## Analyses of wavelength- and polarization-division multiplexed transmission characteristics of optical quadrature-amplitude-modulation signals |

Optics Express, Vol. 19, Issue 19, pp. 17985-17995 (2011)

http://dx.doi.org/10.1364/OE.19.017985

Acrobat PDF (933 KB)

### Abstract

We theoretically study optical transmission characteristics of wavelength-division multiplexed (WDM) and polarization-multiplexed (POLMUX) signals using high-order optical quadrature-amplitude-modulation (QAM) formats up to 256. First, we conduct intensive computer simulations on bit-error rates (BERs) in WDM POLMUX QAM transmission systems and find maximum transmission distances under the influence of nonlinear impairments. Next, to elucidate the physics behind such nonlinear transmission characteristics, we calculate the distribution of constellation points for QAM signals as functions of the the launched power, the transmission distance, and the symbol rate. These results lead to a closed-form formula for BER of any QAM formats. From such formula, we derive simple laws that determine the maximum transmission distance and the optimum power as functions of the QAM order and the symbol rate. These laws can well explain the simulation results.

© 2011 OSA

## 1. Introduction

1. A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of *Optical Fiber Communication Conference* (Optical Society of America, 2010), PDPB8.

4. D.-S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. **24**, 12–21 (2006). [CrossRef]

5. H. Sun, K.-T. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express **16**, 873–879 (2008). [CrossRef] [PubMed]

6. K. Kikuchi, “Ultra long-haul optical transmission characteristics of wavelength-division multiplexed dual-polarisation 16-quadrature-amplitude-modulation signals,” Electron. Lett. **46**, 433–434 (2010). [CrossRef]

## 2. Simulation model

*D*of 19 ps/nm/km (

*β*

_{2}= −24 ps

^{2}/km), the nonlinearity coefficient

*γ*of 1.1 /W/km, and the loss coefficient

*α*of 0.17 dB/km at the wavelength of 1550 nm [8

8. T. Kato, M. Hirano, M. Onishi, and M. Nishimura, “Ultra-low nonlinearity low-loss pure silica core fibre for long-haul WDM transmission,” Electron. Lett. **35**, 1615–1617 (1999). [CrossRef]

*E*and

_{x}*E*denote electric fields of the WDM signal with

_{y}*x*- and

*y*-polarization components, respectively. In these equations, randomly-distributed linear birefringence is assumed along the link [7

7. S. G. Evangelides Jr., L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. **10**, 28–35 (1992). [CrossRef]

^{15}per channel. We have not introduced nonlinearity compensation based on the back-propagation method [11

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

## 3. Simulation results

*n*for each QAM format is determined such that the BER of the polarization tributary of the center WDM channels becomes lower than 10

^{−3}at the optimum launched power, when POLMUX five WDM channels are transmitted.

*P*. Powers of all channels are changed simultaneously. Green, red, black, and blue curves represent BERs when we transmit a single channel, co-polarized three WDM channels, co-polarized five WDM channels, and POLMUX five WDM channels, respectively. The symbol rate is 12.5 Gsymbol/s and the WDM channel spacing 25 GHz. Numbers of spans

_{ave}*n*corresponding to (a)–(d) are 160, 37, 10, and 3, respectively, resulting in total transmission distances of 12,800 km, 2,960 km, 800 km, and 240 km. Note that the optimum power is about −9 dBm ~ −8 dBm, which is almost common in all of the QAM orders. Above this value, the BER performance is degraded by nonlinear impairments stemming from SPM and XPM between WDM channels and POLMUX channels.

*n*as a function of the order of QAM

*m*, when POLMUX five WDM channels are transmitted. We find that

*n*is inversely proportional to

*m*as seen from the solid line and that transmission distances of 64QAM and 256QAM systems are severely limited below 1,000 km.

*n*for 4, 16, 64, and 256QAM transmission as a function of the symbol rate

*B*. Solid lines represent the slope of

*n*∝

*B*

^{−2/3}, which is in good agreement with simulation results in the range above 25 Gsymbol/s. On the other hand, the optimum launched power

*P*to obtain the minimum BER for 4QAM is plotted as a function of the symbol rate

_{opt}*B*in Fig. 5. In higher-order QAM transmission systems, we have similar

*B*-versus-

*P*characteristics within the error range. The solid line represents the slope of

_{opt}*P*∝

_{opt}*B*

^{1/3}. Simulation results differ from this line significantly in the range below 25 Gsymbol/s. The theoretical background for these dependencies on

*B*will be discussed in Sec. 5 in detail.

## 4. Distribution of constellation points spread by nonlinear effects

^{15}sampled data. Black curves in Figs. 7(a), 7(b), and 7(c) show distributions of the real part of the received complex amplitude for launched average powers

*P*of −14 dBm, −9 dBm, and −4 dBm, respectively. On the other hand, red curves are Gaussian distributions fitted to black ones.

_{ave}*P*=−14 dBm, the BER performance is determined by amplified spontaneous emission (ASE) from EDFAs as shown in Fig. 2(a). On the other hand, when

_{ave}*P*=−4 dBm, the BER performance is degraded by nonlinear effects. At an intermediate power such as

_{ave}*P*=−9 dBm, both of ASE and nonlinear effects spread out the distribution of constellation points. Note that distributions are Gaussian in all of the three cases.

_{ave}*P*. In Fig. 8, the blue curve is obtained when

_{ave}*γ*= 0 and ASE is included,

*i.e.*, the linear case, whereas the red curve are obtained when ASE is neglected and fiber nonlinearity is included

*i.e.*, the nonlinear case. The black curve is calculated when both of ASE and fiber nonlinearity are taken into account. Since the black curve is simply equal to the sum of the red and blue curves, we find that the nonlinear impairment stems from waveform distortion due to GVD and fiber nonlinearity rather than ASE noise enhanced by fiber nonlinearity,

*i.e.*, Gordon-Mollenauer phase noise [10

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

*P*. On the other hand, in the nonlinear case, we find that

_{ave}*δ*

*P*proportional to

*P*in all of the channels. Then, SPM and XPM induces the phase fluctuation

_{ave}*γδ*

*P*per unit length in the channel under consideration, which in turn spreads out the distribution of constellation points through GVD; thus,

*γ*

*P*)

_{ave}^{2}.

*P*at −7 dBm. Figure 9 shows

_{ave}*n*, as shown by blue fitted lines. This result shows that evolution of the complex amplitude along the link obeys the two-dimensional random-walk model: from one span to the next, the signal sequence takes a random step from its last position, which results in the linear increase in the variance as a function of the number of spans.

*P*is −7 dBm, and the number of spans 100. The variance is almost constant when the symbol rate is larger than 25 Gsymbol/s; however, it increases significantly when the symbol rate is lower than 25 Gsymbol/s. This tendency is explained in terms of modulation instability [12]. The bandwidth of modulation-instability gain is about 1 GHz in our case. Therefore, when

_{ave}*B*< 25 Gsymbol/s, the modulation-instability effect inside the signal bandwidth cannot be ignored, and the variance of the Gaussian distribution is enhanced.

*L*is the span length and

*C*is a constant. In the five-channel WDM and POLMUX environments,

_{p}*C*= 13.4 [km] above 50 Gsymbol/s; however, we need some corrections for

_{p}*C*at lower symbol rates as shown in Fig. 10.

_{p}13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. **23**, 742–744 (2011). [CrossRef]

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

*n*and

*P*expressed by Eq. (3) is the same as that given in [13

_{ave}13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. **23**, 742–744 (2011). [CrossRef]

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

*B*given by Eq. (3) is different from those derived in [13

13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. **23**, 742–744 (2011). [CrossRef]

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

## 5. BER formula for high-order QAM transmission systems

*δ*and the variance of the distribution

*σ*

^{2}. In such a case, the bit-error rate for QAM signals is approximately given as where erfc(*) is the complementary error function, and

*D*=1, 3/4, 7/12, and 15/32 for differentially encoded 4QAM, 16QAM, 64QAM, and 256QAM, respectively. The average power of the

_{e}*m*-th order QAM signal

*P*(

_{ave}*m*) is given as In the linear region, the variance

*f*denote the frequency of the carrier,

*G*the amplifier gain, and

*n*the spontaneous emission factor of amplifiers. On the other hand, the variance in the nonlinear region is given from Eq. (3) as where we assume that each constellation point of the high-order QAM has the same Gaussian distribution determined from the average power. Since the total variance of the Gaussian distribution is given as the sum of

_{sp}*at a certain value, for example 10*

_{min}^{−3}, the following relation must be satisfied for any values of

*m*,

*n*, and

*B*: where

*C*is a constant.

*n*∝

*m*

^{−1}when

*B*is fixed. On the other hand, for a fixed

*m*, Fig. 4 and Fig. 5 lead to relations of

*n*∝

*B*

^{−2/3}and

*P*∝

_{opt}*B*

^{1/3}, respectively, when

*B*is larger than 50 Gsymbol/s. Deviations of simulation results from these relations at lower symbol rates are due to the fact that

*C*in Eq. (3) is enhanced by modulation instability. It should be noted here that the larger GVD of fibers generally provides us with the better transmission performance because of the smaller bandwidth of modulation-instability gain.

_{p}*C*in Eq. (10) is determined from computer simulations in fixed WDM and POLMUX environments, we can calculate BER of any QAM formats using signal parameters (

_{p}*n*,

*m*,

*B*, and

*P*) and link parameters (

_{ave}*L*,

*G*,

*n*, and

_{sp}*γ*). For five-channel WDM and POLMUX transmission at 12.5 Gsymbol/s, which is discussed in Sec. 3, we need to use

*C*=56.1 [km] read from Fig. 10. Then, BER curves calculated for 4QAM, 16QAM, 64QAM, and 256QAM are respectively shown by the black, red, blue, and green curves in Fig. 11. Numbers of spans are the same as those used in Sec. 3. Even using such a simple formula, we can obtain BER curves for any QAM formats which are in reasonable agreement with full simulation results given by Fig. 2.

_{p}## 6. Conclusions

## Acknowledgments

## References and links

1. | A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of |

2. | A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2. |

3. | 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,” in |

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

5. | H. Sun, K.-T. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express |

6. | K. Kikuchi, “Ultra long-haul optical transmission characteristics of wavelength-division multiplexed dual-polarisation 16-quadrature-amplitude-modulation signals,” Electron. Lett. |

7. | S. G. Evangelides Jr., L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. |

8. | T. Kato, M. Hirano, M. Onishi, and M. Nishimura, “Ultra-low nonlinearity low-loss pure silica core fibre for long-haul WDM transmission,” Electron. Lett. |

9. | M. Seimetz, “Laser linewidth limitations for optical systems with high order modulation employing feed forward digital carrier phase estimation,” in |

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

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

12. | G. P. Agrawal, |

13. | P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. |

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

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

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

(060.2920) Fiber optics and optical communications : Homodyning

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: July 19, 2011

Revised Manuscript: August 8, 2011

Manuscript Accepted: August 19, 2011

Published: August 29, 2011

**Citation**

Kazuro Kikuchi, "Analyses of wavelength- and polarization-division multiplexed transmission characteristics of optical quadrature-amplitude-modulation signals," Opt. Express **19**, 17985-17995 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-17985

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### References

- A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), PDPB8.
- A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.
- 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,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), OMJ5.
- D.-S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol.24, 12–21 (2006). [CrossRef]
- H. Sun, K.-T. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express16, 873–879 (2008). [CrossRef] [PubMed]
- K. Kikuchi, “Ultra long-haul optical transmission characteristics of wavelength-division multiplexed dual-polarisation 16-quadrature-amplitude-modulation signals,” Electron. Lett.46, 433–434 (2010). [CrossRef]
- S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.10, 28–35 (1992). [CrossRef]
- T. Kato, M. Hirano, M. Onishi, and M. Nishimura, “Ultra-low nonlinearity low-loss pure silica core fibre for long-haul WDM transmission,” Electron. Lett.35, 1615–1617 (1999). [CrossRef]
- M. Seimetz, “Laser linewidth limitations for optical systems with high order modulation employing feed forward digital carrier phase estimation,” in 2008 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2008), OTuM2.
- J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett.15, 1351–1353 (1990). [CrossRef] [PubMed]
- E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol.26, 3416–3425 (2008). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989), Chap. 5.
- P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23, 742–744 (2011). [CrossRef]
- X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent OFDM systems,” Opt. Express18, 19039–19054 (2010). [CrossRef] [PubMed]

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