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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 18 — Aug. 29, 2011
  • pp: 16919–16926
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Nonlinearity compensation in multi-rate 28 Gbaud WDM systems employing optical and digital techniques under diverse link configurations

Danish Rafique and Andrew D. Ellis  »View Author Affiliations


Optics Express, Vol. 19, Issue 18, pp. 16919-16926 (2011)
http://dx.doi.org/10.1364/OE.19.016919


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Abstract

Digital back-propagation (DBP) has recently been proposed for the comprehensive compensation of channel nonlinearities in optical communication systems. While DBP is attractive for its flexibility and performance, it poses significant challenges in terms of computational complexity. Alternatively, phase conjugation or spectral inversion has previously been employed to mitigate nonlinear fibre impairments. Though spectral inversion is relatively straightforward to implement in optical or electrical domain, it requires precise positioning and symmetrised link power profile in order to avail the full benefit. In this paper, we directly compare ideal and low-precision single-channel DBP with single-channel spectral-inversion both with and without symmetry correction via dispersive chirping. We demonstrate that for all the dispersion maps studied, spectral inversion approaches the performance of ideal DBP with 40 steps per span and exceeds the performance of electronic dispersion compensation by ~3.5 dB in Q-factor, enabling up to 96% reduction in complexity in terms of required DBP stages, relative to low precision one step per span based DBP. For maps where quasi-phase matching is a significant issue, spectral inversion significantly outperforms ideal DBP by ~3 dB.

© 2011 OSA

1. Introduction

2. Principle of operation

PSI enables the use of SI in circumstances where the effective length (Leff) is less than the amplifier spacing (Lamp) and the dispersion length is sufficiently small, thereby adding dispersion asymmetry. The effect of the added DCF is that of modifying the value of accumulated dispersion exhibited by the pulses during propagation along the nonlinear regions downstream of the SI. Note that an additional DCF must be inserted at the end of the link, or electronic compensation of additional dispersion can be applied [23

23. S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express 15(5), 2120–2126 (2007). [CrossRef] [PubMed]

]. We can see from Fig. 1 that for a conventional uncompensated map, the regions of high nonlinearity correspond to different accumulated dispersion on either side of SI, when SSI is employed. Whilst, the addition of a small piece of DCF applying compensation for (Lamp-Leff) ensures that nonlinear effects occur for similar ranges of accumulated dispersion.

3. Simulation setup

3.1 Transmitter

Figure 2
Fig. 2 Simulation setup for 28 Gbaud PM-mQAM (m = 4, 64) transmission system. Link: Dispersion profile as a function of number of spans for 0% and 90% inline dispersion compensation. Also, spectral inversion and pre-compensated spectral inversion architectures are shown.
illustrates the simulation setup. The transmission system comprised nine 28 Gbaud WDM channels, employing PM-mQAM formats with a channel spacing of 50 GHz. A multi-rate system was employed with a 336 Gb/s central test-channel as PM-64QAM, and neighbouring 112 Gb/s channels as PM-4QAM channels. For all the carriers, both the polarization states were modulated independently using de-correlated 215 and 216 pseudo-random bit sequences (PRBS) with different random number seeds, for x- and y-polarization states, respectively. Each PRBS was de-multiplexed separately into two multi-level output symbol streams which were used to modulate an in-phase and a quadrature phase carrier. The optical transmitters consisted of continuous wave laser sources (5 kHz line-width), followed by two nested Mach-Zehnder Modulator structures for x- and y-polarization states, and the two polarization states were combined using an ideal polarization beam combiner. The simulation conditions ensured 16 samples per symbol with 213 symbols per polarization (98,304 bits in total).

3.2 Transmission link

3.3 Spectral inversion

3.4 Receiver and digital back-propagation

At the coherent receiver the central PM-64QAM channel was de-multiplexed, pre-amplified, coherently-detected using four balanced detectors to give the baseband electrical signal and sampled at 2 samples per symbol. If SI was omitted, transmission impairments were digitally compensated via single-channel DBP (SC-DBP), which was numerically implemented by up-sampling the received signal to 16 samples/symbol and reconstructing the optical field from the in-phase and quadrature samples, followed by split-step Fourier method based solution of the nonlinear Schrödinger equation. Note that we considered a high number of samples per symbol to enable high DBP precision; however it has been shown previously that similar performance may be achieved with only 2 samples/symbol [24

24. D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation format,” IEICE Trans. Commun. E94-B(7), 1815–1822 (2011). [CrossRef]

]. We employed multi-precision DBP where step-size was varied from 1 step per span to 40 steps per span (ideal). The performance with 2 steps per span here is close to that observed for the modified method of [17

17. D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express 19(10), 9453–9460 (2011). [CrossRef] [PubMed]

,18

18. L. Lei, T. Zhenning, D. Liang, Y. Weizhen, O. Shoichiro, T. Takahito, H. Takeshi, and C. R. Jens, “Implementation efficient nonlinear equalizer based on correlated digital backpropagation,” Optical Fiber Communication Conference, OWW3 (2011).

] with 1 step per span and is treated as a lower bound to low complexity. In context of recently proposed DBP simplification algorithms evaluated using a 4QAM baed system, the higher-order formats used here are more prone to fibre nonlinearity (high peak-to-average), necessitating higher numbers of required DBP steps to enable optimum performance [20

20. M. Mussolin, D. Rafique, J. Mårtensson, M. Forzati, J. K. Fischer, L. Molle, M. Nölle, C. Schubert, and A. D. Ellis, “Polarization multiplexed 224 Gb/s 16QAM transmission employing digital back-propagation,” European Conference on Optical Communications, accepted for publication (2011).

]. Also, for comparison, electronic dispersion compensation (EDC) was employed using finite impulse response (FIR) filters (adapted using least mean square algorithm). In all cases, polarization de-multiplexing and residual dispersion compensation was performed using 13 tap FIR filters, followed by carrier phase recovery. Finally, the symbol decisions were made, and the performance assessed by direct error counting (converted into an effective Q-factor (Qeff)). All the fibre propagations were carried out using VPItransmissionMaker®v8.5, and the digital signal processing was performed in MATLAB®7.10.

4. Results and discussions

Typical results of our simulations are shown in Fig. 3
Fig. 3 Qeff of central PM-64QAM channel as a function of launch power for various nonlinear compensation techniques. (a) SSMF with no inline dispersion compensation, (b) NZDSF with no inline dispersion compensation, (c) NZDSF with 90% inline dispersion compensation (hybrid transmission)., showing EDC (squares), SSI (circles), PSI (stars), DBP (1 step, down-triangle), DBP (2 steps, left-triangle), DBP (40 steps, up-triangle). (d) Qeff as a function of number of back-propagation steps per span for SSMF (red) and NZDSF (blue) showing DBP (circles with solid fit), SSI (dotted line,) and PSI (thick solid line).
as a function of signal launch power (PL) for the central PM-64QAM channel. The launch power of all the neighbours was fixed at −1 dBm [5

5. D. Rafique and A. D. Ellis, “Nonlinear penalties in dynamic optical networks employing autonomous transponders,” IEEE Photon. Technol. Lett. 23(17), 1213–1215 (2011). [CrossRef]

]. This is a practical approach for next-generation multi-rate networks, since OSNR requirements for low bit-rate channels are low; and hence lower launch power. This approach leads to reduced inter-channel nonlinearities, as demonstrated in [5

5. D. Rafique and A. D. Ellis, “Nonlinear penalties in dynamic optical networks employing autonomous transponders,” IEEE Photon. Technol. Lett. 23(17), 1213–1215 (2011). [CrossRef]

], however the performance improvements enabled by single channel nonlinear compensation may vary depending on either homogenous [6

6. D. Rafique and A. D. Ellis, “Nonlinear penalties in long-haul optical networks employing dynamic transponders,” Opt. Express 19(10), 9044–9049 (2011). [CrossRef] [PubMed]

] or heterogeneous [5

5. D. Rafique and A. D. Ellis, “Nonlinear penalties in dynamic optical networks employing autonomous transponders,” IEEE Photon. Technol. Lett. 23(17), 1213–1215 (2011). [CrossRef]

] traffic is deployed. Figure 3 depicts the performance after EDC, SSI, PSI and DBP. At lower launch powers, the system is noise-limited and Qeff of all approaches overlap. As we increase the launch power, nonlinear effects become significant and the different approaches demonstrate different optimum launch power, reflecting their dissimilar nonlinear compensation effects. Figure 4
Fig. 4 Constellation maps after SI (top) and EDC (bottom) for 28 Gbaud PM-64QAM at optimum launch power. (a) 0% inline compensation (SSMF), PSI and EDC (b) 0% inline compensation (NZDSF), PSI and EDC (c) 90% inline compensation (NZDSF), SSI and EDC.
qualitatively depicts the difference in performance between SI (top) and EDC (bottom) at the optimum launch power, for various dispersion maps studied in Fig. 3.

Having established that PSI matches, or outperforms 40 steps per span DBP, it remains to analyse the implementation complexity and power consumption of two techniques. Figure 3(d) shows the performance of DBP with varying precision or step size per span, both for SSMF and NZDSF. It can be seen that for SSMF, SSI and PSI enable performance improvements equivalent to DBP employing ~5 steps per span and ~22 steps per span, respectively. This can be seen as SSI and PSI enabling 80% and 96% simplifications to one step per span SC-DBP. Likewise, for low dispersion fibre, SSI and PSI enable performance enhancements corresponding to 12 steps per span and 18 steps per span, respectively. These findings clearly demonstrate that even though significant efforts are put into simplifying standard DBP algorithms, conventional simplistic SI radically outperforms DBP technique. However, these considerable savings in application specific integrated circuit (ASIC) complexity and computational load must of course be traded-off with the complexity of adding PSI at appropriate network node, suggesting that a hybrid solution using SI to compensate the bulk of the SPM, and simplified DBP to accommodate the residual penalties due to varying SI location would offer the optimum configuration.

5. Conclusion

Acknowledgments

This work is supported by Science Foundation Ireland and CTVR II under Grant 06/IN/I969 and 10/CE/I1853, respectively.

References and links

1.

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

2.

X. Liu, S. Chandrasekhar, B. Zhu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “448-Gb/s reduced-guard-interval CO-OFDM transmission over 2000 km of ultra-large-area fiber and five 80-GHz-grid ROADMs,” J. Lightwave Technol. 29(4), 483–490 (2011). [CrossRef]

3.

P. J. Winzer, A. H. Gnauck, S. Chandrasekhar, S. Draving, J. Evangelista, and B. Zhu, “Generation and 1,200-km transmission of 448-Gb/s ETDM 56-Gbaud PDM 16-QAM using a single I/Q modulator,” European Conference on Optical Communications, PD2.2 (2010).

4.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) Coherent Optical Transmission over 160 km with an Optical Bandwidth of 5.4 GHz,” Optical Fiber Communication Conference, OThD5 (2010).

5.

D. Rafique and A. D. Ellis, “Nonlinear penalties in dynamic optical networks employing autonomous transponders,” IEEE Photon. Technol. Lett. 23(17), 1213–1215 (2011). [CrossRef]

6.

D. Rafique and A. D. Ellis, “Nonlinear penalties in long-haul optical networks employing dynamic transponders,” Opt. Express 19(10), 9044–9049 (2011). [CrossRef] [PubMed]

7.

A. Nag, M. Tornatore, and B. Mukherjee, “Optical network design with mixed line rates and multiple modulation formats,” J. Lightwave Technol. 28(4), 466–475 (2010). [CrossRef]

8.

C. Meusburger, D. A. Schupke, and A. Lord, “Optimizing the migration of channels with higher bitrates,” J. Lightwave Technol. 28(4), 608–615 (2010). [CrossRef]

9.

M. Suzuki, I. Morita, N. Edagawa, S. Yamamoto, H. Taga, and S. Akiba, “Reduction of Gordon-Haus timing jitter by periodic dispersion compensation in soliton transmission,” Electron. Lett. 31(23), 2027–2029 (1995). [CrossRef]

10.

D. D. Marcenac, D. Nesset, A. E. Kelly, M. Brierley, A. D. Ellis, D. G. Moodie, and C. W. Ford, “40 Gbit/s transmission over 406 km of NDSF using mid-span spectral inversion by four-wave-mixing in a 2 mm long semiconductor optical amplifier,” Electron. Lett. 33(10), 879–880 (1997). [CrossRef]

11.

I. Brener, B. Mikkelsen, K. Rottwitt, W. Burkett, G. Raybon, J. B. Stark, K. Parameswaran, M. H. Chou, M. M. Fejer, E. E. Chaban, R. Harel, D. L. Philen, and A. Kosinski, “Cancellation of all Kerr nonlinearities in long fiber spans using a LiNbO3 phase conjugator and Raman amplification,” Optical Fiber Communication Conference, 266–PD33–1 (2000).

12.

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

13.

F. M. Eduardo, Z. Xiang, and G. Li, “Electronic phase conjugation for nonlinearity compensation in fiber communication systems,” Optical Fiber Communication Conference, JWA025 (2011).

14.

P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photon. Technol. Lett. 18(9), 995–997 (2006). [CrossRef]

15.

G. Li, E. Mateo, and L. Zhu, “Compensation of nonlinear effects using digital coherent receivers,” Optical Fiber Communication Conference, OWW1 (2011).

16.

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

17.

D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express 19(10), 9453–9460 (2011). [CrossRef] [PubMed]

18.

L. Lei, T. Zhenning, D. Liang, Y. Weizhen, O. Shoichiro, T. Takahito, H. Takeshi, and C. R. Jens, “Implementation efficient nonlinear equalizer based on correlated digital backpropagation,” Optical Fiber Communication Conference, OWW3 (2011).

19.

L. B. Du and A. J. Lowery, “Experimental demonstration of XPM compensation for CO-OFDM systems with periodic dispersion maps,” Optical Fiber Communication Conference, OWW2 (2011).

20.

M. Mussolin, D. Rafique, J. Mårtensson, M. Forzati, J. K. Fischer, L. Molle, M. Nölle, C. Schubert, and A. D. Ellis, “Polarization multiplexed 224 Gb/s 16QAM transmission employing digital back-propagation,” European Conference on Optical Communications, accepted for publication (2011).

21.

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]

22.

M. Shtaif and M. Eiselt, “Analysis of intensity interference caused by cross-phase modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10(7), 979–981 (1997). [CrossRef]

23.

S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express 15(5), 2120–2126 (2007). [CrossRef] [PubMed]

24.

D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation format,” IEICE Trans. Commun. E94-B(7), 1815–1822 (2011). [CrossRef]

25.

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17(4), 1448–1453 (1978). [CrossRef]

26.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000). [CrossRef]

27.

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]

28.

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]

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(190.5040) Nonlinear optics : Phase conjugation

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: June 20, 2011
Revised Manuscript: July 27, 2011
Manuscript Accepted: August 8, 2011
Published: August 15, 2011

Citation
Danish Rafique and Andrew D. Ellis, "Nonlinearity compensation in multi-rate 28 Gbaud WDM systems employing optical and digital techniques under diverse link configurations," Opt. Express 19, 16919-16926 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16919


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References

  1. A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. 28(4), 423–433 (2010). [CrossRef]
  2. X. Liu, S. Chandrasekhar, B. Zhu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “448-Gb/s reduced-guard-interval CO-OFDM transmission over 2000 km of ultra-large-area fiber and five 80-GHz-grid ROADMs,” J. Lightwave Technol. 29(4), 483–490 (2011). [CrossRef]
  3. P. J. Winzer, A. H. Gnauck, S. Chandrasekhar, S. Draving, J. Evangelista, and B. Zhu, “Generation and 1,200-km transmission of 448-Gb/s ETDM 56-Gbaud PDM 16-QAM using a single I/Q modulator,” European Conference on Optical Communications, PD2.2 (2010).
  4. M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) Coherent Optical Transmission over 160 km with an Optical Bandwidth of 5.4 GHz,” Optical Fiber Communication Conference, OThD5 (2010).
  5. D. Rafique and A. D. Ellis, “Nonlinear penalties in dynamic optical networks employing autonomous transponders,” IEEE Photon. Technol. Lett. 23(17), 1213–1215 (2011). [CrossRef]
  6. D. Rafique and A. D. Ellis, “Nonlinear penalties in long-haul optical networks employing dynamic transponders,” Opt. Express 19(10), 9044–9049 (2011). [CrossRef] [PubMed]
  7. A. Nag, M. Tornatore, and B. Mukherjee, “Optical network design with mixed line rates and multiple modulation formats,” J. Lightwave Technol. 28(4), 466–475 (2010). [CrossRef]
  8. C. Meusburger, D. A. Schupke, and A. Lord, “Optimizing the migration of channels with higher bitrates,” J. Lightwave Technol. 28(4), 608–615 (2010). [CrossRef]
  9. M. Suzuki, I. Morita, N. Edagawa, S. Yamamoto, H. Taga, and S. Akiba, “Reduction of Gordon-Haus timing jitter by periodic dispersion compensation in soliton transmission,” Electron. Lett. 31(23), 2027–2029 (1995). [CrossRef]
  10. D. D. Marcenac, D. Nesset, A. E. Kelly, M. Brierley, A. D. Ellis, D. G. Moodie, and C. W. Ford, “40 Gbit/s transmission over 406 km of NDSF using mid-span spectral inversion by four-wave-mixing in a 2 mm long semiconductor optical amplifier,” Electron. Lett. 33(10), 879–880 (1997). [CrossRef]
  11. I. Brener, B. Mikkelsen, K. Rottwitt, W. Burkett, G. Raybon, J. B. Stark, K. Parameswaran, M. H. Chou, M. M. Fejer, E. E. Chaban, R. Harel, D. L. Philen, and A. Kosinski, “Cancellation of all Kerr nonlinearities in long fiber spans using a LiNbO3 phase conjugator and Raman amplification,” Optical Fiber Communication Conference, 266–PD33–1 (2000).
  12. S. L. Jansen, D. Borne, B. Spinnler, S. Calabrò, H. Suche, P. M. Krummrich, W. Sohler, G. D. Khoe, and H. Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24(1), 54–64 (2006). [CrossRef]
  13. F. M. Eduardo, Z. Xiang, and G. Li, “Electronic phase conjugation for nonlinearity compensation in fiber communication systems,” Optical Fiber Communication Conference, JWA025 (2011).
  14. P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photon. Technol. Lett. 18(9), 995–997 (2006). [CrossRef]
  15. G. Li, E. Mateo, and L. Zhu, “Compensation of nonlinear effects using digital coherent receivers,” Optical Fiber Communication Conference, OWW1 (2011).
  16. E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol. 28(6), 939–951 (2010). [CrossRef]
  17. D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express 19(10), 9453–9460 (2011). [CrossRef] [PubMed]
  18. L. Lei, T. Zhenning, D. Liang, Y. Weizhen, O. Shoichiro, T. Takahito, H. Takeshi, and C. R. Jens, “Implementation efficient nonlinear equalizer based on correlated digital backpropagation,” Optical Fiber Communication Conference, OWW3 (2011).
  19. L. B. Du and A. J. Lowery, “Experimental demonstration of XPM compensation for CO-OFDM systems with periodic dispersion maps,” Optical Fiber Communication Conference, OWW2 (2011).
  20. M. Mussolin, D. Rafique, J. Mårtensson, M. Forzati, J. K. Fischer, L. Molle, M. Nölle, C. Schubert, and A. D. Ellis, “Polarization multiplexed 224 Gb/s 16QAM transmission employing digital back-propagation,” European Conference on Optical Communications, accepted for publication (2011).
  21. 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]
  22. M. Shtaif and M. Eiselt, “Analysis of intensity interference caused by cross-phase modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10(7), 979–981 (1997). [CrossRef]
  23. S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express 15(5), 2120–2126 (2007). [CrossRef] [PubMed]
  24. D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation format,” IEICE Trans. Commun. E94-B(7), 1815–1822 (2011). [CrossRef]
  25. R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17(4), 1448–1453 (1978). [CrossRef]
  26. A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000). [CrossRef]
  27. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]
  28. 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]

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