## On nonlinear distortions of highly dispersive optical coherent systems |

Optics Express, Vol. 20, Issue 2, pp. 1022-1032 (2012)

http://dx.doi.org/10.1364/OE.20.001022

Acrobat PDF (1949 KB)

### Abstract

We investigate via experiments and simulations the statistical properties and the accumulation of nonlinear transmission impairments in coherent systems without optical dispersion compensation. We experimentally show that signal distortion due to Kerr nonlinearity can be modeled as additive Gaussian noise, and we demonstrate that its variance has a supra-linear dependence on propagation distance for 100 Gb/s transmissions over both low dispersion and standard single mode fiber. We propose a simple empirical model to account for linear and nonlinear noise accumulation, and to predict system performance for a wide range of distances, signal powers and optical noise levels.

© 2012 OSA

## 1. Introduction

2. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with gaussian-distributed nonlinear noise,” Opt. Express **19**(13), 12781–12788 (2011). [CrossRef] [PubMed]

## 2. Experimental setup

^{15}-1. As is customary in laboratory experiments, polarization division multiplexing is emulated by splitting the signal, delaying one branch, and recombining the signal through a polarization beam combiner. Finally, the two combs are combined using an interleaver to form a single WDM comb of 50GHz-spaced PDM-QPSK signals at 112 Gb/s, accounting for the transport of a 100Gb/s payload with 12% overhead for forward error correction and framing purposes. The experimental setup of the recirculating loop is depicted in Fig. 1(b). Inside the loop, light propagates into three spans of 100 km of either SSMF or NZDSF separated by erbium doped fiber amplifiers (EDFAs). The loop also comprises a polarization scrambler and a wavelength selective switch (WSS) for channel power equalization.

^{rd}order supergaussian. In the receiver, the signal beats with a local oscillator in a dual polarization downconverter before being photodetected. Transmitter laser and local oscillator have the same nominal linewidth of 300 kHz. After photodetection, a 16 GHz, 50 Gsamples/s real time oscilloscope is used to capture traces which are then processed offline. In the offline digital signal processing (DSP), the following steps are applied to the signal: 1) Normalization and resampling to 2 samples/symbol, 2) Chromatic dispersion compensation, 3) Adaptive blind equalization with the constant modulus algorithm [6

6. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express **16**(2), 804–817 (2008). [CrossRef] [PubMed]

6. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express **16**(2), 804–817 (2008). [CrossRef] [PubMed]

^{2}factor. The convergence parameter of the adaptive equalizer is 10

^{−3}[5], and the number of taps in the phase estimation filter was optimized for each configuration.

_{ASE}) before the receiver, measured in 0.1 nm (equivalent to 12.47 GHz). Here and throughout the paper, the signal to noise ratio will be referred to the nominal 0.1 nm bandwidth. As it can be seen, the SNR clearly shows saturation for high values of OSNR

_{ASE}, which is a signature of transceiver imperfections or in-band cross-talk. The SNR can therefore be well modeled as [7

7. F. Forghieri, R. W. Tkach, and D. L. Favin, “Simple model of optical amplifier chains to evaluate penalties in WDM systems,” J. Lightwave Technol. **16**(9), 1570–1576 (1998). [CrossRef]

_{TRX}is a suitable constant depending on the particular practical implementation of the transmitter and receiver. In our case, K

_{TRX}= 1/215 yields very good fit with the measured data, as can be seen in Fig. 2(a). Therefore, given our transceiver, the maximum achievable electrical SNR at the decision gate is ~23 dB.

9. D. Saha and T. G. Birdsall, “Quadrature-quadrature phase-shift keying,” IEEE Trans. Commun. **37**(5), 437–448 (1989). [CrossRef]

*η*is a suitable constant accounting for all deviations of the receiver from the ideal matched filter. With an ideal receiver,

*η*is the ratio of the noise single-sided bandwidth and the signal baud rate,

*i.e. η*= (12.47/2)/28≈0.22. In our case,

*η*= 0.2 gives a very good fit with the measured data, as can be seen in Fig. 2(b). It is important to stress that here, and throughout the paper, the Q

^{2}factor is calculated from the measured bit error ratio as Q

^{2}= 20log10(

## 3. Statistical characterization of measured signals after 15x100km of SSMF

_{L}= −3 dBm, and P

_{H}= + 4 dBm. Figure 3(b) shows the measured OSNR

_{ASE}at the receiver input as a function of power per channel after 15x100 km of SSMF propagation without additional noise loading. The noise figure of our EDFAs slightly varies with the optical power level at the amplifier input, therefore the measured OSNR

_{ASE}deviates from the ideal 1 dB/dB law. As it can be seen, the OSNR

_{ASE}at receiver input is 16 dB and 21.6 dB, for P

_{L}and P

_{H}respectively. These power levels have been chosen because they yield the same bit error ratio after transmission, but for the lowest power P

_{L}the system is chiefly limited by OSNR

_{ASE}, whereas for the highest power P

_{H}the main source of impairments is fiber nonlinearity.

10. J. J. Filliben, “The probability plot correlation coefficient test for normality,” Technometrics **17**(1), 111–117 (1975). [CrossRef]

11. S. W. Looney and T. R. Gulledge, “Use of the correlation coefficient with normal probability plots,” Am. Stat. **39**(1), 75–79 (1985). [CrossRef]

_{L}and P

_{H}respectively. In our case, the average PPCC is well above the 99% confidence level, therefore proving that for both powers the noise statistics are indeed normal. In conclusion, noise can be considered Gaussian both in the linear and in the nonlinear regime, where the major sources of impairments are ASE noise and nonlinearity-induced signal distortions, respectively.

_{L}and Fig. 5(b) for P

_{H}. For these results, more than ten million measured samples have been used. The results shown here are for one polarization only, since for the other polarization the results are essentially identical: the normal hypothesis holds with more than 99% confidence, and the PDF fit is again very good. We can therefore conclude that in our transmission experiment noise is Gaussian and identically distributed in the 4D space of the two polarizations in-phase and quadrature components, both in the linear and nonlinear regimes.

## 4. Performance modeling and prediction at fixed distance

_{NL}P

^{3}, where P is the channel power and a

_{NL}a suitable constant which depends on system parameters and can be obtained numerically, analytically, or by simple measurements [2

2. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with gaussian-distributed nonlinear noise,” Opt. Express **19**(13), 12781–12788 (2011). [CrossRef] [PubMed]

_{lin}= (1/OSNR

_{ASE}+ K

_{TRX}) is the inverse of the linear part, and 1/SNR

_{NL}= a

_{NL}P

^{3}/P is the inverse of the nonlinear signal to noise ratio, and thus depends on the channel power squared.

_{ASE}after propagation, together with the back to back characterization of K

_{TRX}, we can infer SNR

_{lin}. We can therefore obtain the signal to noise ratio due to nonlinearities SNR

_{NL}from (3) and doing so, we have a method for separating the impact of linear noise due to ASE and transceiver imperfections from that of nonlinear noise coming from propagation distortions.

_{tot}and its two components SNR

_{lin}and SNR

_{NL}as a function of signal power in the case of 15x100km of SSMF propagation without noise loading (empty markers). Filled markers are the result of a measurement where we load noise at the receiver, to degrade the OSNR

_{ASE}by roughly 2 dB. Without noise loading, SNR

_{tot}has a maximum at the optimal power per channel P

_{NLT}~1.5dBm (visualized by a vertical line in Fig. 6). The linear part SNR

_{lin}increases as the signal power (and therefore the OSNR

_{ASE}) grows. The nonlinear signal to noise ratio SNR

_{NL}, on the other hand, decreases as the signal power triggers the nonlinear effects. As it can be seen, the measured SNR

_{NL}has a slope of roughly-2 dB/dB, therefore confirming its dependence on 1/P

^{2}. The solid lines show the results of the model, where Eq. (3) has been used. In Eq. (3), we have used the measured OSNR

_{ASE}, K

_{TRX}as characterized in back to back, and a

_{NL}= 3.3e-3 mW

^{−2}. The value of a

_{NL}has been calculated from the measured value of SNR

_{NL}in the case without noise loading, for the power P = 4 dBm/chan, using a

_{NL}= 1/(SNR

_{NL}∙P

^{2}). The same value of a

_{NL}is used also in the second series of measurements. The model in both cases fits very well the measured values.

_{lin}and 1/SNR

_{NL}as fraction of 1/SNR

_{tot}, in the case without noise loading. Around P

_{NLT}, the nonlinear noise power is half the linear noise power as predicted in [2

2. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with gaussian-distributed nonlinear noise,” Opt. Express **19**(13), 12781–12788 (2011). [CrossRef] [PubMed]

_{NL}accounts for ~33% and 1/SNR

_{lin}for ~66% of the total noise-to-signal ratio.

^{2}factor for the same system (markers). The modeled values obtained using (3) and (2), using the same a

_{NL}value characterized before, are shown as solid lines. Figure 7(b) shows the error of the modeled Q

^{2}-factor with respect to the measured value, as function of the power per channel. As it can be seen, the error is always within ± 0.3 dB, indicating excellent agreement between model and measurements also for prediction of bit error ratio.

^{2}factor for this configuration, and Fig. 8(b) shows the model error. As compared to SSMF, there is a reduced tolerance to nonlinear effects, due to the lower local dispersion and effective area. Still, the model holds very well, yielding an error of the modeled Q

^{2}factor with respect to the measured one within ± 0.2 dB.

## 5. Performance modeling and prediction for variable distance

_{NL}for a fixed power and variable number of spans, in order to investigate its dependence on propagation distance. In this configuration 1/SNR

_{NL}is proportional to the nonlinear noise variance, so that we can use 1/SNR

_{NL}to deduce rules on the accumulation of nonlinearity-induced distortions.

_{NL}calculated with Eq. (3) from the measured data (electrical SNR, K

_{TX}and OSNR

_{ASE}) as function of the number of spans. We repeated the experiment for powers from + 1 dBm/channel to + 5 dBm/channel. The slope of the linear fit in log-log scale, shown with a solid line, is almost independent of the channel power. The average slope over the investigated powers and span range is 1.37 dB/dB. That means that the nonlinear noise variance grows as ~

*N*

^{1.37}, where

*N*is the number of spans. For NZDSF propagation, Fig. 9(b) shows the same type of results. Again, it can be seen that nonlinear noise accumulates almost independently of the power per channel, and the average slope is 1.33 dB/dB. Similarly to SSMF, the nonlinear noise variance therefore grows as ~

*N*

^{1.33}.

_{NL}in Eq. (3), and with Eq. (2) we predict system performance as function of distance. Results are shown in Fig. 10(a) for SSMF and Fig. 10(b) for NZDSF. In the figures, markers are experimental results of Q

^{2}factor versus propagation distance for different optical powers per channel. The solid lines show the results of the proposed model, where

*ε*is used as calculated before. Once

*ε*is determined, a single free parameter α

_{NL}is needed for each fiber type, with which system performance can be predicted accurately for a wide range of distances, noise level and channel power. The single parameter actually depends on the characteristics of the optical fiber, as well as on the modulation format which is employed. As it can be seen, there is very good agreement of the model with the measured data.

## 6. Kerr effect-induced signal distortions accumulation

^{18}FFT points. The systems under test are based on 100-km spans of SSMF or NZDSF without optical dispersion compensation. The spans are separated by EDFAs which exactly compensate for the optical loss of the preceding fiber. The nonlinear interaction of ASE noise and signal is known to be negligible in NDM systems [13]; we therefore safely neglect ASE noise in our simulations. We are thus able to estimate the distortions induced by the sole Kerr effect by subtracting the (normalized) optical field launched into the transmission link from the (normalized) optical field at the link output. After having verified that the statistics of such distortions are well approximated by a Gaussian distribution (identically distributed in the 4D space of in-phase and quadrature components for the two polarizations), we express results in terms of the variance of this nonlinear noise.

^{2}for SSMF, and 72 μm

^{2}for NZDSF. The loss coefficient is 0.22 dB/km for both SSMF and NZDSF. We repeated the simulation 20 times, randomly changing the channels SOPs, relative delays and sequences. These random parameters account for a ~1dB spread in the variance of the nonlinear noise for a given fiber type and number of spans. At fixed number of spans still, the variance of nonlinear noise is found to increase from SSMF to NZDSF (1550nm) to NZDSF (1530nm).

- • Partial (positive) correlation,
*i.e.*, constructive interference, between the distortions induced by different spans of the transmission links. In the extreme case of all spans creating an identical distortion*f*, the total distortion after*N*spans would be*N∙f*and its variance*N*var(^{2}*f*), thus yielding a 2 dB/dB growth of the total variance of nonlinear distortion with the number of span. - • Increasing of the noise variance generated by the individual spans along propagation,
*i.e.*, each span generates a nonlinear noise, the variance of which is greater than the variance generated by the preceding span.

*ξ*, we find

*ξ*~0.22. This result agrees with the intuition that in NDM links, chromatic dispersion gradually distorts the optical signal, increasing the occurrence of short power spikes prone to high nonlinear distortions and increasing the number of interpulse four wave mixing interactions.

*f*can be written as the sum of the variance of

_{tot}*f*’s, where

_{k}*f*is the nonlinear distortion induced by the span

_{k}*k.*Grouping the various results presented in this paper, we can write var(

*f*) as:where

_{k}*P*is the channel power,

*C*the cumulated chromatic dispersion at the input of span

_{k}*k*and

*A*is a constant which is independent of the channel power and propagated distance. This model clearly fails for the first span of a transmission link without precompensation, as the zero chromatic dispersion at the span input would yield zero nonlinear distortion regardless of the power. Keeping in mind that the first span must be handled separately, the variance of the total nonlinear distortions can thus be written as:where

*A*

_{1}is a constant to account for the first span noise variance var(

*f*),

_{1}*D*is the local chromatic dispersion,

*L*is the span length, and we have used

_{span}*C*=

_{k}*DL*(

_{span}*k*-1), assuming that all spans are of equal length.

*N*large enough, we thus find var(

*f*) ∝

_{tot}*N*

^{1+ξ}with 1 +

*ξ*= 1.22. This value is very close to the 1.18dB/dB slope found in Fig. 11(a) for SSMF links and the small remaining discrepancy is within the accuracy of the simulation. We have thus proved that the variance growth versus span number in NDM systems is mostly to be attributed to the increase of per-span variance according to the empirical law (5), rather than to correlations among spans.

## 7. Conclusion

^{2}factor with accuracy within ± 0.3 dB over a wide range of distances.

## Acknowledgments

## References and links

1. | A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC 2010, paper P4.07 (2010). |

2. | E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with gaussian-distributed nonlinear noise,” Opt. Express |

3. | G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance prediction for WDM PM-QPSK transmission over uncompensated links”, in Proc. OFC 2011, paper OTh07 (2011). |

4. | 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. |

5. | A. Bononi, E. Grellier, P. Serena, N. Rossi, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant interpulse-four-wave-mixing,” in Proc. ECOC 2011, paper We.7.b.2 (2011). |

6. | S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express |

7. | F. Forghieri, R. W. Tkach, and D. L. Favin, “Simple model of optical amplifier chains to evaluate penalties in WDM systems,” J. Lightwave Technol. |

8. | E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analyitical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC 2011, paper We.7.B.2 (2011). |

9. | D. Saha and T. G. Birdsall, “Quadrature-quadrature phase-shift keying,” IEEE Trans. Commun. |

10. | J. J. Filliben, “The probability plot correlation coefficient test for normality,” Technometrics |

11. | S. W. Looney and T. R. Gulledge, “Use of the correlation coefficient with normal probability plots,” Am. Stat. |

12. | E. Grellier, J.-C. Antona, and S. Bigo, “Global criteria to account for tolerance to nonlinearities of highly dispersive systems,” IEEE Photon. Technol. Lett. |

13. | A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?” in Proc. ECOC 2010, paper Th10E1 (2010). |

14. | A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. OFC 2011, paper OWO7 (2011). |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

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

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: July 5, 2011

Revised Manuscript: August 16, 2011

Manuscript Accepted: September 16, 2011

Published: January 4, 2012

**Citation**

Francesco Vacondio, Olivier Rival, Christian Simonneau, Edouard Grellier, Alberto Bononi, Laurence Lorcy, Jean-Christophe Antona, and Sébastien Bigo, "On nonlinear distortions of highly dispersive optical coherent systems," Opt. Express **20**, 1022-1032 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1022

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### References

- A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC 2010, paper P4.07 (2010).
- E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with gaussian-distributed nonlinear noise,” Opt. Express19(13), 12781–12788 (2011). [CrossRef] [PubMed]
- G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance prediction for WDM PM-QPSK transmission over uncompensated links”, in Proc. OFC 2011, paper OTh07 (2011).
- 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(11), 742–744 (2011). [CrossRef]
- A. Bononi, E. Grellier, P. Serena, N. Rossi, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant interpulse-four-wave-mixing,” in Proc. ECOC 2011, paper We.7.b.2 (2011).
- S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express16(2), 804–817 (2008). [CrossRef] [PubMed]
- F. Forghieri, R. W. Tkach, and D. L. Favin, “Simple model of optical amplifier chains to evaluate penalties in WDM systems,” J. Lightwave Technol.16(9), 1570–1576 (1998). [CrossRef]
- E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analyitical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC 2011, paper We.7.B.2 (2011).
- D. Saha and T. G. Birdsall, “Quadrature-quadrature phase-shift keying,” IEEE Trans. Commun.37(5), 437–448 (1989). [CrossRef]
- J. J. Filliben, “The probability plot correlation coefficient test for normality,” Technometrics17(1), 111–117 (1975). [CrossRef]
- S. W. Looney and T. R. Gulledge, “Use of the correlation coefficient with normal probability plots,” Am. Stat.39(1), 75–79 (1985). [CrossRef]
- E. Grellier, J.-C. Antona, and S. Bigo, “Global criteria to account for tolerance to nonlinearities of highly dispersive systems,” IEEE Photon. Technol. Lett.22(10), 685–687 (2010). [CrossRef]
- A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?” in Proc. ECOC 2010, paper Th10E1 (2010).
- A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. OFC 2011, paper OWO7 (2011).

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