## Folded digital backward propagation for dispersion-managed fiber-optic transmission |

Optics Express, Vol. 19, Issue 7, pp. 5953-5959 (2011)

http://dx.doi.org/10.1364/OE.19.005953

Acrobat PDF (1050 KB)

### Abstract

In periodically dispersion managed long-haul transmission systems, waveform distortion is dominated by chromatic dispersion. As a result of the periodic waveform evolution, the nonlinear behavior also repeats itself in every dispersion period. It is shown that, under the weakly nonlinear assumption, nonlinear effects accumulated in a large number (*K*) of spans can be approximated by nonlinear effects accumulated in a single span with the same dispersion map and *K* times the nonlinearity. Thus, significant savings in computational load can be achieved in digital compensation of fiber nonlinearity using folded digital backward propagation (DBP). Simulation results show that the required computation for DBP of dispersion managed transoceanic transmission systems can be reduced by up to 2 orders of magnitude with negligible penalty using folded DBP.

© 2011 OSA

## 1. Introduction

1. K. Mukasa, K. Imamura, I. Shimotakahara, T. Yagi, and K. Kokura, “Dispersion compensating fiber used as a transmission fiber: inverse/reverse dispersion fiber,” Opt. Fiber. Commun. **3**(5), 292–339 (2006). [CrossRef]

2. C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E.-D. Schmidt, T. Wuth, J. Geyer, E. De Man, Khoe Giok-Djan, and H. de Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol. **26**(1), 64–72 (2008). [CrossRef]

5. Q. Lin and G. P. Agrawa, “Effects of polarization-mode dispersion on cross-phase modulation in dispersion-managed wavelength-division-multiplexed systems,” J. Lightwave Technol. **22**(4), 977–987 (2004). [CrossRef]

6. B. C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photon. Technol. Lett. **5**(10), 1250–1253 (1993). [CrossRef]

1. K. Mukasa, K. Imamura, I. Shimotakahara, T. Yagi, and K. Kokura, “Dispersion compensating fiber used as a transmission fiber: inverse/reverse dispersion fiber,” Opt. Fiber. Commun. **3**(5), 292–339 (2006). [CrossRef]

7. T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, and K. Kasahara, “Comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise,” J. Lightwave Technol. **21**(9), 1933–1943 (2003). [CrossRef]

8. K.-P. Ho and J. M. Kahn, “Electronic compensation technique to mitigate nonlinear phase noise,” J. Lightwave Technol. **22**(3), 779–783 (2004). [CrossRef]

9. K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. **18**(2), 403–405 (2006). [CrossRef]

11. 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]

13. L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. **46**(16), 1140–1141 (2010). [CrossRef]

14. 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]

15. 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**(20), 16124–16137 (2008). [CrossRef] [PubMed]

16. 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]

17. 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 **18**(14), 15144–15154 (2010). [CrossRef] [PubMed]

18. 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 **18**(16), 17075–17088 (2010). [CrossRef] [PubMed]

## 2. Theory

*L*is one period of the dispersion map. For long-haul fiber-optic transmission, an optimum power exists as a result of the tradeoff between optical signal to noise ratio (OSNR) and nonlinear impairments. The total nonlinear phase shift at the optimum power level is on the order of 1 radian [19

19. 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]

*j*

^{th}fiber span can be expressed aswhere

*D*is the linear operator for dispersion, fiber loss and amplifier gain,

*ε*(to be set to unity) is a parameter indicating that the nonlinear perturbation is small for the reasons given above. The boundary conditions are where

*ε*, we haveEquating to zero the successive terms of the series, we have The boundary conditions are andFirst, we assume that dispersion is completely compensated in each span. As a result, at the end of the first span,andwhere

*2*spans is approximately the same as the nonlinear distortion accumulated in

*1*span with the same dispersion map and

*twice*the nonlinearity. It follows that, assuming weak nonlinearity and periodic dispersion management, the optical field after

*K*spans of propagation can be written aswhich is the solution of the NLSEThe nonlinear term

*γ*. So the NLSE describing the optical propagation in a fiber span with the same dispersion map and

*K*times the nonlinearity can be written asThe equivalence of Eqs. (19) and (20) suggests that DBP for

*K*spans can be folded into a single span with

*K*times the nonlinearity. Assuming that the step size for the split-step implementation of DBP is unchanged, the computational load for the folded DBP can be saved by the folding factor

*K*.

## 3. Simulation

^{2}. The corresponding parameters for the IDF fiber are 0.23 dB/km, −44 ps/nm/km, 0.003/nm and 31 μm

^{2}, respectively. The RDPS is determined by the proportion of SLA fiber to IDF fiber in each span. A piece of fiber at the receiver was used to compensate for the residual dispersion. After de-multiplexing and coherent detection, DSP was performed in Matlab.

13. L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. **46**(16), 1140–1141 (2010). [CrossRef]

17. 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 **18**(14), 15144–15154 (2010). [CrossRef] [PubMed]

20. O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. **21**(1), 61–68 (2003). [CrossRef]

21. K. Nakkeeran, A. B. Moubissi, and P. Tchofo Dinda, “Analytical design of dispersion-managed fiber system with map strength 1.65,” Phys. Lett. A **308**(5-6), 417–425 (2003). [CrossRef]

## 4. Conclusion and discussion

14. 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]

## References and links

1. | K. Mukasa, K. Imamura, I. Shimotakahara, T. Yagi, and K. Kokura, “Dispersion compensating fiber used as a transmission fiber: inverse/reverse dispersion fiber,” Opt. Fiber. Commun. |

2. | C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E.-D. Schmidt, T. Wuth, J. Geyer, E. De Man, Khoe Giok-Djan, and H. de Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol. |

3. | A. Pilipetski, “Nonlinearity management and compensation in transmission systems,” in |

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

5. | Q. Lin and G. P. Agrawa, “Effects of polarization-mode dispersion on cross-phase modulation in dispersion-managed wavelength-division-multiplexed systems,” J. Lightwave Technol. |

6. | B. C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photon. Technol. Lett. |

7. | T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, and K. Kasahara, “Comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise,” J. Lightwave Technol. |

8. | K.-P. Ho and J. M. Kahn, “Electronic compensation technique to mitigate nonlinear phase noise,” J. Lightwave Technol. |

9. | K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. |

10. | L. B. Du and A. J. Lowery, “Fiber nonlinearity compensation for CO-OFDM systems with periodic dispersion maps,” |

11. | 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 |

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

13. | L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. |

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

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

17. | 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 |

18. | 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 |

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

20. | O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. |

21. | K. Nakkeeran, A. B. Moubissi, and P. Tchofo Dinda, “Analytical design of dispersion-managed fiber system with map strength 1.65,” Phys. Lett. A |

22. | V. A. J. M. Sleiffer, D. van den Borne, M. S. Alfiad, S. L. Jansen, and H. de Waardt, “Dispersion management in long-haul 111-Gb/s POLMUX-RZ-DQPSK transmission systems,” in |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

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

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 7, 2011

Revised Manuscript: February 7, 2011

Manuscript Accepted: February 21, 2011

Published: March 16, 2011

**Citation**

Likai Zhu and Guifang Li, "Folded digital backward propagation for dispersion-managed fiber-optic transmission," Opt. Express **19**, 5953-5959 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-5953

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### References

- K. Mukasa, K. Imamura, I. Shimotakahara, T. Yagi, and K. Kokura, “Dispersion compensating fiber used as a transmission fiber: inverse/reverse dispersion fiber,” Opt. Fiber. Commun. 3(5), 292–339 (2006). [CrossRef]
- C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E.-D. Schmidt, T. Wuth, J. Geyer, E. De Man, Khoe Giok-Djan, and H. de Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol. 26(1), 64–72 (2008). [CrossRef]
- A. Pilipetski, “Nonlinearity management and compensation in transmission systems,” in Proceedings of OFC/NFOEC 2010, paper OTuL5.
- R. 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]
- Q. Lin and G. P. Agrawa, “Effects of polarization-mode dispersion on cross-phase modulation in dispersion-managed wavelength-division-multiplexed systems,” J. Lightwave Technol. 22(4), 977–987 (2004). [CrossRef]
- B. C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photon. Technol. Lett. 5(10), 1250–1253 (1993). [CrossRef]
- T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, and K. Kasahara, “Comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise,” J. Lightwave Technol. 21(9), 1933–1943 (2003). [CrossRef]
- K.-P. Ho and J. M. Kahn, “Electronic compensation technique to mitigate nonlinear phase noise,” J. Lightwave Technol. 22(3), 779–783 (2004). [CrossRef]
- K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006). [CrossRef]
- L. B. Du and A. J. Lowery, “Fiber nonlinearity compensation for CO-OFDM systems with periodic dispersion maps,” Proceedings of OFC/NFOEC 2009, paper OTuO1.
- 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. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]
- L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. 46(16), 1140–1141 (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]
- 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(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. Express 18(14), 15144–15154 (2010). [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. Express 18(16), 17075–17088 (2010). [CrossRef] [PubMed]
- 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]
- O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21(1), 61–68 (2003). [CrossRef]
- K. Nakkeeran, A. B. Moubissi, and P. Tchofo Dinda, “Analytical design of dispersion-managed fiber system with map strength 1.65,” Phys. Lett. A 308(5-6), 417–425 (2003). [CrossRef]
- V. A. J. M. Sleiffer, D. van den Borne, M. S. Alfiad, S. L. Jansen, and H. de Waardt, “Dispersion management in long-haul 111-Gb/s POLMUX-RZ-DQPSK transmission systems,” in Proceedings of LEOS Annual Meeting Conference 2009, pp.569–570.

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