Abstract

By extending a well-established time-domain perturbation approach to dual-polarization propagation, we provide an analytical framework to predict the nonlinear interference (NLI) variance, i.e., the variance induced by nonlinearity on the sampled field, and the nonlinear threshold (NLT) in coherent transmissions with dominant intrachannel-four-wave-mixing (IFWM). Such a framework applies to non dispersion managed (NDM) very long-haul coherent optical systems at nowadays typical baudrates of tens of Gigabaud, as well as to dispersion-managed (DM) systems at even higher baudrates, whenever IFWM is not removed by nonlinear equalization and is thus the dominant nonlinearity. The NLI variance formula has two fitting parameters which can be calibrated from simulations. From the NLI variance formula, analytical expressions of the NLT for both DM and NDM systems are derived and checked against recent NLT Monte-Carlo simulations.

© 2011 OSA

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References

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  1. A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical Characterization of PM-QPSK Signals after Propagation in Uncompensated Fiber Links,” Proc. ECOC’10, paper P4.07 (2010).
  2. P. Ramantanis and Y. Frignac, “Pattern-dependent nonlinear impairments on QPSK signals in dispersion-managed optical transmission systems,” Proc. ECOC’10, paper Mo.1.C.4 (2010).
  3. G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance Prediction for WDM PM-QPSK Transmission over Uncompensated Links,” Proc. OFC’11, paper OThO7 (2011).
  4. 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, 742–744 (2011).
    [CrossRef]
  5. E. Grellier and A. Bononi, “Quality Parameter for Coherent Transmissions with Gaussian-distributed Nonlinear Noise,” Opt. Express 19, 12781–12788 (2011).
    [CrossRef] [PubMed]
  6. F. Vacondio, C. Simonneau, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. ECOC’11, paper We.7.B.1 (2011).
  7. F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published).
    [PubMed]
  8. 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,” Proc. ECOC’11, paper We.7.B.2 (2011).
  9. A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” Proc. OFC’11, paper OWO7 (2011).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282, 2421–2425 (2009).
    [CrossRef]
  14. A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of Intrachannel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett. 12, 392–394 (2000).
    [CrossRef]
  15. A. Mecozzi, C. B. Clausen, and M. Shtaif, “System Impact of Intra-Channel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett.  12, 1633–1635 (2000).
    [CrossRef]
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    [CrossRef]
  18. A. Bononi, P. Serena, and N. Rossi, “Modeling Nonlinearity in Coherent Transmissions with Dominant Interpulse-Four-Wave-Mixing,” Proc. ECOC’11, paper We.7.B.4 (2011).
  19. For interleaved RZ (iRZ) we would need two different support pulses for each polarization, so here iRZ is excluded.
  20. A. Bononi, P. Serena, and A. Orlandini, “A Unified Design Framework for Single-Channel Dispersion-Managed Terrestrial Systems,” J. Lightwave Technol. 26, 3617–3631 (2008).
    [CrossRef]
  21. J. C. Antona and S. Bigo, “Physical design and performance estimation of heterogeneous optical transmission systems,” C. R. Phys. 9, 963–984 (2008).
    [CrossRef]
  22. G. A. Korn and T. A. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 2000).
  23. C. R. Menyuk and B. Marks, “Interaction of Polarization Mode Dispersion and Nonlinearity in Optical Fiber Transmission Systems,” J. Lightwave Technol. 24, 2806–2826 (2006).
    [CrossRef]

2011 (2)

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical Modeling of Non-Linear Propagation in Uncompensated Optical Transmission Links,” IEEE Photon. Technol. Lett.  23, 742–744 (2011).
[CrossRef]

E. Grellier and A. Bononi, “Quality Parameter for Coherent Transmissions with Gaussian-distributed Nonlinear Noise,” Opt. Express 19, 12781–12788 (2011).
[CrossRef] [PubMed]

2010 (1)

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron.  16, 1217–1226 (2010).
[CrossRef]

2009 (1)

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282, 2421–2425 (2009).
[CrossRef]

2008 (4)

E. Ip and J. M. Kahn, “Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008).
[CrossRef]

A. Bononi, P. Serena, and M. Bertolini, “Unified Analysis of Weakly-Nonlinear Dispersion-Managed Optical Transmission Systems from Perturbative Approach,” C. R. Phys. 9, 947–962 (2008).
[CrossRef]

A. Bononi, P. Serena, and A. Orlandini, “A Unified Design Framework for Single-Channel Dispersion-Managed Terrestrial Systems,” J. Lightwave Technol. 26, 3617–3631 (2008).
[CrossRef]

J. C. Antona and S. Bigo, “Physical design and performance estimation of heterogeneous optical transmission systems,” C. R. Phys. 9, 963–984 (2008).
[CrossRef]

2006 (2)

2000 (2)

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of Intrachannel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett. 12, 392–394 (2000).
[CrossRef]

A. Mecozzi, C. B. Clausen, and M. Shtaif, “System Impact of Intra-Channel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett.  12, 1633–1635 (2000).
[CrossRef]

1998 (1)

Antona, J. C.

J. C. Antona and S. Bigo, “Physical design and performance estimation of heterogeneous optical transmission systems,” C. R. Phys. 9, 963–984 (2008).
[CrossRef]

Antona, J.-C.

F. Vacondio, C. Simonneau, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. ECOC’11, paper We.7.B.1 (2011).

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published).
[PubMed]

Bayvel, P.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron.  16, 1217–1226 (2010).
[CrossRef]

Behrens, C.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron.  16, 1217–1226 (2010).
[CrossRef]

Bertolini, M.

A. Bononi, P. Serena, and M. Bertolini, “Unified Analysis of Weakly-Nonlinear Dispersion-Managed Optical Transmission Systems from Perturbative Approach,” C. R. Phys. 9, 947–962 (2008).
[CrossRef]

Bigo, S.

J. C. Antona and S. Bigo, “Physical design and performance estimation of heterogeneous optical transmission systems,” C. R. Phys. 9, 963–984 (2008).
[CrossRef]

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published).
[PubMed]

F. Vacondio, C. Simonneau, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. ECOC’11, paper We.7.B.1 (2011).

Bononi, A.

E. Grellier and A. Bononi, “Quality Parameter for Coherent Transmissions with Gaussian-distributed Nonlinear Noise,” Opt. Express 19, 12781–12788 (2011).
[CrossRef] [PubMed]

A. Bononi, P. Serena, and A. Orlandini, “A Unified Design Framework for Single-Channel Dispersion-Managed Terrestrial Systems,” J. Lightwave Technol. 26, 3617–3631 (2008).
[CrossRef]

A. Bononi, P. Serena, and M. Bertolini, “Unified Analysis of Weakly-Nonlinear Dispersion-Managed Optical Transmission Systems from Perturbative Approach,” C. R. Phys. 9, 947–962 (2008).
[CrossRef]

A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” Proc. OFC’11, paper OWO7 (2011).

F. Vacondio, C. Simonneau, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. ECOC’11, paper We.7.B.1 (2011).

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published).
[PubMed]

A. Bononi, P. Serena, and N. Rossi, “Modeling Nonlinearity in Coherent Transmissions with Dominant Interpulse-Four-Wave-Mixing,” Proc. ECOC’11, paper We.7.B.4 (2011).

Bosco, G.

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, 742–744 (2011).
[CrossRef]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical Characterization of PM-QPSK Signals after Propagation in Uncompensated Fiber Links,” Proc. ECOC’10, paper P4.07 (2010).

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance Prediction for WDM PM-QPSK Transmission over Uncompensated Links,” Proc. OFC’11, paper OThO7 (2011).

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,” Proc. ECOC’11, paper We.7.B.2 (2011).

Brandt-Pearce, M.

Carena, A.

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, 742–744 (2011).
[CrossRef]

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance Prediction for WDM PM-QPSK Transmission over Uncompensated Links,” Proc. OFC’11, paper OThO7 (2011).

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical Characterization of PM-QPSK Signals after Propagation in Uncompensated Fiber Links,” Proc. ECOC’10, paper P4.07 (2010).

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,” Proc. ECOC’11, paper We.7.B.2 (2011).

Chen, Z.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282, 2421–2425 (2009).
[CrossRef]

Cigliutti, R.

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance Prediction for WDM PM-QPSK Transmission over Uncompensated Links,” Proc. OFC’11, paper OThO7 (2011).

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,” Proc. ECOC’11, paper We.7.B.2 (2011).

Clausen, C. B.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of Intrachannel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett. 12, 392–394 (2000).
[CrossRef]

A. Mecozzi, C. B. Clausen, and M. Shtaif, “System Impact of Intra-Channel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett.  12, 1633–1635 (2000).
[CrossRef]

Curri, V.

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, 742–744 (2011).
[CrossRef]

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance Prediction for WDM PM-QPSK Transmission over Uncompensated Links,” Proc. OFC’11, paper OThO7 (2011).

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical Characterization of PM-QPSK Signals after Propagation in Uncompensated Fiber Links,” Proc. ECOC’10, paper P4.07 (2010).

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,” Proc. ECOC’11, paper We.7.B.2 (2011).

Dou, L.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282, 2421–2425 (2009).
[CrossRef]

Forghieri, F.

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, 742–744 (2011).
[CrossRef]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical Characterization of PM-QPSK Signals after Propagation in Uncompensated Fiber Links,” Proc. ECOC’10, paper P4.07 (2010).

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance Prediction for WDM PM-QPSK Transmission over Uncompensated Links,” Proc. OFC’11, paper OThO7 (2011).

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,” Proc. ECOC’11, paper We.7.B.2 (2011).

Frignac, Y.

P. Ramantanis and Y. Frignac, “Pattern-dependent nonlinear impairments on QPSK signals in dispersion-managed optical transmission systems,” Proc. ECOC’10, paper Mo.1.C.4 (2010).

Gao, Y.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282, 2421–2425 (2009).
[CrossRef]

Grellier, E.

E. Grellier and A. Bononi, “Quality Parameter for Coherent Transmissions with Gaussian-distributed Nonlinear Noise,” Opt. Express 19, 12781–12788 (2011).
[CrossRef] [PubMed]

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published).
[PubMed]

Hellerbrand, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron.  16, 1217–1226 (2010).
[CrossRef]

Ip, E.

Kahn, J. M.

Killey, R. I.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron.  16, 1217–1226 (2010).
[CrossRef]

Korn, G. A.

G. A. Korn and T. A. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 2000).

Korn, T. A.

G. A. Korn and T. A. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 2000).

Lorcy, L.

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published).
[PubMed]

F. Vacondio, C. Simonneau, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. ECOC’11, paper We.7.B.1 (2011).

Makovejs, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron.  16, 1217–1226 (2010).
[CrossRef]

Marks, B.

Mecozzi, A.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of Intrachannel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett. 12, 392–394 (2000).
[CrossRef]

A. Mecozzi, C. B. Clausen, and M. Shtaif, “System Impact of Intra-Channel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett.  12, 1633–1635 (2000).
[CrossRef]

Menyuk, C. R.

Millar, D. S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron.  16, 1217–1226 (2010).
[CrossRef]

Nespola, A.

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,” Proc. ECOC’11, paper We.7.B.2 (2011).

Orlandini, A.

Peddanarappagari, K. V.

Poggiolini, P.

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, 742–744 (2011).
[CrossRef]

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance Prediction for WDM PM-QPSK Transmission over Uncompensated Links,” Proc. OFC’11, paper OThO7 (2011).

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical Characterization of PM-QPSK Signals after Propagation in Uncompensated Fiber Links,” Proc. ECOC’10, paper P4.07 (2010).

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,” Proc. ECOC’11, paper We.7.B.2 (2011).

Ramantanis, P.

P. Ramantanis and Y. Frignac, “Pattern-dependent nonlinear impairments on QPSK signals in dispersion-managed optical transmission systems,” Proc. ECOC’10, paper Mo.1.C.4 (2010).

Rival, O.

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published).
[PubMed]

Rossi, N.

A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” Proc. OFC’11, paper OWO7 (2011).

A. Bononi, P. Serena, and N. Rossi, “Modeling Nonlinearity in Coherent Transmissions with Dominant Interpulse-Four-Wave-Mixing,” Proc. ECOC’11, paper We.7.B.4 (2011).

Savory, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron.  16, 1217–1226 (2010).
[CrossRef]

Serena, P.

A. Bononi, P. Serena, and M. Bertolini, “Unified Analysis of Weakly-Nonlinear Dispersion-Managed Optical Transmission Systems from Perturbative Approach,” C. R. Phys. 9, 947–962 (2008).
[CrossRef]

A. Bononi, P. Serena, and A. Orlandini, “A Unified Design Framework for Single-Channel Dispersion-Managed Terrestrial Systems,” J. Lightwave Technol. 26, 3617–3631 (2008).
[CrossRef]

A. Bononi, P. Serena, and N. Rossi, “Modeling Nonlinearity in Coherent Transmissions with Dominant Interpulse-Four-Wave-Mixing,” Proc. ECOC’11, paper We.7.B.4 (2011).

A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” Proc. OFC’11, paper OWO7 (2011).

Shtaif, M.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “System Impact of Intra-Channel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett.  12, 1633–1635 (2000).
[CrossRef]

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of Intrachannel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett. 12, 392–394 (2000).
[CrossRef]

Simonneau, C.

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published).
[PubMed]

F. Vacondio, C. Simonneau, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. ECOC’11, paper We.7.B.1 (2011).

Taiba, M. T.

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical Characterization of PM-QPSK Signals after Propagation in Uncompensated Fiber Links,” Proc. ECOC’10, paper P4.07 (2010).

Torrengo, E.

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,” Proc. ECOC’11, paper We.7.B.2 (2011).

Vacondio, F.

F. Vacondio, C. Simonneau, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. ECOC’11, paper We.7.B.1 (2011).

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published).
[PubMed]

Wei, X.

Xu, A.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282, 2421–2425 (2009).
[CrossRef]

Zeolla, D.

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,” Proc. ECOC’11, paper We.7.B.2 (2011).

Zhang, F.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282, 2421–2425 (2009).
[CrossRef]

C. R. Phys. (2)

A. Bononi, P. Serena, and M. Bertolini, “Unified Analysis of Weakly-Nonlinear Dispersion-Managed Optical Transmission Systems from Perturbative Approach,” C. R. Phys. 9, 947–962 (2008).
[CrossRef]

J. C. Antona and S. Bigo, “Physical design and performance estimation of heterogeneous optical transmission systems,” C. R. Phys. 9, 963–984 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron (1)

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron.  16, 1217–1226 (2010).
[CrossRef]

IEEE Photon. Technol. Lett (2)

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical Modeling of Non-Linear Propagation in Uncompensated Optical Transmission Links,” IEEE Photon. Technol. Lett.  23, 742–744 (2011).
[CrossRef]

A. Mecozzi, C. B. Clausen, and M. Shtaif, “System Impact of Intra-Channel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett.  12, 1633–1635 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of Intrachannel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett. 12, 392–394 (2000).
[CrossRef]

J. Lightwave Technol. (4)

Opt. Commun. (1)

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282, 2421–2425 (2009).
[CrossRef]

Opt. Express (2)

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published).
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[CrossRef] [PubMed]

Opt. Lett. (1)

Other (9)

G. A. Korn and T. A. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 2000).

A. Bononi, P. Serena, and N. Rossi, “Modeling Nonlinearity in Coherent Transmissions with Dominant Interpulse-Four-Wave-Mixing,” Proc. ECOC’11, paper We.7.B.4 (2011).

For interleaved RZ (iRZ) we would need two different support pulses for each polarization, so here iRZ is excluded.

F. Vacondio, C. Simonneau, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. ECOC’11, paper We.7.B.1 (2011).

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical Characterization of PM-QPSK Signals after Propagation in Uncompensated Fiber Links,” Proc. ECOC’10, paper P4.07 (2010).

P. Ramantanis and Y. Frignac, “Pattern-dependent nonlinear impairments on QPSK signals in dispersion-managed optical transmission systems,” Proc. ECOC’10, paper Mo.1.C.4 (2010).

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance Prediction for WDM PM-QPSK Transmission over Uncompensated Links,” Proc. OFC’11, paper OThO7 (2011).

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,” Proc. ECOC’11, paper We.7.B.2 (2011).

A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” Proc. OFC’11, paper OWO7 (2011).

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Figures (6)

Fig. 1
Fig. 1

(Left) aNL [mW−2] versus spans N from Eq. (10) (solid) and simulations (symbols). DP-QPSK on Nx100 km SMF links, R=28 Gbaud. (Right) 1dB NLT vs. symbol rate R for: theory 1 = NLT – 1.05 dBm, with NLT as in Eq. (2) (solid lines); simulations from [9] (symbols). DM30 = DM with 30 ps/nm RDPS.

Fig. 2
Fig. 2

(Left) aNL versus symbol rate R for DP-QPSK 20x100 km NDM link. Symbols: simulations. Red line: theory (11) with ( η p , μ ) = ( 3 88 , 6 ) as in Fig. 1. Magenta line: theory (10) with optimized ( η p , μ ) = ( 3 31 , 0.02 ) . (Right) aNL versus spans N for 28 Gbaud DP-QPSK NDM link. Symbols: simulations. Red and Magenta lines: theory.

Fig. 3
Fig. 3

(Left) example of SNR [dB] vs. P [dBm] “Bell” curve and its roots at S0 = 12 dB, along with graphical definition of constrained NLT NLT at 1.76 dB of penalty; (Right) SNR penalty S P = S d B S 0 d B at reference S0 vs. P [dBm]. It is shown in Eq. (18) that the constrained NLT at 1dB of penalty is below NLT by 1.05 dB.

Fig. 4
Fig. 4

Colored dots represent IFWM points on (m,n) plane when infinitely many precursors and postcursors are taken into account. They all have degeneracy factor df = 2, except those on the m = n line (partially degenerate) which have degeneracy df = 1. Points on axes (IXPM) and (0,0) point (pure SPM) should not be included in IFWM count.

Fig. 5
Fig. 5

aNL versus baudrate in 20x100 km SMF NDM coherent link with SP- and DP-QPSK single channel transmission. a N L p c is the per-component NLI coefficient in DP, with a N L p c = 4 a N L D P . The figure shows convergence of a N L p c to the value 3 2 a N L S P theoretically predicted when IFWM is dominant.

Fig. 6
Fig. 6

PWDD versus normalized cumulated dispersion on SMF fiber link (D = 17 ps/nm/km, α = 0.2 dB/km) and transmission at R = 28 Gbaud, corresponding to strength �� = 0.35, for (Left) N = 20 span NDM system with span length zA = 100 km (Right) N = 120 span DM system at 30 ps/nm RDPS and span length zA = 50 km.

Tables (1)

Tables Icon

Table 1 List of Main Symbols Used in the Paper

Equations (46)

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S = P N A + N N L
P ^ N L T = 1 ( 3 S 0 a N L ) 1 / 2
n N L ( t ) = j P Φ N L ( P ) η ( t 1 t 2 ) U ( t + t 1 ) U ( t + t 1 + t 2 ) U ( t + t 2 ) d t 1 d t 2
c N L = j L < γ G > m , n , l s m s l s n η ( ( m l ) ( n l ) )
a N L = η p 8 ( L < γ G > ) 2 m = 1 n = 1 | η ( m n ) | 2
A lim m = 1 n = 1 | η ( m n ) | 2 1 1 | η ( t 1 t 2 ) | 2 d t 1 d t 2 = 1 ln ( τ ) | η ( τ ) | 2 d τ .
A lim 1 τ M ln ( τ ) | η ( τ ) | 2 d τ ln ( τ M ) 1 τ M | η ( τ ) | 2 d τ ln ( τ M ) 0 | η ( τ ) | 2 d τ .
D E N | η ( τ ) | 2 d τ = J 2 ( c ) d c 2 π
N U M τ 2 | η ( τ ) | 2 d τ = c 2 [ J ( c ) + c J ( c ) ] 2 d c 2 π .
a N L η p 8 ( L < γ G > ) 2 D E N 2 ln ( μ N U M D E N ) .
a N L η P ( γ α ) 2 N π | 𝒮 | ln ( 4 μ 5 ( α z A N ) 2 | 𝒮 | )
η ( t 1 t 2 ) p ( t 1 m ) p ( t 2 n ) p ( t 1 + t 2 l ) d t 1 d t 2 η ( ( m l ) ( n l ) )
P N L T = ( N A 2 a N L ) 1 3
S N L T P N L T 3 2 N A = ( 3 3 a N L ( N A 2 ) 2 ) 1 3
N ^ A = 2 ( 3 S 0 ) 3 / 2 a N L 1 / 2 .
y 1 = 2 p 3 cos ( α 3 ) y 2 , 3 = 2 p 3 cos ( α 3 ± π 3 )
P M = 3 S 0 N ^ A cos ( arcos ( N A / N ^ A ) 3 ) P m = 3 S 0 N ^ A cos ( 2 π arcos ( N A / N ^ A ) 3 ) .
S P M = 3 N ^ A N A cos ( arcos ( N A / N ^ A ) 3 ) S P m = 3 N ^ A N A cos ( 2 π arcos ( N A / N ^ A ) 3 ) .
P ^ N L T P ^ 1 = 3 2 S 0 N ^ A 3 S 0 N ^ A cos ( 2 π arcos ( x 1 ) 3 ) 1.273
A ( z , t ) z = g ( z ) 2 A ( z , t ) + j β 2 ( z ) R 2 2 2 A ( z , t ) t 2 j γ ( z ) | A 2 | A ( z , t )
A ˜ ( z , ω ) z = g ( z ) j ω 2 β 2 ( z ) R 2 2 A ˜ ( z , ω ) j γ ( z ) A ˜ ( z , ω + ω 1 ) A ˜ ( z , ω + ω 1 + ω 2 ) A ˜ ( z , ω + ω 2 ) d ω 1 2 π d ω 2 2 π
A ˜ ( z , ω ) = P 0 e ln G ( z ) + j C ( z ) ω 2 2 U ˜ ( z , ω )
U ˜ ( z , ω ) z = j γ ( z ) P 0 G ( z ) e j C ( z ) ω 1 ω 2 U ˜ ( z , ω + ω 1 ) . U ˜ ( z , ω + ω 1 + ω 2 ) U ˜ ( z , ω + ω 2 ) d ω 1 2 π d ω 2 2 π .
< γ G > 1 L 0 L γ ( s ) G ( s ) d s .
U ˜ ( z , ω ) z = j Φ N L ( P 0 ) γ ( z ) G ( z ) e j C ( z ) ω 1 ω 2 L < γ G > U ˜ ( z , ω + ω 1 ) . U ˜ ( z , ω + ω 1 + ω 2 ) U ˜ ( z , ω + ω 2 ) d ω 1 2 π d ω 2 2 π .
U ˜ ( L , ω ) = U ˜ ( 0 , ω ) j Φ N L ( P 0 ) 0 L γ ( s ) G ( s ) e j C ( s ) ω 1 ω 2 d s L < γ G > . U ˜ ( 0 , ω + ω 1 ) U ˜ ( 0 , ω + ω 1 + ω 2 ) U ˜ ( 0 , ω + ω 2 ) d ω 1 2 π d ω 2 2 π
η ˜ ( w ) 0 L γ ( s ) G ( s ) e j C ( s ) w d s 0 L γ ( s ) G ( s ) d s
U ˜ N L ( ω ) j Φ N L ( P 0 ) η ˜ ( ω 1 ω 2 ) U ˜ ( 0 , ω + ω 1 ) U ˜ ( 0 , ω + ω 1 + ω 2 ) U ˜ ( 0 , ω + ω 2 ) d ω 1 2 π d ω 2 2 π .
r ˜ ( ω ) = P 0 e ln G ( L ) + j ξ tot ω 2 2 [ U ˜ ( 0 , ω ) + U ˜ N L ( ω ) ]
c N L = j L < γ G > m , n s m s m + n s n η ( m n ) = j L < γ G > m , n ( X m ( X n X m + n * + Y n Y m + n * ) Y m ( X n X m + n * + Y n Y m + n * ) ) η ( n m )
a N L = Var [ c N L ] = ( L < γ G > ) 2 m , n Var [ S m n ] | η ( m n ) | 2
a N L ( L < γ G > ) 2 = 2 ( ( m , n ) : m n 1 8 d f 2 | η ( m n ) | 2 + ( m , n ) 1 8 | η ( m n ) | 2 )
a N L D P ( L < γ G > ) 2 = 2 ( 2 8 d f 2 + 4 8 ) m = 1 n = 1 | η ( m n ) | 2
a N L p c ( L < γ G > ) 2 = ( 2 d f 2 + 4 ) m = 1 n = 1 | η ( m n ) | 2 .
a N L S P ( L < γ G > ) 2 = ( 2 d f 2 ) m = 1 n = 1 | η ( m n ) | 2 .
| η ( τ ) | 2 d τ = J 2 ( c ) d c 2 π
| η ( τ ) | 2 d τ = | 1 | ω 1 | J ( 1 ω ) | 2 d ω 2 π = J 2 ( 1 ω ) d ω ω 2 1 2 π
τ 2 | η ( τ ) | 2 d τ = c 2 [ J ( c ) + c J ( c ) ] 2 d c 2 π .
J N D M ( c ) = 1 N k = 0 N 1 J s ( c k ξ s )
N U M = k = 0 N 1 0 c 2 ( J s ( c k ξ s ) + c J s ( c k ξ s ) ) 2 d c 2 π N 2 = 2 4 ξ s 4 + 2 2 ξ s 2 𝒮 2 + 2 1 ξ s 𝒮 3 + 𝒮 4 8 π N 2 𝒮 3
D E N = k = 0 N 1 0 ( J s ( c k ξ s ) ) 2 d c 2 π N 2 = 1 4 π N 𝒮 .
τ rms N D M ( 2 N 4 5 ξ s 4 + 2 N 2 3 ξ s 2 𝒮 2 + 2 N 2 ξ s 𝒮 3 2 𝒮 2 ) 1 2 = 1 5 ( α z A N ) 2 𝒮
J D M ( c ) = { 1 e c ξ pre 𝒮 ξ in if ξ pre < c < ξ pre + ξ in e c ξ pre ξ in 𝒮 e c ξ pre 𝒮 ξ in if c > ξ pre + ξ in
( J + c J ) 2 = { [ 1 e c ξ pre 𝒮 ( 1 c 𝒮 ) ] 2 ξ in 2 if ξ pre < c < ξ pre + ξ in ( 1 c 𝒮 ) 2 e 2 ( c ξ pre ) 𝒮 ( 1 e ξ in 𝒮 ) 2 ξ in 2 if c > ξ pre + ξ in
N U M = 1 2 π ξ in 2 6 𝒮 { e ξ s 𝒮 [ 6 ( ξ in + ξ pre ) 4 + 12 ( ξ in + ξ pre ) 3 𝒮 + 30 ( ξ in + ξ pre ) 2 𝒮 2 + 54 ( ξ in + ξ pre ) 𝒮 3 + + 51 𝒮 4 ] + [ 3 ξ in 4 + 6 ξ pre 4 + 12 ξ pre 3 𝒮 + 30 ξ pre 2 𝒮 2 + 54 ξ pre 𝒮 3 + 51 𝒮 4 + 2 ξ in 3 ( 6 ξ pre + 𝒮 ) + + 3 ξ in ( 2 ξ pre + 𝒮 ) ( 2 ξ pre 2 + 𝒮 2 ) + 3 ξ in 2 ( 6 ξ pre 2 + 2 ξ pre 𝒮 + 𝒮 2 ) ] } .
D E N = ξ in 𝒮 ( 1 e ξ in 𝒮 ) 2 π ξ in 2

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