## Nonlinear phase noise in coherent optical OFDM transmission systems

Optics Express, Vol. 18, Issue 7, pp. 7347-7360 (2010)

http://dx.doi.org/10.1364/OE.18.007347

Acrobat PDF (871 KB)

### Abstract

We derive an analytical formula to estimate the variance of nonlinear phase noise caused by the interaction of amplified spontaneous emission (ASE) noise with fiber nonlinearity such as self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM) in coherent orthogonal frequency division multiplexing (OFDM) systems. The analytical results agree very well with numerical simulations, enabling the study of the nonlinear penalties in long-haul coherent OFDM systems without extensive numerical simulation. Our results show that the nonlinear phase noise induced by FWM is significantly larger than that induced by SPM and XPM, which is in contrast to traditional WDM systems where ASE-FWM interaction is negligible in quasi-linear systems. We also found that fiber chromatic dispersion can reduce the nonlinear phase noise. The variance of the total phase noise increases linearly with the bit rate, and does not depend significantly on the number of subcarriers for systems with moderate fiber chromatic dispersion.

© 2010 OSA

## 1. Introduction

1. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. **42**(10), 587–588 (2006). [CrossRef]

5. S. Jasen, I. Morita, T. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM transmission with 2-b/s/Hz spectral efficiency over 1000 km of SSMF,” J. Lightwave Technol. **27**(3), 177–188 (2009). [CrossRef]

6. Y. Yang, Y. Ma, and W. Shieh, “Performance impact of inline chromatic dispersion compensation for 107-Gb/s coherent optical OFDM,” IEEE Photon. Technol. Lett. **21**(15), 1042–1044 (2009). [CrossRef]

7. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express **15**(20), 13282–13287 (2007). [CrossRef] [PubMed]

8. 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]

9. A. J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM,” Opt. Express **15**(20), 12965–12970 (2007). [CrossRef] [PubMed]

14. 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]

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

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

32. S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. **27**(21), 4722–4733 (2009). [CrossRef]

27. M. Hanna, D. Boivin, P. Lacourt, and J. Goedgebuer, “Calculation of optical phase jitter in dispersion-managed systems by the use of the moment method,” J. Opt. Soc. Am. B **21**(1), 24–28 (2004). [CrossRef]

## 2. Mathematical analysis for the nonlinear phase noise in coherent OFDM systems

*γ*is the nonlinear coefficient,

8. 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]

*N*is the total number of subcarriers,

### 2.1 SPM and XPM induced nonlinear phase noise

*κ*= 1, 2, …,

*M*, where

*M*is the total number of fiber spans. Consider the noise added by the amplifier located at

*N*degrees of freedom (DOF) instead of infinite degrees of freedom. For a linear system that employs matched filter at the receiver, 2

*N*DOFs accurately describe the noise process [32

32. S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. **27**(21), 4722–4733 (2009). [CrossRef]

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

*N*DOFs) is sufficient to describe the noise field even in a nonlinear system. The total field immediately after the amplifier located at

### 2.2 FWM induced nonlinear phase noise

34. 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]

14. 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]

### 2.3 Total phase noise

*l*

^{th}subcarrier in an OFDM system including the linear phase noise and nonlinear phase noise (induced by interaction between ASE and SPM, XPM and FWM) is as follows:where the first, second, and third terms on the RHS of Eq. (35) are given by Eqs. (33), (15), and (34), respectively.

## 3. Results and discussions

14. 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]

*D*of 1 ps/nm/km and a total launch power of 0 dBm. Here we use only one subcarrier (

*N*= 1) to carry data while the total number of subcarriers is 8 (8th-folder oversampling), so that the nonlinear phase noise model that includes SPM effects alone can be validated. The subcarrier carrying data is located at the central of the OFDM spectrum, corresponding to the first subcarrier

_{e}*U*

_{0}in Fig. 1 due to FFT operation. The signal spectrum before entering into the fiber span is shown in Fig. 2. And in Fig. 3, the solid lines show the analytical linear phase noise and nonlinear phase noise variance induced by SPM only, the dashed line with circles show the numerical simulation results for the variance of linear phase noise and SPM induced nonlinear phase noise, as a function of fiber propagation distance. As can be seen, the agreement is quite good.

*D =*17 ps/nm/km and D = 0 ps/nm/km. Solid lines and circles show the analytical results and the numerical simulation results, respectively. As can be seen from Fig. 7, the nonlinear tolerance increases significantly as the fiber chromatic dispersion parameter increases. It is also shown in Fig. 7 that the total variance of the phase noise initially decreases with launch power since the linear phase noise is dominant at low launch powers. However, as the launch power increases beyond −2 dBm, the variance increases with D = 0 ps/nm/km since nonlinear phase noise becomes dominant at higher powers.

*D*= 17 ps/nm/km and

*D*= 0 ps/nm/km. The total launch power is −3 dBm. Solid lines show the analytical results, while filled circles show the numerical simulation results. From Fig. 8, we see that the variance of the total phase noise scales linearly with the bit rate. This could be explained by the fact that, with the increase of the bit rate, the OFDM symbol time

*BER*as a function of the phase noise variance

27. M. Hanna, D. Boivin, P. Lacourt, and J. Goedgebuer, “Calculation of optical phase jitter in dispersion-managed systems by the use of the moment method,” J. Opt. Soc. Am. B **21**(1), 24–28 (2004). [CrossRef]

## 4. Conclusions

## References and links

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

2. | A. Lowery, L. Du, and J. Armstrong, “Performance of optical OFDM in ultra-long-haul WDM lightwave systems,” J. Lightwave Technol. |

3. | J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. |

4. | A. Sano, E. Yamada, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, and Y. Takatori, “No-guard-interval coherent optical ODFM for 100-Gb/s long-haul WDM transmission,” J. Lightwave Technol. |

5. | S. Jasen, I. Morita, T. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM transmission with 2-b/s/Hz spectral efficiency over 1000 km of SSMF,” J. Lightwave Technol. |

6. | Y. Yang, Y. Ma, and W. Shieh, “Performance impact of inline chromatic dispersion compensation for 107-Gb/s coherent optical OFDM,” IEEE Photon. Technol. Lett. |

7. | A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express |

8. | 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 |

9. | A. J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM,” Opt. Express |

10. | L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express |

11. | X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express |

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

13. | W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express |

14. | 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 |

15. | E. Ip and J. Kahn, “Compensation of dispersion and nonlinear effects using digital back-propagation,” J. Lightwave Technol. |

16. | E. Yamazaki, H. Masuda, A. Sano, T. Yoshimatsu, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, M. Matsui, and Y. Takatori, “Multi-staged nonlinear compensation in coherent receiver for 16,340-km transmission of 111-Gb/s no-guard-interval co-OFDM,” |

17. | J. Gordon and L. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. |

18. | P. Winzer and R. Essiambre, “Advanced modulation formats for high capacity optical transport networks,” J. Lightwave Technol. |

19. | A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. |

20. | K.-P. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc. Am. B |

21. | K.-P. Ho, “Asymptotic probability density of nonlinear phase noise,” Opt. Lett. |

22. | A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. |

23. | A. G. Green, P. P. Mitra, and L. G. Wegener, “Effect of chromatic dispersion on nonlinear phase noise,” Opt. Lett. |

24. | S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. |

25. | C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. |

26. | C. McKinstrie and C. Xie, “Phase jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. |

27. | M. Hanna, D. Boivin, P. Lacourt, and J. Goedgebuer, “Calculation of optical phase jitter in dispersion-managed systems by the use of the moment method,” J. Opt. Soc. Am. B |

28. | K.-P. Ho and H.-C. Wang, “Effect of dispersion on nonlinear phase noise,” Opt. Lett. |

29. | K.-P. Ho, “Error probability of DPSK signals with cross-phase modulation induced nonlinear phase noise,” IEEE J. Sel. Top. Quantum Electron. |

30. | X. Zhu, S. Kumar, and X. Li, “Analysis and comparison of impairments in differential phase-shift keying and on-off keying transmission systems based on the error probability,” Appl. Opt. |

31. | A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. |

32. | S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. |

33. | G. P. Agrawal, |

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

35. | J. G. Proakis, |

**OCIS Codes**

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

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 27, 2010

Revised Manuscript: March 8, 2010

Manuscript Accepted: March 11, 2010

Published: March 24, 2010

**Citation**

Xianming Zhu and Shiva Kumar, "Nonlinear phase noise in coherent optical OFDM transmission systems," Opt. Express **18**, 7347-7360 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-7347

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### References

- W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–588 (2006). [CrossRef]
- A. Lowery, L. Du, and J. Armstrong, “Performance of optical OFDM in ultra-long-haul WDM lightwave systems,” J. Lightwave Technol. 25(1), 131–138 (2007). [CrossRef]
- J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009). [CrossRef]
- A. Sano, E. Yamada, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, and Y. Takatori, “No-guard-interval coherent optical ODFM for 100-Gb/s long-haul WDM transmission,” J. Lightwave Technol. 27(16), 3705–3713 (2009). [CrossRef]
- S. Jasen, I. Morita, T. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM transmission with 2-b/s/Hz spectral efficiency over 1000 km of SSMF,” J. Lightwave Technol. 27(3), 177–188 (2009). [CrossRef]
- Y. Yang, Y. Ma, and W. Shieh, “Performance impact of inline chromatic dispersion compensation for 107-Gb/s coherent optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1042–1044 (2009). [CrossRef]
- A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15(20), 13282–13287 (2007). [CrossRef] [PubMed]
- 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]
- A. J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM,” Opt. Express 15(20), 12965–12970 (2007). [CrossRef] [PubMed]
- L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express 16(24), 19920–19925 (2008). [CrossRef] [PubMed]
- X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express 16(26), 21944–21957 (2008). [CrossRef] [PubMed]
- X. Liu, F. Buchali, and R. 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]
- W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008). [CrossRef] [PubMed]
- 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. Ip and J. Kahn, “Compensation of dispersion and nonlinear effects using digital back-propagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]
- E. Yamazaki, H. Masuda, A. Sano, T. Yoshimatsu, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, M. Matsui, and Y. Takatori, “Multi-staged nonlinear compensation in coherent receiver for 16,340-km transmission of 111-Gb/s no-guard-interval co-OFDM,” ECOC2009, paper 9.4.6.
- J. Gordon and L. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]
- P. Winzer and R. Essiambre, “Advanced modulation formats for high capacity optical transport networks,” J. Lightwave Technol. 24(12), 4711–4728 (2006). [CrossRef]
- A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994). [CrossRef]
- K.-P. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc. Am. B 20(9), 1875–1879 (2003). [CrossRef]
- K.-P. Ho, “Asymptotic probability density of nonlinear phase noise,” Opt. Lett. 28(15), 1350–1352 (2003). [CrossRef] [PubMed]
- A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. 29(7), 673–675 (2004). [CrossRef] [PubMed]
- A. G. Green, P. P. Mitra, and L. G. Wegener, “Effect of chromatic dispersion on nonlinear phase noise,” Opt. Lett. 28(24), 2455–2457 (2003). [CrossRef] [PubMed]
- S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30(24), 3278–3280 (2005). [CrossRef]
- C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27(21), 1887–1889 (2002). [CrossRef]
- C. McKinstrie and C. Xie, “Phase jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8(3), 616–625 (2002). [CrossRef]
- M. Hanna, D. Boivin, P. Lacourt, and J. Goedgebuer, “Calculation of optical phase jitter in dispersion-managed systems by the use of the moment method,” J. Opt. Soc. Am. B 21(1), 24–28 (2004). [CrossRef]
- K.-P. Ho and H.-C. Wang, “Effect of dispersion on nonlinear phase noise,” Opt. Lett. 31(14), 2109–2111 (2006). [CrossRef] [PubMed]
- K.-P. Ho, “Error probability of DPSK signals with cross-phase modulation induced nonlinear phase noise,” IEEE J. Sel. Top. Quantum Electron. 10(2), 421–427 (2004). [CrossRef]
- X. Zhu, S. Kumar, and X. Li, “Analysis and comparison of impairments in differential phase-shift keying and on-off keying transmission systems based on the error probability,” Appl. Opt. 45(26), 6812–6822 (2006). [CrossRef] [PubMed]
- A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. 25(8), 2002–2032 (2007). [CrossRef]
- S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber optics, New York: Academic, 3rd Edition, 2001.
- 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]
- J. G. Proakis, Digital Communications, New York: McGraw-Hill, 4th Edition, 2000, pp. 269–274.

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