## Probability based modeling of cross phase modulation in 16-QAM dispersion compensated coherent systems |

Optics Express, Vol. 19, Issue 14, pp. 13334-13342 (2011)

http://dx.doi.org/10.1364/OE.19.013334

Acrobat PDF (924 KB)

### Abstract

A probability-based model is developed to describe cross phase modulation in multichannel multilevel amplitude/phase modulated coherent systems. Standard deviation of nonlinear phase-shift is evaluated in 16-QAM coherent systems accordingly and by numerical simulation for different values of chromatic dispersion and symbol rate. Furthermore, an error analysis is provided to evaluate the accuracy of the model which demonstrates maximum relative error of 12% in the field of interest.

© 2011 OSA

## 1. Introduction

1. C. Xie, “Interchannel nonlinearities in coherent polarization-division multiplexed quadrature-phase-shift- keying systems,” IEEE Photon. Technol. Lett. **21**, 274–276 (2009). [CrossRef]

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

4. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express **16**(2), 880–888 (2008). [CrossRef] [PubMed]

5. E. 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 **16**, 16124–16137 (2008). [CrossRef] [PubMed]

6. Z. Tao, L. Li, L. Liu, W. Yan, H. Nakashima, T. Tanimura, S. Oda, T. Hoshida, and J. C. Rasmussen, “Improvements to digital carrier phase recovery algorithm for high-performance optical coherent receivers,” IEEE J. Sel. Top. Quantum Electron. **16**(5), 1201–1209 (2010). [CrossRef]

7. K. P. Ho, “Cross-phase modulation-induced nonlinear phase noise for quadriphase-shift-keying signals,” in *Impact of Nonlinearities on Fiber Optic Communications*, S. Kumar ed. (Springer, 2011), pp. 325–341. [CrossRef]

8. G. Bellotti, M. Varani, C. Francia, and A. Bononi, “Intensity distortion induced by cross-phase modulation and chromatic dispersion in optical-fiber transmissions with dispersion compensation,” IEEE Photon. Technol. Lett. **10**(12), 1745–1747 (1998). [CrossRef]

10. R. Q. Hui, K. R. Demarest, and C. T. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightwave Technol. **17**(6), 1018–1026 (1999). [CrossRef]

12. H. Kim, “Cross-phase-modulation-induced nonlinear phase noise in WDM direct-detection DPSK systems,” J. Lightwave Technol. **21**(8), 1770–1774 (2003). [CrossRef]

13. Y. Mori, C. Zhang, K. Igarashi, K. Katoh, and K. Kikuchi, “Unrepeated 200-km transmission of 40-Gbit/s 16-QAM signals using digital coherent receiver,” Opt. Express **17**(3), 1435–1441 (2009). [CrossRef] [PubMed]

14. P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. **28**(4), 547–556 (2010). [CrossRef]

15. V. Tavassoli and T. E. Darcie, “An analytical method for performance evaluation of a DQPSK channel in presence of OOK signal,” Proc. SPIE **7386**, 73861C (2009). [CrossRef]

17. Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “A simplified model for nonlinear cross-phase modulation in hybrid optical coherent system,” Opt. Express **17**, 13860–13868 (2009). [CrossRef] [PubMed]

## 2. Principles and limits of the model

*ϕ*, prompted by the optical power

_{NL}*P*of any other channel is equal to 2

_{i}*γP*where

_{i}z*γ*is fiber nonlinear parameter and

*z*is the length of interaction.

*P*changes with time and degrades from its ideal shape because of filters and fiber impairments such as CD. The solution for Δ

_{i}*ϕ*is generally achieved by solving the Coupled Nonlinear Schrödinger Equations (CNLSE) numerically and the bit error rate is calculated according to deviations of nonlinear phase shift. These solutions are complicated and offer little insight. We seek a more intuitive approach.

_{NL}### 2.1. Assumptions

17. Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “A simplified model for nonlinear cross-phase modulation in hybrid optical coherent system,” Opt. Express **17**, 13860–13868 (2009). [CrossRef] [PubMed]

### 2.2. Variance of the phase shift

*P*

_{1},

*P*

_{2}and

*P*

_{3}are present, as deduced from the constellation shown in Fig. 1(a). A point on a probe signal encounters a random sequence of three power levels which decay due to fiber attenuation,

*α*, as shown in Fig. 1(b) and therefore, nonlinear impact of the first colliding symbol is greater than the second one and so on. The nonlinear impact at the end of the fiber is negligible and we would only consider the interaction along the effective length of fiber. Waveform distortion due to CD shown in Fig. 1(b) at different positions is considered a secondary effect in the model and is neglected. In building the phase shift sample space, it is important to notice that the occurrence probability of the three power levels are not the same. Half of the constellation points correspond to the middle power,

*P*

_{2}, a quarter of constellation points correspond to high power,

*P*

_{3}, and another quarter correspond to low power,

*P*

_{1}. Consequently the occurrence probability of

*P*

_{1}or

*P*

_{3}is 0.25 while probability of

*P*

_{2}is 0.5.

*L*, it experiences XPM from

_{e}*N*symbols of pump signal, which is equal to where

_{W}*B*is symbol rate,

_{S}*D*is the fiber dispersion coefficient, Δ

*ν*is channel spacing,

*λ*is optical wavelength and c is the speed of light in vacuum.

*m*th walk-off symbol as: where

*L*=

_{s}*L*/

_{e}*N*is symbol walk-off length. The nonlinear phase shift caused by

_{W}*m*th symbol,

*δϕ*(

*m*), is

*P*

_{1}Φ(

*m*) with a probability of 1/4,

*P*

_{2}Φ(

*m*) with a probability of 1/2 and

*P*

_{3}Φ(

*m*) with a probability of 1/4. Simple calculations give us the following relationship for variance of the total nonlinear phase shift: where

*N*results in the following approximation for variance of nonlinear phase shift: It should be noted that we have not considered any phase recovery method which may enhance the system performance.

_{W}*N*. As the graph shows, the relative error is less than 10% for values of

_{W}*N*bigger than 3.

_{W}*N*is number of spans and

_{s}*σ*is standard deviation of phase shift in one span.

_{ϕ}### 2.3. Accuracy of model

*N*since the probe sample point passes over the pump symbol faster, but it could be of more concern for smaller values of

_{W}*N*.

_{W}*σ*is given by Eq. (3) and

_{ϕ}*σ*is the standard deviation of nonlinear phase shift given by CNLSE simulation.

_{ϕS}## 3. Results and comparison

*.*

^{TM}^{2}.km) dispersion slope, 2.6×10

^{−20}m

^{2}/W nonlinear index and effective core area of 80

*μ*m

^{2}is used. ASE is neglected, signals are dispersion compensated and no distributed amplification is performed. Channel spacing is set to 50 GHz and the probe is at 1551.66 nm. The bandwidth of the optical filters are twice the symbol rate, and the probe launched power is kept at a low value to minimize the effect of SPM on

*σ*.

_{ϕ}*σ*. Similar graphs are shown in Fig. 3(b) for 15 Gbaud and in Fig. 3(c) for 20 Gbaud. Graphs of Fig. (3) reveal that for any of the three symbol rates, the difference between results from simulation and the model are smaller for bigger values of dispersion. Also it can be seen that for any value of dispersion, the difference between the results from simulation and the model are smaller for bigger values of symbol rate. These facts are in agreement with the prediction of subsection 2.3 that the model is a better approximation for larger

_{ϕ}*N*.

_{W}*σ*is given for two sybmol rates (10 and 20 Gbaud) and launched power of 26 mW. The graph shows that agreement between the model and simulation breaks down for very small values of dispersion. The model is no longer valid for small values of

_{ϕ}*N*as we previously mentioned, and since the number of walk-off symbols is small for small dispersions the break down is expected. On the other hand, agreement between the model and simulation gets better for larger dispersions since pump/probe synchronization at the beginning of the fiber is less of an issue for larger values of

_{W}*N*.

_{W}*Err*, given in Eq. (8) with symbol rate and fiber dispersion in a contour diagram. The values of

*Err*are less than 12% and can get as samll as 2% in the given range of dispersion parameters and symbol rates. Higher values of dispersion and symbol rate product, or equivalently higher number of walk-off symbols,

*N*, results in

_{W}*Err*closers to zero and better match between the model and simulation.

## 4. Conclusion

## References and links

1. | C. Xie, “Interchannel nonlinearities in coherent polarization-division multiplexed quadrature-phase-shift- keying systems,” IEEE Photon. Technol. Lett. |

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

3. | D. S. Ly-Gagnon, S. Tsukarnoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. |

4. | X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express |

5. | E. 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 |

6. | Z. Tao, L. Li, L. Liu, W. Yan, H. Nakashima, T. Tanimura, S. Oda, T. Hoshida, and J. C. Rasmussen, “Improvements to digital carrier phase recovery algorithm for high-performance optical coherent receivers,” IEEE J. Sel. Top. Quantum Electron. |

7. | K. P. Ho, “Cross-phase modulation-induced nonlinear phase noise for quadriphase-shift-keying signals,” in |

8. | G. Bellotti, M. Varani, C. Francia, and A. Bononi, “Intensity distortion induced by cross-phase modulation and chromatic dispersion in optical-fiber transmissions with dispersion compensation,” IEEE Photon. Technol. Lett. |

9. | A. V. T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. |

10. | R. Q. Hui, K. R. Demarest, and C. T. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightwave Technol. |

11. | K. P. Ho, |

12. | H. Kim, “Cross-phase-modulation-induced nonlinear phase noise in WDM direct-detection DPSK systems,” J. Lightwave Technol. |

13. | Y. Mori, C. Zhang, K. Igarashi, K. Katoh, and K. Kikuchi, “Unrepeated 200-km transmission of 40-Gbit/s 16-QAM signals using digital coherent receiver,” Opt. Express |

14. | P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. |

15. | V. Tavassoli and T. E. Darcie, “An analytical method for performance evaluation of a DQPSK channel in presence of OOK signal,” Proc. SPIE |

16. | X. Li, F. Zhang, Z. Chen, and A. Xu, “Suppression of XPM and XPM-induced nonlinear phase noise for RZ-DPSK signals in 40 Gbit/s WDM transmission systems with optimum dispersion mapping,” Opt. Express |

17. | Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “A simplified model for nonlinear cross-phase modulation in hybrid optical coherent system,” Opt. Express |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 20, 2011

Revised Manuscript: May 18, 2011

Manuscript Accepted: May 20, 2011

Published: June 27, 2011

**Citation**

Vahid Tavassoli and Thomas E. Darcie, "Probability based modeling of cross phase modulation in 16-QAM dispersion compensated coherent systems," Opt. Express **19**, 13334-13342 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13334

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### References

- C. Xie, “Interchannel nonlinearities in coherent polarization-division multiplexed quadrature-phase-shift- keying systems,” IEEE Photon. Technol. Lett. 21, 274–276 (2009). [CrossRef]
- E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16, 753–791 (2008). [CrossRef] [PubMed]
- D. S. Ly-Gagnon, S. Tsukarnoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. 24(1), 12–21 (2006). [CrossRef]
- X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008). [CrossRef] [PubMed]
- E. 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 16, 16124–16137 (2008). [CrossRef] [PubMed]
- Z. Tao, L. Li, L. Liu, W. Yan, H. Nakashima, T. Tanimura, S. Oda, T. Hoshida, and J. C. Rasmussen, “Improvements to digital carrier phase recovery algorithm for high-performance optical coherent receivers,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1201–1209 (2010). [CrossRef]
- K. P. Ho, “Cross-phase modulation-induced nonlinear phase noise for quadriphase-shift-keying signals,” in Impact of Nonlinearities on Fiber Optic Communications , S. Kumar ed. (Springer, 2011), pp. 325–341. [CrossRef]
- G. Bellotti, M. Varani, C. Francia, and A. Bononi, “Intensity distortion induced by cross-phase modulation and chromatic dispersion in optical-fiber transmissions with dispersion compensation,” IEEE Photon. Technol. Lett. 10(12), 1745–1747 (1998). [CrossRef]
- A. V. T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 17(2), 178–190 (1999). [CrossRef]
- R. Q. Hui, K. R. Demarest, and C. T. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightwave Technol. 17(6), 1018–1026 (1999). [CrossRef]
- K. P. Ho, Phase Modulated Optical Communication Systems (Springer, 2005).
- H. Kim, “Cross-phase-modulation-induced nonlinear phase noise in WDM direct-detection DPSK systems,” J. Lightwave Technol. 21(8), 1770–1774 (2003). [CrossRef]
- Y. Mori, C. Zhang, K. Igarashi, K. Katoh, and K. Kikuchi, “Unrepeated 200-km transmission of 40-Gbit/s 16-QAM signals using digital coherent receiver,” Opt. Express 17(3), 1435–1441 (2009). [CrossRef] [PubMed]
- P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010). [CrossRef]
- V. Tavassoli and T. E. Darcie, “An analytical method for performance evaluation of a DQPSK channel in presence of OOK signal,” Proc. SPIE 7386, 73861C (2009). [CrossRef]
- X. Li, F. Zhang, Z. Chen, and A. Xu, “Suppression of XPM and XPM-induced nonlinear phase noise for RZ-DPSK signals in 40 Gbit/s WDM transmission systems with optimum dispersion mapping,” Opt. Express 15, 18247–18252 (2007). [CrossRef] [PubMed]
- Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “A simplified model for nonlinear cross-phase modulation in hybrid optical coherent system,” Opt. Express 17, 13860–13868 (2009). [CrossRef] [PubMed]

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