Abstract

In this paper, we investigate the performance limits of electronic chromatic dispersion compensation (EDC) and digital backpropagation (DBP) for a single-channel non-dispersion-managed fiber-optical link. A known analytical method to derive the performance of the system with EDC is extended to derive a first-order approximation for the performance of the system with DBP. In contrast to the cubic growth of the variance of the nonlinear noise-like interference, often called nonlinear noise, with input power for EDC, a quadratic growth is observed with DBP using this approximation. Finally, we provide numerical results to verify the accuracy of the proposed approach and compare it with existing analytical models.

© 2013 OSA

Full Article  |  PDF Article

Errata

Erik Agrell, Alex Alvarado, Giuseppe Durisi, and Magnus Karlsson, "Capacity of a Nonlinear Optical Channel With Finite Memory," J. Lightwave Technol. 32, 2862-2876 (2014)
https://www.osapublishing.org/jlt/abstract.cfm?uri=jlt-32-16-2862

References

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    [CrossRef]
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    [CrossRef]
  3. A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightw. Technol., 28(4), 423–433 (2010).
    [CrossRef]
  4. E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightw. Technol., 27(22), 5115–5126 (2009).
    [CrossRef]
  5. A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett., 29(7), 673–675 (2004).
    [CrossRef] [PubMed]
  6. 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. Lett., 91(20), 203901 (2003).
    [CrossRef] [PubMed]
  7. M. I. Yousefi and F. R. Kschischang, “On the per-sample capacity of nondispersive optical fibers,” IEEE Trans. Inf. Theory, 57(11), 7522–7541 (2011).
    [CrossRef]
  8. K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005).
  9. L. Beygi, E. Agrell, M. Karlsson, and P. Johannisson, “Signal statistics in fiber-optical channels with polarization multiplexing and self-phase modulation,” J. Lightw. Technol., 29(16), 2379–2386 (2011).
    [CrossRef]
  10. A. P. T. Lau, T. S. R. Shen, W. Shieh, and K.-P. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express, 18(16), 17 239–17 251 (2010).
    [CrossRef]
  11. K.-P. Ho, A. P. T. Lau, and W. Shieh, “Equalization-enhanced phase noise induced timing jitter,” Opt. Lett., 36(4), 585–587 (2011).
    [CrossRef] [PubMed]
  12. B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with applications to fiber optics,” IEEE Trans. Inf. Theory, 57(9), 5720–5736 (2011).
    [CrossRef]
  13. A. T. Lau, S. Rabbani, and J. M. Kahn, “On the statistics of intrachannel four-wave mixing in phase-modulated optical communication systems,” J. Lightw. Technol., 26(14), 2128–2135 (2008).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  32. A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. Optic. Fiber Commun. Conf., p. OWO7, Mar.2011.
  33. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express, 19(13), 12 781–12 788 (2011).
    [CrossRef]

2013

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

2012

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]

P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. Lightw. Technol., 30(24), 3857–3879 (2012).
[CrossRef]

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

A. Mecozzi and R.-J. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol., 30(12), 2011–2024 (2012).
[CrossRef]

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (2012).
[CrossRef]

T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express, 20(27), pp. 28 779–28 785 (2012).
[CrossRef]

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express, 20(2), 1022–1032 (2012).
[CrossRef] [PubMed]

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-Four-Wave-Mixing,” Opt. Express, 20(7), 7777–7791 (2012).
[CrossRef] [PubMed]

2011

E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express, 19(13), 12 781–12 788 (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. Express, 19(26), B790–B798 (2011).
[CrossRef]

M. I. Yousefi and F. R. Kschischang, “On the per-sample capacity of nondispersive optical fibers,” IEEE Trans. Inf. Theory, 57(11), 7522–7541 (2011).
[CrossRef]

L. Beygi, E. Agrell, M. Karlsson, and P. Johannisson, “Signal statistics in fiber-optical channels with polarization multiplexing and self-phase modulation,” J. Lightw. Technol., 29(16), 2379–2386 (2011).
[CrossRef]

K.-P. Ho, A. P. T. Lau, and W. Shieh, “Equalization-enhanced phase noise induced timing jitter,” Opt. Lett., 36(4), 585–587 (2011).
[CrossRef] [PubMed]

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with applications to fiber optics,” IEEE Trans. Inf. Theory, 57(9), 5720–5736 (2011).
[CrossRef]

2010

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

A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightw. Technol., 28(4), 423–433 (2010).
[CrossRef]

A. P. T. Lau, T. S. R. Shen, W. Shieh, and K.-P. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express, 18(16), 17 239–17 251 (2010).
[CrossRef]

E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightw. Technol., 28(6), 939–951 (2010).
[CrossRef]

2009

E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightw. Technol., 27(22), 5115–5126 (2009).
[CrossRef]

2008

A. T. Lau, S. Rabbani, and J. M. Kahn, “On the statistics of intrachannel four-wave mixing in phase-modulated optical communication systems,” J. Lightw. Technol., 26(14), 2128–2135 (2008).
[CrossRef]

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital Backpropagation,” J. Lightw. Technol., 26(20), 3416–3425 (2008).
[CrossRef]

2007

D. J. Costello and G. D. Forney, “Channel coding: The road to channel capacity,” Proc. IEEE, 95(6), 1150–1177 (2007).
[CrossRef]

2005

K.-P. Ho and H.-C. Wang, “Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-DPSK signals in dispersive transmission systems,” IEEE Photon. Technol. Lett., 17(7), 1426–1428 (2005).
[CrossRef]

2004

A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett., 29(7), 673–675 (2004).
[CrossRef] [PubMed]

2003

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. Lett., 91(20), 203901 (2003).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, 2002).
[CrossRef]

G. P. Agrawal, Nonlinear fiber optics, 4th ed. (Academic Press, 2007).

Agrell, E.

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]

L. Beygi, E. Agrell, M. Karlsson, and P. Johannisson, “Signal statistics in fiber-optical channels with polarization multiplexing and self-phase modulation,” J. Lightw. Technol., 29(16), 2379–2386 (2011).
[CrossRef]

E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightw. Technol., 27(22), 5115–5126 (2009).
[CrossRef]

L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The Limits of Digital Backpropagation in Nonlinear Coherent Fiber-Optical Links,” in Proceedings of European Conference and Exhibition on Optical Communication, Amsterdam, The Netherlands, September 2012, P4.14.

Antona, J.-C.

Beygi, L.

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]

L. Beygi, E. Agrell, M. Karlsson, and P. Johannisson, “Signal statistics in fiber-optical channels with polarization multiplexing and self-phase modulation,” J. Lightw. Technol., 29(16), 2379–2386 (2011).
[CrossRef]

L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The Limits of Digital Backpropagation in Nonlinear Coherent Fiber-Optical Links,” in Proceedings of European Conference and Exhibition on Optical Communication, Amsterdam, The Netherlands, September 2012, P4.14.

Bigo, S.

Bononi, A.

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express, 20(2), 1022–1032 (2012).
[CrossRef] [PubMed]

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-Four-Wave-Mixing,” Opt. Express, 20(7), 7777–7791 (2012).
[CrossRef] [PubMed]

E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express, 19(13), 12 781–12 788 (2011).
[CrossRef]

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. Optic. Fiber Commun. Conf., p. OWO7, Mar.2011.

Bosco, G.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (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. Express, 19(26), B790–B798 (2011).
[CrossRef]

Brandt-Pearce, M.

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

Carena, A.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (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. Express, 19(26), B790–B798 (2011).
[CrossRef]

Cigliutti, R.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (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. Express, 19(26), B790–B798 (2011).
[CrossRef]

Costello, D. J.

D. J. Costello and G. D. Forney, “Channel coding: The road to channel capacity,” Proc. IEEE, 95(6), 1150–1177 (2007).
[CrossRef]

Cotter, D.

A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightw. Technol., 28(4), 423–433 (2010).
[CrossRef]

Curri, V.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (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. Express, 19(26), B790–B798 (2011).
[CrossRef]

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. Lett., 91(20), 203901 (2003).
[CrossRef] [PubMed]

Ellis, A. D.

A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightw. Technol., 28(4), 423–433 (2010).
[CrossRef]

Essiambre, R.-J.

A. Mecozzi and R.-J. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol., 30(12), 2011–2024 (2012).
[CrossRef]

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with applications to fiber optics,” IEEE Trans. Inf. Theory, 57(9), 5720–5736 (2011).
[CrossRef]

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

Fischer, J. K.

T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express, 20(27), pp. 28 779–28 785 (2012).
[CrossRef]

Forghieri, F.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (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. Express, 19(26), B790–B798 (2011).
[CrossRef]

Forney, G. D.

D. J. Costello and G. D. Forney, “Channel coding: The road to channel capacity,” Proc. IEEE, 95(6), 1150–1177 (2007).
[CrossRef]

Foschini, G. J.

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

Goebel, B.

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with applications to fiber optics,” IEEE Trans. Inf. Theory, 57(9), 5720–5736 (2011).
[CrossRef]

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

Grellier, E.

Hanik, N.

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with applications to fiber optics,” IEEE Trans. Inf. Theory, 57(9), 5720–5736 (2011).
[CrossRef]

Ho, K.-P.

K.-P. Ho, A. P. T. Lau, and W. Shieh, “Equalization-enhanced phase noise induced timing jitter,” Opt. Lett., 36(4), 585–587 (2011).
[CrossRef] [PubMed]

A. P. T. Lau, T. S. R. Shen, W. Shieh, and K.-P. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express, 18(16), 17 239–17 251 (2010).
[CrossRef]

K.-P. Ho and H.-C. Wang, “Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-DPSK signals in dispersive transmission systems,” IEEE Photon. Technol. Lett., 17(7), 1426–1428 (2005).
[CrossRef]

K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005).

Ip, E.

E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightw. Technol., 28(6), 939–951 (2010).
[CrossRef]

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital Backpropagation,” J. Lightw. Technol., 26(20), 3416–3425 (2008).
[CrossRef]

Jiang, Y.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (2012).
[CrossRef]

Johannisson, P.

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

L. Beygi, E. Agrell, M. Karlsson, and P. Johannisson, “Signal statistics in fiber-optical channels with polarization multiplexing and self-phase modulation,” J. Lightw. Technol., 29(16), 2379–2386 (2011).
[CrossRef]

L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The Limits of Digital Backpropagation in Nonlinear Coherent Fiber-Optical Links,” in Proceedings of European Conference and Exhibition on Optical Communication, Amsterdam, The Netherlands, September 2012, P4.14.

Kahn, J. M.

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital Backpropagation,” J. Lightw. Technol., 26(20), 3416–3425 (2008).
[CrossRef]

A. T. Lau, S. Rabbani, and J. M. Kahn, “On the statistics of intrachannel four-wave mixing in phase-modulated optical communication systems,” J. Lightw. Technol., 26(14), 2128–2135 (2008).
[CrossRef]

Karlsson, M.

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

L. Beygi, E. Agrell, M. Karlsson, and P. Johannisson, “Signal statistics in fiber-optical channels with polarization multiplexing and self-phase modulation,” J. Lightw. Technol., 29(16), 2379–2386 (2011).
[CrossRef]

E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightw. Technol., 27(22), 5115–5126 (2009).
[CrossRef]

L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The Limits of Digital Backpropagation in Nonlinear Coherent Fiber-Optical Links,” in Proceedings of European Conference and Exhibition on Optical Communication, Amsterdam, The Netherlands, September 2012, P4.14.

Kramer, G.

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with applications to fiber optics,” IEEE Trans. Inf. Theory, 57(9), 5720–5736 (2011).
[CrossRef]

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

Kschischang, F. R.

M. I. Yousefi and F. R. Kschischang, “On the per-sample capacity of nondispersive optical fibers,” IEEE Trans. Inf. Theory, 57(11), 7522–7541 (2011).
[CrossRef]

Lau, A. P. T.

K.-P. Ho, A. P. T. Lau, and W. Shieh, “Equalization-enhanced phase noise induced timing jitter,” Opt. Lett., 36(4), 585–587 (2011).
[CrossRef] [PubMed]

A. P. T. Lau, T. S. R. Shen, W. Shieh, and K.-P. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express, 18(16), 17 239–17 251 (2010).
[CrossRef]

Lau, A. T.

A. T. Lau, S. Rabbani, and J. M. Kahn, “On the statistics of intrachannel four-wave mixing in phase-modulated optical communication systems,” J. Lightw. Technol., 26(14), 2128–2135 (2008).
[CrossRef]

Lorcy, L.

Mecozzi, A.

A. Mecozzi and R.-J. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol., 30(12), 2011–2024 (2012).
[CrossRef]

A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett., 29(7), 673–675 (2004).
[CrossRef] [PubMed]

Nespola, A.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (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. Express, 19(26), B790–B798 (2011).
[CrossRef]

Nölle, M.

T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express, 20(27), pp. 28 779–28 785 (2012).
[CrossRef]

Poggiolini, P.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (2012).
[CrossRef]

P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. Lightw. Technol., 30(24), 3857–3879 (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. Express, 19(26), B790–B798 (2011).
[CrossRef]

Proakis, J. G.

J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw-Hill, 2008).

Rabbani, S.

A. T. Lau, S. Rabbani, and J. M. Kahn, “On the statistics of intrachannel four-wave mixing in phase-modulated optical communication systems,” J. Lightw. Technol., 26(14), 2128–2135 (2008).
[CrossRef]

Rival, O.

Rossi, N.

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-Four-Wave-Mixing,” Opt. Express, 20(7), 7777–7791 (2012).
[CrossRef] [PubMed]

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. Optic. Fiber Commun. Conf., p. OWO7, Mar.2011.

Salehi, M.

J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw-Hill, 2008).

Sasaki, T.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (2012).
[CrossRef]

Schubert, C.

T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express, 20(27), pp. 28 779–28 785 (2012).
[CrossRef]

Serena, P.

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-Four-Wave-Mixing,” Opt. Express, 20(7), 7777–7791 (2012).
[CrossRef] [PubMed]

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. Optic. Fiber Commun. Conf., p. OWO7, Mar.2011.

Shen, T. S. R.

A. P. T. Lau, T. S. R. Shen, W. Shieh, and K.-P. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express, 18(16), 17 239–17 251 (2010).
[CrossRef]

Shieh, W.

K.-P. Ho, A. P. T. Lau, and W. Shieh, “Equalization-enhanced phase noise induced timing jitter,” Opt. Lett., 36(4), 585–587 (2011).
[CrossRef] [PubMed]

A. P. T. Lau, T. S. R. Shen, W. Shieh, and K.-P. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express, 18(16), 17 239–17 251 (2010).
[CrossRef]

Simonneau, C.

Song, H.

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

Tanimura, T.

T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express, 20(27), pp. 28 779–28 785 (2012).
[CrossRef]

Torrengo, E.

Turitsyn, K. S.

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. Lett., 91(20), 203901 (2003).
[CrossRef] [PubMed]

Turitsyn, S. K.

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. Lett., 91(20), 203901 (2003).
[CrossRef] [PubMed]

Vacondio, F.

Wang, H.-C.

K.-P. Ho and H.-C. Wang, “Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-DPSK signals in dispersive transmission systems,” IEEE Photon. Technol. Lett., 17(7), 1426–1428 (2005).
[CrossRef]

Winzer, P. J.

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with applications to fiber optics,” IEEE Trans. Inf. Theory, 57(9), 5720–5736 (2011).
[CrossRef]

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

Wymeersch, H.

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]

L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The Limits of Digital Backpropagation in Nonlinear Coherent Fiber-Optical Links,” in Proceedings of European Conference and Exhibition on Optical Communication, Amsterdam, The Netherlands, September 2012, P4.14.

Yamamoto, Y.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (2012).
[CrossRef]

Yousefi, M. I.

M. I. Yousefi and F. R. Kschischang, “On the per-sample capacity of nondispersive optical fibers,” IEEE Trans. Inf. Theory, 57(11), 7522–7541 (2011).
[CrossRef]

Yurkevich, I. V.

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. Lett., 91(20), 203901 (2003).
[CrossRef] [PubMed]

Zeolla, D.

Zhao, J.

A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightw. Technol., 28(4), 423–433 (2010).
[CrossRef]

IEEE Photon. Technol. Lett

K.-P. Ho and H.-C. Wang, “Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-DPSK signals in dispersive transmission systems,” IEEE Photon. Technol. Lett., 17(7), 1426–1428 (2005).
[CrossRef]

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett., 24(14), 1230–1232 (2012).
[CrossRef]

IEEE Trans. Commun

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]

IEEE Trans. Inf. Theory

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with applications to fiber optics,” IEEE Trans. Inf. Theory, 57(9), 5720–5736 (2011).
[CrossRef]

M. I. Yousefi and F. R. Kschischang, “On the per-sample capacity of nondispersive optical fibers,” IEEE Trans. Inf. Theory, 57(11), 7522–7541 (2011).
[CrossRef]

J. Lightw. Technol

L. Beygi, E. Agrell, M. Karlsson, and P. Johannisson, “Signal statistics in fiber-optical channels with polarization multiplexing and self-phase modulation,” J. Lightw. Technol., 29(16), 2379–2386 (2011).
[CrossRef]

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

A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightw. Technol., 28(4), 423–433 (2010).
[CrossRef]

E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightw. Technol., 27(22), 5115–5126 (2009).
[CrossRef]

A. T. Lau, S. Rabbani, and J. M. Kahn, “On the statistics of intrachannel four-wave mixing in phase-modulated optical communication systems,” J. Lightw. Technol., 26(14), 2128–2135 (2008).
[CrossRef]

P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. Lightw. Technol., 30(24), 3857–3879 (2012).
[CrossRef]

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

A. Mecozzi and R.-J. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol., 30(12), 2011–2024 (2012).
[CrossRef]

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

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital Backpropagation,” J. Lightw. Technol., 26(20), 3416–3425 (2008).
[CrossRef]

E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightw. Technol., 28(6), 939–951 (2010).
[CrossRef]

Opt. Express

T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express, 20(27), pp. 28 779–28 785 (2012).
[CrossRef]

E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express, 19(13), 12 781–12 788 (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. Express, 19(26), B790–B798 (2011).
[CrossRef]

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express, 20(2), 1022–1032 (2012).
[CrossRef] [PubMed]

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-Four-Wave-Mixing,” Opt. Express, 20(7), 7777–7791 (2012).
[CrossRef] [PubMed]

A. P. T. Lau, T. S. R. Shen, W. Shieh, and K.-P. Ho, “Equalization-enhanced phase noise for 100Gb/s transmission and beyond with coherent detection,” Opt. Express, 18(16), 17 239–17 251 (2010).
[CrossRef]

Opt. Lett

K.-P. Ho, A. P. T. Lau, and W. Shieh, “Equalization-enhanced phase noise induced timing jitter,” Opt. Lett., 36(4), 585–587 (2011).
[CrossRef] [PubMed]

A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett., 29(7), 673–675 (2004).
[CrossRef] [PubMed]

Phys. Rev. Lett

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. Lett., 91(20), 203901 (2003).
[CrossRef] [PubMed]

Proc. IEEE

D. J. Costello and G. D. Forney, “Channel coding: The road to channel capacity,” Proc. IEEE, 95(6), 1150–1177 (2007).
[CrossRef]

Other

A. Bononi and P. Serena, “An alternative derivation of Johannisson’s regular perturbation model,” 2012. [Online]. Available: http://arXiv:1207.4729

K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005).

G. P. Agrawal, Nonlinear fiber optics, 4th ed. (Academic Press, 2007).

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, 2002).
[CrossRef]

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. Optic. Fiber Commun. Conf., p. OWO7, Mar.2011.

L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The Limits of Digital Backpropagation in Nonlinear Coherent Fiber-Optical Links,” in Proceedings of European Conference and Exhibition on Optical Communication, Amsterdam, The Netherlands, September 2012, P4.14.

J. G. Proakis and M. Salehi, Digital Communications, 5th ed. (McGraw-Hill, 2008).

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Figures (5)

Fig. 1
Fig. 1

A baseband continuous-time model based on the SSFM for a fiber-optical link with N spans of SMF fiber (i = 1,...,N), each consisting of M segments (m = 1,...,M), and electronic linear (EDC) and nonlinear (DBP) pre-compensation.

Fig. 2
Fig. 2

(a) Nonlinear pre-compensation based on the DBP [27] (h−1[n] is the inverse of the filter h[n]). (b) A baseband discrete-time model for the SMF.

Fig. 3
Fig. 3

The discrete-time model of segment 1 from span 2 and the first span together with their corresponding pre-compensation units. The gain of the EDFA unit is assumed to be canceled out by the compensation unit. The channel deterministic impairments are fully compensated for the first span because the first amplifier is assumed to be added at the beginning of the second span. All the impairments for the first span are deterministic and there is no noise interaction involved in the signal propagation in this span.

Fig. 4
Fig. 4

(a) The SERs of two fiber-optical links with EDC and PM QPSK versus transmitted power per polarization P, consisting of 90 spans of length 80 km at 42.7 Gbaud and 30 spans of length 120 km at 32 Gbaud. The analytical results using the introduced model in Eqs. (3)(5) as well as the model in [15, Eqs. (7), (13), and (23)]. (b) The SERs of two systems consisting of 70 spans of length 120 km with a QPSK signal and 100 spans of length 80 km with a 16-QAM signal, both at 32 Gbaud.

Fig. 5
Fig. 5

(a) The discrete-time model of segment 2 and its pre-compensation unit from span 2 together with the simplified model of segment 1. (b) The simplified model for segments 1 and 2 and their corresponding pre-compensation units.

Tables (1)

Tables Icon

Table 1 The variance of the additive Gaussian noise Wx and Wy introduced in Eq. (2) with EDC and DBP consisting of the linear (ASE) and nonlinear noise-like interference.

Equations (17)

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

j U ( t , z ) z β 2 2 2 U ( t , z ) t 2 + γ ( U ( t , z ) U ( t , z ) ) ( U ( t , z ) + j α 2 U ( t , z ) = 0 ,
R = ζ S + W ,
| ζ EDC | 2 1 3 N 1 + ε ϕ s 2 tanh ( α 4 L D ) .
ε = 3 10 log ( 1 + 6 α L asinh ( π 2 2 α L D ) ) .
σ NL 2 = ( 1 | ζ EDC | 2 ) P 3 N 1 + ε γ 2 α 2 tanh ( α 4 L D ) P 3 .
| ζ DBP m , i | 2 1 6 ( i 1 ) A 4 ( m 1 ) α 2 L eff 2 ϕ s ϕ n .
| ζ DBP | 2 = i = 1 N m = 1 M | ζ DBP m , i | 2 1 6 α 2 L eff 2 ϕ s ϕ n i = 1 N m = 1 M ( i 1 ) A 4 ( m 1 ) .
| ζ DBP | 2 1 3 N 1 + ε ( N 1 ) ϕ s ϕ n tanh ( α 4 L D ) .
Var { W x } = Var { W y } ( 1 | ζ | 2 ) P + N σ 2 = 3 N 1 + ε ( N 1 ) γ 2 α 2 σ 2 tanh ( α 4 L D ) P 2 + N σ 2 ,
U 1 , 2 = A ( U 0 , 2 e j γ L eff U 0 , 2 2 ) * h = A ( ( V ˜ 0 , 2 + Z 1 ) e j γ L eff ( V ˜ 0 , 2 + Z 1 2 V ˜ 0 , 2 2 ) ) * h ,
W 1 , 2 = A ( ( V ˜ 0 , 2 + Z 1 ) e j γ L eff ( V ˜ 0 , 2 + Z 1 2 V ˜ 0 , 2 2 ) ζ DBP 1 , 2 V ˜ 0 , 2 ) * h .
𝔼 { W 1 , 2 } = A 𝔼 { ( V ˜ 0 , 2 + Z 1 ) e j γ L eff ( V ˜ 0 , 2 + Z 1 2 V ˜ 0 , 2 2 ) ζ DBP 1 , 2 V ˜ 0 , 2 } * h = 0 .
| ζ DBP 1 , 2 | 2 1 6 α 2 L eff 2 ϕ n ϕ s .
U 2 , 2 = A ( U 1 , 2 e j γ L eff U 1 , 2 2 ) * h = A ( ( ζ DBP 1 , 2 V ˜ 1 , 2 + W 1 , 2 ) e j γ L eff ( ζ DBP 1 , 2 V ˜ 1 , 2 + W 1 , 2 2 V ˜ 1 , 2 2 ) ) * h ,
W 2 , 2 = A ( ( ζ DBP 1 , 2 V ˜ 1 , 2 + W 1 , 2 ) e j γ L eff ( ζ DBP 1 , 2 V ˜ 1 , 2 + W 1 , 2 2 V ˜ 1 , 2 2 ) ζ DBP 2 , 2 ζ DBP 1 , 2 V ˜ 1 , 2 ) * h .
| ζ DBP 2 , 2 | 2 1 6 A 4 α 2 L eff 2 ϕ s ϕ n .
| ζ DBP m , i | 2 1 6 ( i 1 ) A 4 ( m 1 ) α 2 L eff 2 ϕ s ϕ n ,

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