## Influence of PMD on fiber nonlinearity compensation using digital back propagation |

Optics Express, Vol. 20, Issue 13, pp. 14406-14418 (2012)

http://dx.doi.org/10.1364/OE.20.014406

Acrobat PDF (1026 KB)

### Abstract

With ideal nonlinearity compensation using digital back propagation (DBP), the transmission performance of an optical fiber channel has been considered to be limited by nondeterministic nonlinear signal-ASE interaction. In this paper, we conduct theoretical and numerical study on nonlinearity compensation using DBP in the presence of polarization-mode dispersion (PMD). Analytical expressions of transmission performance with DBP are derived and substantiated by numerical simulations for polarization-division-multiplexed systems under the influence of PMD effects. We find that nondeterministic distributed PMD impairs the effectiveness of DBP-based nonlinearity compensation much more than nonlinear signal-ASE interaction, and is therefore the fundamental limitation to single-mode fiber channel capacity.

© 2012 OSA

## 1. Introduction

1. E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express **16**(2), 753–791 (2008). [CrossRef] [PubMed]

3. W. Shieh and C. Athaudage, “coherent optical orthogonal frequency division multiplexing,” Electron. Lett. **42**(10), 587–589 (2006). [CrossRef]

4. A. Chowdhury, G. Raybon, R. J. Essiambre, J. H. Sinsky, A. Adamiecki, J. Leuthold, C. R. Doerr, and S. Chandrasekhar, “Compensation of intrachannel nonlinearities in 40-Gb/s pseudolinear systems using optical-phase conjugation,” J. Lightwave Technol. **23**(1), 172–177 (2005). [CrossRef]

5. S. L. Jansen, D. van den Borne, B. Spinnler, S. Calabrò, H. Suche, P. M. Krummrich, W. Sohler, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra-long-haul phase-shift-keyed transmission,” J. Lightwave Technol. **24**(1), 54–64 (2006). [CrossRef]

6. W. Shieh and Y. Tang, “Ultrahigh-speed signal transmission over nonlinear and dispersive fiber optic channel: the multicarrier advantage,” IEEE Photon. J. **2**(3), 276–283 (2010). [CrossRef]

7. X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. **27**(16), 3632–3640 (2009). [CrossRef]

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

17. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. **2**(5), 816–832 (2010). [CrossRef]

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

17. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. **2**(5), 816–832 (2010). [CrossRef]

18. 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]

19. D. Rafique and A. D. 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]

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

17. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. **2**(5), 816–832 (2010). [CrossRef]

19. D. Rafique and A. D. 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]

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

**2**(5), 816–832 (2010). [CrossRef]

## 2. Theoretical derivations of nonlinear transmission performance using DBP

21. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express **16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

22. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express **18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

_{fi, fj}and

_{ fk}would produce a mixing product at frequency of

_{fg=fi+fj−fk}. For PDM systems, the propagation equation for the FWM component at frequency

_{fg}, is given by [23

23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. **3**(2), 158–173 (2011). [CrossRef]

_{cg'}is the FWM component invoked by subcarrier

_{ci},

_{cj}and

_{ck}after transmission of

*M*fiber spans, and can be expressed as [22

22. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express **18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. **3**(2), 158–173 (2011). [CrossRef]

_{α},

_{L},

_{ζ}and

_{β2}are fiber loss coefficient, fiber length per span, residual dispersion ratio per span and group-velocity dispersion parameter respectively, superscript ‘ + ’ stands for Hermitian conjugate.

_{γ=8γ0/9}is the third-order nonlinear coefficient for fiber with randomly varying birefringence eigen axis. For systems with

*N*subcarriers, the power of FWM noise generated at

_{fg}can be obtained as [22

22. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express **18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. **3**(2), 158–173 (2011). [CrossRef]

_{PNL,M}is the average FWM noise power generated at

_{fg}and

_{Pi,j,k}is the power of interfering subcarrier.

_{η1}and

_{η2}stand for FWM efficiency for single span and multi-span FWM interference effect [22

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

24. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. **17**(11), 801–803 (1992). [CrossRef] [PubMed]

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

**3**(2), 158–173 (2011). [CrossRef]

_{f=jΔf}and

_{f1=(j−k)Δf}, the nonlinear noise intensity can be expressed as [22

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

**3**(2), 158–173 (2011). [CrossRef]

_{B}is the signal bandwidth and

_{B0}is defined as

_{|β2|/(2π2αB)}.

_{I}is the signal power density and

_{he}is the multi-span FWM interference enhancement factor, which can be approximated as 1 for dispersion uncompensated systems. Then the nonlinear transmission performance such as SNR, maximum SNR, optimum launch power density, spectral efficiency (SE) can be obtained using the closed form solution of Eq. (4) according to [22

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

**3**(2), 158–173 (2011). [CrossRef]

### 2.1 Nonlinear Signal-Noise interaction

_{Ns}spans (‘

*A*’ to ‘

*C’*) and digitally back propagated through

_{Ns}virtual fiber spans (‘

*C'*’ to ‘

*A'*’) with inverse value of fiber parameters used in forward propagation, removing any of the deterministic nonlinear effect. However, with ASE noise added at each amplifier in the forward propagation, the distributive nonlinear signal-ASE interaction cannot be compensated after

_{Ns}spans of digital back propagation. For instance, considering the ASE noise (

_{n0}) generated at

*M*fiber link (Location ‘

_{th}*B’*), the signal and the noise

_{n0}are transmitted from

*M*span (‘

_{th}*B*’) to the receiver (‘

*C*’) after

_{Ns−M}spans forward transmission. With exactly

_{Ns−M}spans of back propagation (from ‘

*C'*’ to ‘

*B'*’), the nonlinear interaction between

_{n0}and signal can be removed. However, after back propagated by further

_{M}spans (from ‘

*B'*’ to ‘

*A'*’), extra nonlinear signal noise interaction is generated, equivalent to the nonlinear signal-ASE interaction of

_{M}spans. The uncompensated nonlinear components generated by

*M*span EDFA can be expressed as

_{th}_{si,j,k=ci,j,k+ni,j,k}is the subcarrier signal contaminated with ASE noise introduced at the

*M*amplifier,

_{th}_{ni,j,k}denoting the ASE noise added to subcarrier

_{i, j or k}, and

_{ci,j,k}is the information symbol for subcarrier

_{i, j}or

_{k}after the first

_{M}spans without added ASE noises. We now carry out analysis using the same procedures as in [22

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

**3**(2), 158–173 (2011). [CrossRef]

_{fg}can be expressed aswhere,

_{P}and

_{Pn}denote power of OFDM signal and ASE noise at each subcarrier. Following the similar derivations to Eqs. (3) and (4) [22

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

**3**(2), 158–173 (2011). [CrossRef]

*M*amplifier can be expressed aswhere,

_{th}_{In=2n0=MeαLh⋅υ⋅NF}is the optical ASE noise density accounting for both polarizations,

_{h}is the Planck constant,

_{υ}is the light frequency, and

_{NF}is the noise figure for each amplifier.

_{B}is entire signal bandwidth, and

_{B0}and

_{he}are expressed as [22

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

**3**(2), 158–173 (2011). [CrossRef]

_{he,M}is the multi-span FWM interference enhancement factor and can be approximated as 1 for dispersion uncompensated systems. Then the overall nonlinear signal-noise beating density caused by all the amplifiers along the transmission links can be expressed asApproximating

_{he}as 1 for dispersion uncompensated systems,

_{IN−ASE}can be further simplified as followingBy introducing a fiber characteristic dependent factor

_{I0,N−ASE}, the concise expressions of nonlinear transmission performance under the influence of nonlinear signal-ASE interaction can be obtained as

_{SNR}is the signal-to-noise ratio,

_{IN−ASEopt},

_{SNRN−ASEmax}and

_{SN−ASEmax}denote optimal launch power density, maximum SNR and maximum spectral efficiency, respectively.

### 2.2 PMD-induced uncompensated nonlinear noise

_{i},

_{j}and

_{k}can be expressed aswhere,

_{ci,j,kp}is the OFDM subcarriers influenced by PMD after transmission of

_{M}spans. It is observed from Eq. (12) that the generated FWM components depend on the polarization states of interfering subcarriers, which change stochastically due to PMD. An intuitive illustration of the PMD influence is as shown in Fig. 2 .

*M*fiber span can be expressed as followingwhere,

_{th}_{((ck+ci)cj+(ck+cj)ci)}denotes the deterministic signals that can be removed by DBP if there is no PMD. Following the derivations in Appendix B, we obtain

_{E{}}’ stands for the ensemble average over fiber polarization state, to be distinguished with ‘

_{〈〉}’ which is the ensemble average over transmitted subcarrier constellations of the two polarizations.

_{Δτm¯≡E{Δτm}=L⋅MDP}is the average DGD after

_{M}spans,

_{DP}denotes fiber PMD parameter and

_{Δτm2¯=3π⋅Δτm¯2/8}is used assuming Maxwellian distribution of

_{Δτm}. We have used over bar as a shorthand for

_{E{}}.

_{Δϖ=2πf1}is the angle frequency difference and

_{f1}denotes the frequency difference between

_{fj}and

_{fk}.

_{fg}of

*M*span averaged over fiber polarization state is given by

_{th}_{Pg,M0}is the generated FWM noise power for systems without DBP-based nonlinear compensation and

_{RPMD}is the normalized residual FWM power ratio after DBP. Similar to the derivation of Eq. (4), nonlinear noise density generated at

*M*span can be obtained by carrying out the following integrations

_{th}_{Ei}is a special function defined as

_{Ei(x)=∫−∞x(et/t)dt}[25].

*PMD*’ stands for quantities being influenced by PMD. Finally, the overall nonlinear noise intensity under the influence of both nonlinear signal-ASE interaction and distributed nondeterministic PMD can be given by

## 3. Simulation results and discussions

**2**(5), 816–832 (2010). [CrossRef]

26. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. **14**(2), 148–157 (1996). [CrossRef]

27. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. **15**(9), 1735–1746 (1997). [CrossRef]

_{ps/km}are shown in Fig. 4 .

_{ps/km}, the maximum SNR difference between distributed nonlinear signal-ASE interaction and PMD impaired systems is 3.8 dB, showing much more severe impact of PMD than nonlinear signal-ASE interaction. Compared to systems without nonlinearity compensation, the maximum SNR of DBP systems with PMD parameter of 0.05 and 0.1

_{ps/km}are improved by 4.3 and 2.7 dB respectively.

## 4. Conclusion

## Appendix A

_{U=(Ux,Uy)T=[(sk+si)sj+(sk+sj)si−(ck+ci)cj−(ck+cj)ci]}, and expanding

_{U}into two polarization components, we obtain

_{P}and

_{Pn}are the power of signal and ASE noise over each subcarrier frequency respectively. By using

_{〈|Ux|2〉=〈|Uy|2〉}, the ensemble average of signal power becomes

## Appendix B

_{U=(Ux,Uy)T=(ckp+cip)cjp+(ckp+cjp)cip−(ck+ci)cj−(ck+cj)ci}, and expanding

_{U}into the two polarization components, we obtain

_{Ux}can be expressed as

_{A}and

_{B}are comprised of polarization dependent and independent terms respectively. The average power of

_{Ux}is expressed as

_{ci}as reference subcarrier and assuming its polarization is unchanged along the transmission, we have

29. A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. **20**(5), 811–821 (2002). [CrossRef]

30. A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett. **26**(10), 675–677 (2001). [CrossRef] [PubMed]

_{j}and

_{k}can be proved as:

_{Δτ2¯=3πΔτ¯2/8}with

_{Δτ¯}denoting the average DGD and

_{Δϖ=2π(fj−fk)}is the angle frequency difference between

_{fj}and

_{fk}. We have omitted the relatively lengthy steps to derive Eq. (29). We also note that the statistics of Jones matrix

_{Mk(j)}defined in this paper is different than that of conventional Jones matrix in [29

29. A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. **20**(5), 811–821 (2002). [CrossRef]

30. A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett. **26**(10), 675–677 (2001). [CrossRef] [PubMed]

_{Mk(j)}as a 2x2 identity matrix for the

_{ith}subcarrier (the reference subcarrier). Substituting Eqs. (28) and (29) into Eq. (26), we obtain

_{U}is obtained as

## References and links

1. | E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express |

2. | S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express |

3. | W. Shieh and C. Athaudage, “coherent optical orthogonal frequency division multiplexing,” Electron. Lett. |

4. | A. Chowdhury, G. Raybon, R. J. Essiambre, J. H. Sinsky, A. Adamiecki, J. Leuthold, C. R. Doerr, and S. Chandrasekhar, “Compensation of intrachannel nonlinearities in 40-Gb/s pseudolinear systems using optical-phase conjugation,” J. Lightwave Technol. |

5. | S. L. Jansen, D. van den Borne, B. Spinnler, S. Calabrò, H. Suche, P. M. Krummrich, W. Sohler, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra-long-haul phase-shift-keyed transmission,” J. Lightwave Technol. |

6. | W. Shieh and Y. Tang, “Ultrahigh-speed signal transmission over nonlinear and dispersive fiber optic channel: the multicarrier advantage,” IEEE Photon. J. |

7. | X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. |

8. | E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. |

9. | E. F. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express |

10. | E. F. Mateo and G. Li, “Compensation of interchannel nonlinearities using enhanced coupled equations for digital backward propagation,” Appl. Opt. |

11. | E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express |

12. | E. F. Mateo, X. Zhou, and G. Li, “Improved digital backward propagation for the compensation of inter-channel nonlinear effects in polarization-multiplexed WDM systems,” Opt. Express |

13. | L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express |

14. | L. B. Du, B. Schmidt, and A. Lowery, “Efficient digital backpropagation for PDM-CO-OFDM optical transmission systems,” OFC’ 2010, paper OTuE2. |

15. | E. Ip and J. M. Kahn, “Nonlinear impairment compensation using backpropagation,” in Optical Fiber, New Developments, C. Lethien, Ed., In-Tech, Vienna Austria, December (2009). |

16. | E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol. |

17. | F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. |

18. | R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. |

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

20. | G. Gao, X. Chen, W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation,” OFC’2012, paper OMA3. |

21. | M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express |

22. | X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express |

23. | W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. |

24. | K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. |

25. | A. Milton and I. Stegun, |

26. | P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. |

27. | D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. |

28. | P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature |

29. | A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. |

30. | A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett. |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.2330) Fiber optics and optical communications : Fiber optics communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 16, 2012

Revised Manuscript: May 14, 2012

Manuscript Accepted: May 14, 2012

Published: June 13, 2012

**Citation**

Guanjun Gao, Xi Chen, and William Shieh, "Influence of PMD on fiber nonlinearity compensation using digital back propagation," Opt. Express **20**, 14406-14418 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14406

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### References

- E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express16(2), 753–791 (2008). [CrossRef] [PubMed]
- S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express15(5), 2120–2126 (2007). [CrossRef] [PubMed]
- W. Shieh and C. Athaudage, “coherent optical orthogonal frequency division multiplexing,” Electron. Lett.42(10), 587–589 (2006). [CrossRef]
- A. Chowdhury, G. Raybon, R. J. Essiambre, J. H. Sinsky, A. Adamiecki, J. Leuthold, C. R. Doerr, and S. Chandrasekhar, “Compensation of intrachannel nonlinearities in 40-Gb/s pseudolinear systems using optical-phase conjugation,” J. Lightwave Technol.23(1), 172–177 (2005). [CrossRef]
- S. L. Jansen, D. van den Borne, B. Spinnler, S. Calabrò, H. Suche, P. M. Krummrich, W. Sohler, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra-long-haul phase-shift-keyed transmission,” J. Lightwave Technol.24(1), 54–64 (2006). [CrossRef]
- W. Shieh and Y. Tang, “Ultrahigh-speed signal transmission over nonlinear and dispersive fiber optic channel: the multicarrier advantage,” IEEE Photon. J.2(3), 276–283 (2010). [CrossRef]
- X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol.27(16), 3632–3640 (2009). [CrossRef]
- E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol.26(20), 3416–3425 (2008). [CrossRef]
- E. F. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express16(20), 16124–16137 (2008). [CrossRef] [PubMed]
- E. F. Mateo and G. Li, “Compensation of interchannel nonlinearities using enhanced coupled equations for digital backward propagation,” Appl. Opt.48(25), F6–F10 (2009). [CrossRef] [PubMed]
- E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express18(14), 15144–15154 (2010). [CrossRef] [PubMed]
- E. F. Mateo, X. Zhou, and G. Li, “Improved digital backward propagation for the compensation of inter-channel nonlinear effects in polarization-multiplexed WDM systems,” Opt. Express19(2), 570–583 (2011). [CrossRef] [PubMed]
- L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express18(16), 17075–17088 (2010). [CrossRef] [PubMed]
- L. B. Du, B. Schmidt, and A. Lowery, “Efficient digital backpropagation for PDM-CO-OFDM optical transmission systems,” OFC’ 2010, paper OTuE2.
- E. Ip and J. M. Kahn, “Nonlinear impairment compensation using backpropagation,” in Optical Fiber, New Developments, C. Lethien, Ed., In-Tech, Vienna Austria, December (2009).
- E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol.28(6), 939–951 (2010). [CrossRef]
- F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J.2(5), 816–832 (2010). [CrossRef]
- 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]
- D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express19(4), 3449–3454 (2011). [CrossRef] [PubMed]
- G. Gao, X. Chen, W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation,” OFC’2012, paper OMA3.
- M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008). [CrossRef] [PubMed]
- X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010). [CrossRef] [PubMed]
- W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J.3(2), 158–173 (2011). [CrossRef]
- K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett.17(11), 801–803 (1992). [CrossRef] [PubMed]
- A. Milton and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Abramowitz and Stegun, eds. (Dover, 1964).
- P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996). [CrossRef]
- D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15(9), 1735–1746 (1997). [CrossRef]
- P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature411(6841), 1027–1030 (2001). [CrossRef] [PubMed]
- A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol.20(5), 811–821 (2002). [CrossRef]
- A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett.26(10), 675–677 (2001). [CrossRef] [PubMed]

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