Jaouen, “Fifth-order Volterra series based nonlinear equalizer for long-haul high data rate optical fiber communications,” Asilomar Conference ACSSC (2014).
M. Sorokina, S. Sygletos, and S. Turitsyn, “Sparse identification for nonlinear optical communication systems: SINO method,” Opt. Express 24(26), 30433–30443 (2016).
[Crossref]
Z. Li, W.-R. Peng, F. Zhu, and Y. Bai, “Optimum quantization of perturbation coefficients for perturbative fiber nonlinearity mitigation,” Tech. Digest of European Conference on Optical Communication paper We.1.3.4. (2014).
A. Ghazisaeidi and R.-J. Essiambre, “Calculation of coefficients of perturbative nonlinear pre-compensation for Nyquist pulses,” Tech. Digest of European Conference on Optical Communication paper We.1.3.3. (2014).
S. Verdu and T. S. Han, “A general formula for channel capacity,” IEEE Trans. Inf. Theory 40(4) 1147–1157 (1994).
[Crossref]
B. Picinbono, “Second-order complex random vectors and normal distributions,” IEEE Transactions on Signal Processing 442637–2640 (1996).
[Crossref]
E. Agrell and M. Karlsson, “Satellite constellations: towards the nonlinear channel capacity,” 25th IEEE Photonics Conference pp. 316–317 (2012).
R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “On shaping gain in the nonlinear fiber-optic channel,” IEEE International Symposium on Information Theory (ISIT), Honolulu, HI, USA, July 2014.
D. Rafique and A. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (2011).
[Crossref] [PubMed]
M. A. Sorokina and S. K. Turitsyn, “Regeneration limit of classical Shannon capacity,” Nat. Commun. 5, 3861 (2014).
[Crossref] [PubMed]
T. Freckmann, R. J. Essiambre, P. J. Winzer, G. J. Foschini, and G. Kramer, “Fiber capacity limits with optimized ring constellations,” IEEE Photonics Technology Letters 21(20), 1496–1498 (2009).
[Crossref]

2016 (4)

T. Fehenberger, A. Alvarado Segovia, G. Bocherer, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photon. Technol. Lett. 28(7) 786–789 (2016).
[Crossref]

M. Secondini, chapter in “Roadmap of optical communications,” by E. Agrell et al. Journal of Optics 18 (6), 063002 (2016).
[Crossref]

F. Buchali, G. Bocherer, W. Idler, L. Schmalen, P. Schulte, and F. Steiner, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: an experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016).
[Crossref]

M. Sorokina, S. Sygletos, and S. Turitsyn, “Sparse identification for nonlinear optical communication systems: SINO method,” Opt. Express 24(26), 30433–30443 (2016).
[Crossref]

2015 (4)

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Inter-channel nonlinear interference noise in WDM systems: modeling and mitigation,” J. Lightwave Technol. 33(5), 1044–1053 (2015).
[Crossref]

Z. Tao, Y. Zhao, Y. Fan, L. Dou, T. Hoshida, and J. C. Rasmussen, “Analytical intrachannel nonlinear models to predict the nonlinear noise waveform,” J. Lightwave Technol. 33(10), 2011–2018 (2015).
[Crossref]

E. Temprana, E. Myslivets, B.P.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[Crossref] [PubMed]

P. J. Winzer, “Scaling optical fiber networks: challenges and solutions,” Optics and Photonics News, 26, 28–35 (2015).
[Crossref]

2014 (3)

R. Dar, M. Shtaif, and M. Feder, “New bounds on the capacity of the nonlinear fiber-optic channel,” Optics Letters 39, 398–401 (2014).
[Crossref] [PubMed]

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightwave Technol. 32(16), 2862–2876 (2014).
[Crossref]

M. A. Sorokina and S. K. Turitsyn, “Regeneration limit of classical Shannon capacity,” Nat. Commun. 5, 3861 (2014).
[Crossref] [PubMed]

2013 (3)

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

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(23), 3839–3852 (2013).
[Crossref]

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]

2012 (2)

L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “A discrete-time model for uncompensated single-channel fiber-optical links,” IEEE Trans. Commun. 60(11), 3440–3450 (2012).
[Crossref]

H. Song and M. Brandt-Pearce, “A 2-D discrete-time model of physical impairments in wavelength-division multiplexing systems,” J. Lightwave Technol. 30(5), 713–726 (2012).
[Crossref]

2011 (2)

D. Rafique and A. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (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]

2010 (2)

D. J. Richardson, “Filling the light pipe,” Science 330, 327–328 (2010).
[Crossref] [PubMed]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

2009 (2)

M. Secondini, E. Forestieri, and C. R. Menyuk, “A combined regular-logarithmic perturbation method for signal-noise interaction in amplified optical systems”, J. Lightwave Technol. 27(16), 3358–3369 (2009).
[Crossref]

T. Freckmann, R. J. Essiambre, P. J. Winzer, G. J. Foschini, and G. Kramer, “Fiber capacity limits with optimized ring constellations,” IEEE Photonics Technology Letters 21(20), 1496–1498 (2009).
[Crossref]

2008 (1)

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).
[Crossref] [PubMed]

2006 (1)

M. H. Taghavi, G. C. Papen, and P.H. Siegel, “On the multiuser capacity of WDM in a nonlinear optical fiber: coherent communication,” IEEE Trans. Inf. Theory, 52(11), 5008–5022 (2006).
[Crossref]

2003 (1)

K. S. Turitsyn, S.A. Derevyanko, I.V. Yurkevich, and S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Letters 91, 203901 (2003).
[Crossref]

2002 (1)

E. E. Narimanov and P. Mitra, “The channel capacity of a fiber optics communication system: perturbation theory,” J. Lightwave Technol. 20(3), 530–537 (2002).
[Crossref]

2001 (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
[Crossref] [PubMed]

1997 (1)

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers”, J. Lightwave Technol. 15(12), 2232–2241 (1997).
[Crossref]

1996 (1)

B. Picinbono, “Second-order complex random vectors and normal distributions,” IEEE Transactions on Signal Processing 442637–2640 (1996).
[Crossref]

1994 (1)

S. Verdu and T. S. Han, “A general formula for channel capacity,” IEEE Trans. Inf. Theory 40(4) 1147–1157 (1994).
[Crossref]

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).
[Crossref]

Agrell, E.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightwave Technol. 32(16), 2862–2876 (2014).
[Crossref]

E. Agrell, G. Durisi, and P. Johannisson, “Information-theory-friendly models for fiberoptic channels: A primer,” IEEE Information Theory Workshop (2015).

E. Agrell and M. Karlsson, “Satellite constellations: towards the nonlinear channel capacity,” 25th IEEE Photonics Conference pp. 316–317 (2012).

Alic, N.

Alvarado, A.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightwave Technol. 32(16), 2862–2876 (2014).
[Crossref]

Alvarado Segovia, A.

Amari, A.

A. Amari, P. Ciblat, and Y. Jaouen, “Fifth-order Volterra series based nonlinear equalizer for long-haul high data rate optical fiber communications,” Asilomar Conference ACSSC (2014).

Ataie, V.

Bai, Y.

Z. Li, W.-R. Peng, F. Zhu, and Y. Bai, “Optimum quantization of perturbation coefficients for perturbative fiber nonlinearity mitigation,” Tech. Digest of European Conference on Optical Communication paper We.1.3.4. (2014).

Batshon, H. G.

I. B. Djordjevic, H. G. Batshon, L. Xu, and T. Wang, “Coded polarization-multiplexed iterative polar modulation (PM-IPM) for beyond 400 Gb/s serial optical transmission,” in Proc. Optical Fiber Communication Conference, Los Angeles, CA, Mar. 2010, p. OMK2

Beygi, L.

Bocherer, G.

Bosco, G.

Brandt-Pearce, M.

H. Song and M. Brandt-Pearce, “A 2-D discrete-time model of physical impairments in wavelength-division multiplexing systems,” J. Lightwave Technol. 30(5), 713–726 (2012).
[Crossref]

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers”, J. Lightwave Technol. 15(12), 2232–2241 (1997).
[Crossref]

Buchali, F.

Carena, A.

Ciblat, P.

A. Amari, P. Ciblat, and Y. Jaouen, “Fifth-order Volterra series based nonlinear equalizer for long-haul high data rate optical fiber communications,” Asilomar Conference ACSSC (2014).

Curri, V.

Dar, R.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Inter-channel nonlinear interference noise in WDM systems: modeling and mitigation,” J. Lightwave Technol. 33(5), 1044–1053 (2015).
[Crossref]

R. Dar, M. Shtaif, and M. Feder, “New bounds on the capacity of the nonlinear fiber-optic channel,” Optics Letters 39, 398–401 (2014).
[Crossref] [PubMed]

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “On shaping gain in the nonlinear fiber-optic channel,” IEEE International Symposium on Information Theory (ISIT), Honolulu, HI, USA, July 2014.

M. Shtaif, R. Dar, A. Mecozzi, and M. Feder, “Nonlinear interference noise in WDM systems and approaches for its cancelation,” in Optical Communication (ECOC 2014), 39th European Conference and Exhibition on Optical Communications, paper We1.3.1.

Derevyanko, S.A.

K. S. Turitsyn, S.A. Derevyanko, I.V. Yurkevich, and S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Letters 91, 203901 (2003).
[Crossref]

Djordjevic, I. B.

Dou, L.

Durisi, G.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightwave Technol. 32(16), 2862–2876 (2014).
[Crossref]

E. Agrell, G. Durisi, and P. Johannisson, “Information-theory-friendly models for fiberoptic channels: A primer,” IEEE Information Theory Workshop (2015).

Ellis, A.

M. Sorokina, A. Ellis, and S. Turitsyn, “Optical information capacity processing,” chapter in All-Optical Signal Processing325–354 (2015).

Essiambre, R. J.

Essiambre, R.-J.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).
[Crossref] [PubMed]

A. Ghazisaeidi and R.-J. Essiambre, “Calculation of coefficients of perturbative nonlinear pre-compensation for Nyquist pulses,” Tech. Digest of European Conference on Optical Communication paper We.1.3.3. (2014).

Fan, Y.

Feder, M.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Inter-channel nonlinear interference noise in WDM systems: modeling and mitigation,” J. Lightwave Technol. 33(5), 1044–1053 (2015).
[Crossref]

R. Dar, M. Shtaif, and M. Feder, “New bounds on the capacity of the nonlinear fiber-optic channel,” Optics Letters 39, 398–401 (2014).
[Crossref] [PubMed]

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “On shaping gain in the nonlinear fiber-optic channel,” IEEE International Symposium on Information Theory (ISIT), Honolulu, HI, USA, July 2014.

Fehenberger, T.

Forestieri, E.

M. Secondini and E. Forestieri, “Scope and limitations of the nonlinear Shannon limit,” online version of 24.10.2016 http://ieeexplore.ieee.org/document/7637002/ .

Forghieri, F.

Foschini, G. J.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).
[Crossref] [PubMed]

Freckmann, T.

Ghazisaeidi, A.

Goebel, B.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

Han, T. S.

S. Verdu and T. S. Han, “A general formula for channel capacity,” IEEE Trans. Inf. Theory 40(4) 1147–1157 (1994).
[Crossref]

Hanik, N.

Hoshida, T.

Idler, W.

Jaouen, Y.

A. Amari, P. Ciblat, and Y. Jaouen, “Fifth-order Volterra series based nonlinear equalizer for long-haul high data rate optical fiber communications,” Asilomar Conference ACSSC (2014).

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]

E. Agrell, G. Durisi, and P. Johannisson, “Information-theory-friendly models for fiberoptic channels: A primer,” IEEE Information Theory Workshop (2015).

Karlsson, M.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightwave Technol. 32(16), 2862–2876 (2014).
[Crossref]

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]

E. Agrell and M. Karlsson, “Satellite constellations: towards the nonlinear channel capacity,” 25th IEEE Photonics Conference pp. 316–317 (2012).

Kramer, G.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).
[Crossref] [PubMed]

Kschischang, F. R.

C. Pan and F. R. Kschischang, “Probabilistic 16-QAM Shaping in WDM Systems,” Journal of Lightwave Technology (in press).

Kuo, B.P.-P.

Kurtzke, C.

A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” Tech. Digest of European Conference on Optical Communication paper MoC2.4. (1993).

Li, Z.

Liu, L.

Mecozzi, A.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Inter-channel nonlinear interference noise in WDM systems: modeling and mitigation,” J. Lightwave Technol. 33(5), 1044–1053 (2015).
[Crossref]

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S. Verdu and T. S. Han, “A general formula for channel capacity,” IEEE Trans. Inf. Theory 40(4) 1147–1157 (1994).
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IEEE Trans. Inf. Theory, (1)

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M. Secondini, chapter in “Roadmap of optical communications,” by E. Agrell et al. Journal of Optics 18 (6), 063002 (2016).
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Nat. Commun. (1)

M. A. Sorokina and S. K. Turitsyn, “Regeneration limit of classical Shannon capacity,” Nat. Commun. 5, 3861 (2014).
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Nature (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
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Opt. Express (3)

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

M. Sorokina, S. Sygletos, and S. Turitsyn, “Sparse identification for nonlinear optical communication systems: SINO method,” Opt. Express 24(26), 30433–30443 (2016).
[Crossref]

Optics and Photonics News (1)

P. J. Winzer, “Scaling optical fiber networks: challenges and solutions,” Optics and Photonics News, 26, 28–35 (2015).
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Optics Letters (1)

R. Dar, M. Shtaif, and M. Feder, “New bounds on the capacity of the nonlinear fiber-optic channel,” Optics Letters 39, 398–401 (2014).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).
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Phys. Rev. Letters (1)

K. S. Turitsyn, S.A. Derevyanko, I.V. Yurkevich, and S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Letters 91, 203901 (2003).
[Crossref]

Science (2)

D. J. Richardson, “Filling the light pipe,” Science 330, 327–328 (2010).
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Other (15)

M. Sorokina, A. Ellis, and S. Turitsyn, “Optical information capacity processing,” chapter in All-Optical Signal Processing325–354 (2015).

E. Agrell, G. Durisi, and P. Johannisson, “Information-theory-friendly models for fiberoptic channels: A primer,” IEEE Information Theory Workshop (2015).

C. Pan and F. R. Kschischang, “Probabilistic 16-QAM Shaping in WDM Systems,” Journal of Lightwave Technology (in press).

M. Secondini and E. Forestieri, “Scope and limitations of the nonlinear Shannon limit,” online version of 24.10.2016 http://ieeexplore.ieee.org/document/7637002/ .

M. Sorokina, S. Sygletos, and S. K. Turitsyn, “Ripple distribution for nonlinear fiber-optic channels,” https://arxiv.org/abs/1610.06937

M. Schetzen, “The Volterra and Wiener theories of nonlinear systems”. (Krieger, 2006).

A. Amari, P. Ciblat, and Y. Jaouen, “Fifth-order Volterra series based nonlinear equalizer for long-haul high data rate optical fiber communications,” Asilomar Conference ACSSC (2014).

E. Agrell and M. Karlsson, “Satellite constellations: towards the nonlinear channel capacity,” 25th IEEE Photonics Conference pp. 316–317 (2012).

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “On shaping gain in the nonlinear fiber-optic channel,” IEEE International Symposium on Information Theory (ISIT), Honolulu, HI, USA, July 2014.

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Fig. 1 Fiber-optic communication system. a) The fundamental building blocks of a communication system where data is coded to a discrete set of symbols X_k (panel b) and transformed into continuous time form E(t, 0) (panel c) to be transmitted via the fiber channel. During transmission the signal is governed by the NLSE (panel d), which results in distortions: dispersion, nonlinearity, noise (panel e), which are reflected in the conditional pdf P(Y|X), which we can use to optimize constellation and coding (panel f), to receive the maximum achievable transmission rate – channel capacity C. The received signal E(t, L) is processed and sampled Y_k and, finally, decoded to receive the data (panel g).

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Fig. 2 Coupling matrix. For various transmission distances a) L = 100 km, b) L = 500 km, c) L = 1000 km and the corresponding received constellation diagrams below (for input power 6dBm and span length 100km). We see that the coupling matrix reflects the strength of the inter symbol interaction and its effect on the signal distortion.

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Fig. 3 Capacity and ripple distribution. a) Capacity lower bounds for uncompensated signal-signal (S-S) distortions I^S−S (GN model (red, dotted) and the proposed model (green)) and compensated deterministic distortions with account of signal-noise (S-N) interference. Previous lower bound (denoted by I₀) decreases to zero, whereas ripple distribution as input pdf (Eq. (15)) allows to achieve higher monotonically increasing bounds, denoted by I₁ and I₂ for the increased number of ripples in an input pdf as shown in panels b) and c) correspondingly (Gaussian pdf is shown in red dashed lines for comparison).

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Tables (2)

Table 1 Review of previous channel models

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Table 2: Statistical properties of ripple distribution

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Equations (22)

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

\frac{\partial E}{\partial z} = - \frac{α}{2} E - i \frac{β_{2}}{2} \frac{\partial^{2} E}{\partial^{2} t} + i γ {| E |}^{2} E + η (t, z),

\begin{matrix} Y_{k}^{'} = \frac{L_{d} ∊}{L} η_{k} + ε V {[Y]}_{k}, \\ V {[Y]}_{k} = Ψ_{s} (ξ) \sum_{m, n = - M}^{M} Y_{k + m} (ξ) Y_{k + n} (ξ) Y_{k + m + n}^{*} (ξ) {\tilde{C}}_{m n} (ξ) \end{matrix}

\begin{array}{l} {\tilde{C}}_{m n} & = i ∭ d ω d ω_{1} d ω_{2} e^{- i ω_{1} ω_{2} β_{2} L_{d} ξ - i ω_{1} m T - i ω_{2} n T} \times \\ f^{*} (ω) f (ω_{1} + ω) f (ω_{2} + ω) f^{*} (ω_{1} + ω_{2} + ω) \end{array}

Y = \hat{X} + M ζ + L ζ^{*}, ζ = \int d ξ η (ξ)

Y_{k} = {\hat{X}}_{k} + \sum_{m} M_{k, m} ζ_{m} + L_{k, m} ζ_{m}^{*}

{\hat{X}}_{k} = Y_{k}^{(0)} + \sum_{N_{o} = 1}^{\infty} ε^{N_{o}} Y_{k}^{(N_{o})}, Y_{k}^{(0)} = X_{k}

Y_{k}^{(N_{o})} = \sum_{\begin{array}{l} i, j, l = 0 \\ i + j + l = N_{o} - 1 \end{array}}^{N_{o} - 1} \sum_{m, n} C_{m n} Y_{k + n}^{(i)} Y_{k + m}^{(j)} {(Y_{k + n + m}^{(l)})}^{*}

M_{k, m} = ∊ δ_{k, m} + ∊ ε \sum_{n} K_{n, m - k} ({\hat{X}}_{k + n} {\hat{X}}_{m + n}^{*} + {\hat{X}}_{m + n} {\hat{X}}_{k + n}^{*}),

L_{k, m} = ∊ ε \sum_{n} K_{n, m - k - n} {\hat{X}}_{k + n} {\hat{X}}_{m - n}

K_{m n} = \int d z \sqrt{Ψ_{n} (z) Ψ_{s} (z)} {\tilde{C}}_{m n} (z)

Y_{k θ} = {\hat{X}}_{k θ} + \sum_{m, α} M_{k θ m α} ζ_{m α} + L_{k θ m α} ζ_{m α}^{*}

C = lim_{d = \dim (X) \to \infty} sup \frac{1}{d} I (X, Y)

\begin{array}{l} P (Y | X) = {(π)}^{- d} {(| Γ | | P |)}^{- 1 / 2} \\ exp [- \frac{1}{2} [{(y - \hat{x})}^{H}, {(y - \hat{x})}^{T}] {(\begin{matrix} Γ & ϒ \\ ϒ^{H} & Γ^{*} \end{matrix})}^{- 1} (\begin{array}{l} y - \hat{x} \\ y^{*} - {\hat{x}}^{*} \end{array})] \end{array}

\begin{matrix} Γ = E [(Y - \hat{X}) {(Y - \hat{X})}^{H}], ϒ = E [(Y - \hat{X}) {(Y - \hat{X})}^{T}] \\ P = Γ^{*} - ϒ^{H} Γ^{- 1} ϒ \end{matrix}

Γ = (M M^{H} + L L^{H}), ϒ = (M L^{T} + L M^{T})

P_{X} = \prod_{i = 1}^{d} P_{X_{i}}, P_{X_{i}} (x_{i} = {r_{i}, φ_{i}}) = \sum_{α = 1}^{q} \frac{r_{i} p_{α}}{π σ_{α}} e^{- \frac{r_{i}^{2} + ρ_{α}^{2}}{S_{α}}} I_{0} (\frac{2 r_{i} ρ_{α}}{σ_{α}})

P_{X} = \prod_{i = 1}^{d} \frac{r_{i}}{π σ_{1}} e^{- \frac{r_{i}^{2}}{σ_{1}}}

I_{0} = {log}_{2} (1 + \frac{σ_{1}}{∊^{2} + C_{nl} σ_{1}^{2}})

\begin{matrix} 1 = \int d x P_{x} ≃ p_{1}^{d} + d p_{1}^{d - 1} p_{2} \\ d = \int d x {‖ x ‖}^{2} P_{x} ≃ d p_{1}^{d} σ_{1} + d p_{1}^{d - 1} p_{2} ((d - 1) σ_{1} + σ_{2} + ρ^{2}) \end{matrix}

\begin{array}{l} I_{1} = lim_{\begin{array}{l} δ \to 0 \\ d \to \infty \end{array}} p_{1}^{d} {log}_{2} (1 + \frac{σ_{1}}{∊^{2} + 6 ε^{2} ∊^{2} \sum_{m, n} {| K_{m n} |}^{2} σ_{1}^{2}}) - p_{1}^{d} {log}_{2} p_{1} - p_{0}^{d - 1} p_{2} {log}_{2} (p_{1}^{d - 1} p_{2}) + \frac{p_{1}^{d - 1} p_{2}}{d} \sum_{l = - d / 2}^{d / 2} \\ [{log}_{2} (\frac{2 \sqrt{π ρ^{2} σ^{2}}}{∊^{2} + 6 ε^{2} ∊^{2} \sum_{m, n \neq l} {| K_{m n} |}^{2} σ_{1}^{2} + 12 ε^{2} ∊^{2} \sum_{m} {| K_{m l} |}^{2} σ_{1} (σ_{2} + ρ^{2}) + 6 ε^{2} ∊^{2} {| K_{l l} |}^{2} {(σ_{2} + ρ^{2})}^{2}}) + \\ (d - 1) {log}_{2} (1 + \frac{σ_{1}}{∊^{2} + 6 ε^{2} ∊^{2} \sum_{m, n \neq l} {| K_{m n} |}^{2} σ_{1}^{2} + 12 ε^{2} ∊^{2} \sum_{m} {| K_{m l} |}^{2} σ_{1} (σ_{2} + ρ^{2}) + 6 ε^{2} ∊^{2} {| K_{l l} |}^{2} {(σ_{2} + ρ^{2})}^{2}})] \end{array}

I_{1} = (1 - δ) {log}_{2} (1 + \frac{1}{2 ∊^{- 1} \sqrt{C_{nl}}}) + δ {log}_{2} (\frac{2 \sqrt{π ρ^{2} σ_{2}}}{σ_{1}}) + δ {log}_{2} e - δ {log}_{2} δ - δ (σ_{2} + ρ^{2} - σ_{1}) ∊^{- 1} C_{nl}^{1 / 2} {log}_{2} e

I_{1} = {log}_{2} (1 + \frac{1}{2 ∊ \sqrt{C_{nl}}}) + \sqrt{2 π ρ^{2} ∊^{- 1} C_{nl}^{1 / 2}} e^{- ρ^{2} ∊^{- 1} C_{nl}^{1 / 2}} {log}_{2} e

model and estimations on capacity	relevance to MCM
signal-signal interactions
infinite memoryGaussian noise [1,2,26,27] – 1	Y_k = X_k + ξ_k + ζ_k, $〈 ξ_{k} ξ_{k}^{*} 〉 = ε^{2} \sum_{m, n \neq 0} {\| C_{m n} \|}^{2}$
infinite memoryGaussian noise model [28] – 2	Y_k = X_k Φ + ζ_k $Φ = 1 + \sum_{m, n} {\| C_{m, n} \|}^{2} ε^{2}$
finite memoryGaussian noise [15] – 3non-decreasing capacity	Y_k = X_k + ξ_k + ζ_k $〈 ξ_{k} ξ_{k}^{*} 〉 = 2 ε^{2} \sum_{m, n \neq 0} {\| C_{m n} \|}^{2} {(2 M + 1)}^{- 1} \times \sum_{i = k - M}^{k + M} {\| X_{i} \|}^{2}$
Nonlinear interference noise (NLIN) [29,30]	1^st-order representation of MCM
NLIN with Gaussian approximation [20] which results in concave capacity lower bound	Y_k = X_ke^iθ_k + ζ_kθ_k, ζ_k - Gaussian variables
NLIN for phase noise [31]	$Y_{k} = X_{k} + \sum_{m}^{M} X_{k + m} C_{m} + ζ_{k}$ $C_{m} = ε {lim}_{M \to \infty} \sum_{n = - M}^{M} 〈 {\| X \|}^{2} 〉 C_{m n}$
nonlinear phase noise – 1 [5,32] capacity equals to linear Shannon limit	Y_k = X_k + εX_k \|X_k\|²C₀₀ + ζ_kfor small nonlinearity Y_k = X_k e^iγL\|X_k\|² + ζ_k
signal-noise interactions
nonlinear phase noise – 2 [32]capacity is monotonically increasing [32]	${(A_{k})}_{z}^{'} = ε C_{00} A_{k} {\| A_{k} \|}^{2} + η_{k}$ , memoryless model
nonlinear signal-noise [33]infinite memory approximation	continuous-time modelpdf is derived in continuous-time approximation

	Ripple	Normal

pdf	$\frac{r}{π σ} e^{- \frac{r^{2} + ρ^{2}}{σ}} I_{0} (\frac{2 r ρ}{σ})$	$\frac{1}{π σ} exp (- \frac{{(x - x_{0})}^{2} + {(y - y_{0})}^{2}}{σ})$
mean	0	$ρ = \sqrt{x_{0}^{2} + y_{0}^{2}}$
mode	ρ	$ρ = \sqrt{x_{0}^{2} + y_{0}^{2}}$
2-nd order non-central moment	ρ² + σ	ρ² + σ
4-th order non-central moment	2σ² + 4σρ² + ρ⁴	3σ² + 6ρ²σ + ρ⁴
Kurtosis	$1 + σ \frac{σ + 2 ρ^{2}}{{(σ + ρ^{2})}^{2}}$	$1 + 2 σ \frac{σ + 2 ρ^{2}}{{(σ + ρ^{2})}^{2}}$

model and estimations on capacity	relevance to MCM
signal-signal interactions
infinite memoryGaussian noise [1,2,26,27] – 1	Y_k = X_k + ξ_k + ζ_k, $〈 ξ_{k} ξ_{k}^{*} 〉 = ε^{2} \sum_{m, n \neq 0} {\| C_{m n} \|}^{2}$
infinite memoryGaussian noise model [28] – 2	Y_k = X_k Φ + ζ_k $Φ = 1 + \sum_{m, n} {\| C_{m, n} \|}^{2} ε^{2}$
finite memoryGaussian noise [15] – 3non-decreasing capacity	Y_k = X_k + ξ_k + ζ_k $〈 ξ_{k} ξ_{k}^{*} 〉 = 2 ε^{2} \sum_{m, n \neq 0} {\| C_{m n} \|}^{2} {(2 M + 1)}^{- 1} \times \sum_{i = k - M}^{k + M} {\| X_{i} \|}^{2}$
Nonlinear interference noise (NLIN) [29,30]	1^st-order representation of MCM
NLIN with Gaussian approximation [20] which results in concave capacity lower bound	Y_k = X_ke^iθ_k + ζ_kθ_k, ζ_k - Gaussian variables
NLIN for phase noise [31]	$Y_{k} = X_{k} + \sum_{m}^{M} X_{k + m} C_{m} + ζ_{k}$ $C_{m} = ε {lim}_{M \to \infty} \sum_{n = - M}^{M} 〈 {\| X \|}^{2} 〉 C_{m n}$
nonlinear phase noise – 1 [5,32] capacity equals to linear Shannon limit	Y_k = X_k + εX_k \|X_k\|²C₀₀ + ζ_kfor small nonlinearity Y_k = X_k e^iγL\|X_k\|² + ζ_k
signal-noise interactions
nonlinear phase noise – 2 [32]capacity is monotonically increasing [32]	${(A_{k})}_{z}^{'} = ε C_{00} A_{k} {\| A_{k} \|}^{2} + η_{k}$ , memoryless model
nonlinear signal-noise [33]infinite memory approximation	continuous-time modelpdf is derived in continuous-time approximation

	Ripple	Normal

pdf	$\frac{r}{π σ} e^{- \frac{r^{2} + ρ^{2}}{σ}} I_{0} (\frac{2 r ρ}{σ})$	$\frac{1}{π σ} exp (- \frac{{(x - x_{0})}^{2} + {(y - y_{0})}^{2}}{σ})$
mean	0	$ρ = \sqrt{x_{0}^{2} + y_{0}^{2}}$
mode	ρ	$ρ = \sqrt{x_{0}^{2} + y_{0}^{2}}$
2-nd order non-central moment	ρ² + σ	ρ² + σ
4-th order non-central moment	2σ² + 4σρ² + ρ⁴	3σ² + 6ρ²σ + ρ⁴
Kurtosis	$1 + σ \frac{σ + 2 ρ^{2}}{{(σ + ρ^{2})}^{2}}$	$1 + 2 σ \frac{σ + 2 ρ^{2}}{{(σ + ρ^{2})}^{2}}$