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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 2 — Jan. 16, 2012
  • pp: 1022–1032
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On nonlinear distortions of highly dispersive optical coherent systems

Francesco Vacondio, Olivier Rival, Christian Simonneau, Edouard Grellier, Alberto Bononi, Laurence Lorcy, Jean-Christophe Antona, and Sébastien Bigo  »View Author Affiliations


Optics Express, Vol. 20, Issue 2, pp. 1022-1032 (2012)
http://dx.doi.org/10.1364/OE.20.001022


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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

The objective of this paper is to investigate the statistical nature of distortions induced by nonlinear effects, and to clarify how such distortions accumulate with distance in coherent systems with no optical dispersion management (NDM), for both standard single mode fiber (SSMF) and nonzero dispersion shifted fiber (NZDSF). We first experimentally assess the behavior of a complete wavelength division multiplexed (WDM) 100 Gb/s polarization division multiplexed (PDM) quadrature phase shift keying (QPSK) system over a 50-GHz grid, including all propagation impairments, transmitter and receiver imperfections, and realistic digital signal processing. In this scenario, we experimentally confirm that an additive Gaussian noise channel can very well approximate the fiber channel both in linear and nonlinear propagation regimes [1

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).

]. We experimentally show the precision of a recently proposed analytical model [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]

5

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).

] for evaluating the performance of such systems at a fixed propagation distance, and the dependence of nonlinear noise variance to channel power. We then demonstrate that for both SSMF and NZDSF the nonlinear noise variance has a faster-than-linear dependence on propagation distance. We propose a model accounting for this dependence and assess its precision in predicting performance for a wide range of propagation distances. Finally, we turn to simulations to gain further insight into the mechanism responsible for the accumulation of nonlinear distortions in the propagation of a WDM-PDM-QPSK signal at 100 Gb/s over a 50-GHz grid. We show the impact of the cumulated dispersion at the input of each span on the total nonlinear variance, and its importance on the overall supra-linear accumulation of nonlinear distortions.

2. Experimental setup

We investigate the respective impact of amplified spontaneous emission (ASE) noise and Kerr nonlinear distortion in a recirculating loop testbed. Figure 1(a)
Fig. 1 Experimental setup of (a) transmitter and (b) recirculating loop.
depicts the setup of the transmitter used in the experiment. Two combs of 40 lasers spaced by 100 GHz are independently modulated with I/Q modulators. The I/Q modulators are driven with 28 Gbaud pseudo random bit sequences (PRBS) of length 215-1. 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.

At the loop output, we can load additional ASE noise through an optical noise source. The channel under test is selected with a passband optical filter, and enters a coherent receiver. The optical filter 3-dB bandwidth is 50 GHz, and its shape is close to a 3rd 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]

], 4) Phase and frequency estimation and correction [6

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

], 5) Electrical signal to noise ratio (SNR) evaluation 6) Symbol identification, bit error ratio (BER) evaluation, and conversion to Q2 factor. The convergence parameter of the adaptive equalizer is 10−3 [5

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).

], and the number of taps in the phase estimation filter was optimized for each configuration.

Figure 2(a)
Fig. 2 (a) Back to back electrical SNR at the decision gate as function of the optical signal to noise ratio (b) Back to back BER-equivalent Q2 factor as function of the electrical SNR at the decision gate.
shows with markers the measured electrical SNR in back to back at the decision gate (expressed in an equivalent two-sided bandwidth of 12.47 GHz) as function of the optical signal to noise ratio (OSNRASE) 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 OSNRASE, 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]

,8

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).

]:
1/SNR=1/OSNRASE+KTRX
(1)
where KTRX is a suitable constant depending on the particular practical implementation of the transmitter and receiver. In our case, KTRX = 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.

The bit error ratio can finally be calculated from the SNR (linear scale) as [9

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

]:
BER=0.5erfc(ηSNR)
(2)
where erfc(.) is the complementary error function, and 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 Q2 factor is calculated from the measured bit error ratio as Q2 = 20log10(2erfcinv(2∙BER)), where erfcinv is the inverse of the complementary error function.

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

In this section, we characterize the statistics of noise coming from ASE, nonlinear signal distortions, transmitter imperfections, all transmission impairments, and realistic receiver and digital signal processing. We measure the noise directly from the received QPSK constellation after propagation over 15x100 km of SSMF. As it is shown in Fig. 3(a)
Fig. 3 (a) Typical measured QPSK constellation from which SNR and BER are measured (b) Measured OSNRASE in 0.1nm after 15 spans of SSMF, as a function of channel power (without noise loading).
, we identify the samples conditioned on the known transmitted symbols (different colors), we calculate their average (stars), which we will subtract from the signal, in order to remove modulation and obtain a zero-mean complex noise. We thoroughly tested the statistics of the noise for two exemplary channel powers: PL = −3 dBm, and PH = + 4 dBm. Figure 3(b) shows the measured OSNRASE 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 OSNRASE deviates from the ideal 1 dB/dB law. As it can be seen, the OSNRASE at receiver input is 16 dB and 21.6 dB, for PL and PH respectively. These power levels have been chosen because they yield the same bit error ratio after transmission, but for the lowest power PL the system is chiefly limited by OSNRASE, whereas for the highest power PH the main source of impairments is fiber nonlinearity.

To study the noise statistics, we have applied Filliben’s probability plot correlation coefficient (PPCC) test for normality [10

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

,11

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]

] to the measured signals. In this test, for each data set we calculate the correlation coefficient between the quantiles of the measured data and the quantiles of a theoretical Gaussian distribution. With a data set of size 1000 samples, when the correlation coefficient is on average higher than 0.9984 or 0.9979 (shown as red dashed lines in the plots) the hypothesis that the data is normal holds with more than 99% or 95% confidence, respectively. We have applied the test to more than 2500 different subsets of our signal, of size 1000 samples each (the test has been done independently for the real part and imaginary part of the recovered signal). Figure 4(a) and (b)
Fig. 4 Probability plot correlation coefficient for normality testing calculated over 1000 samples. The test is repeated more than 2500 times, and for real and imaginary parts. (a) PL = −3 dBm (b) PH = + 4 dBm.
shows the test results, for PL and PH 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.

The normality of the noise is also visually confirmed by the excellent fit over more than 6 orders of magnitude of the probability density functions (PDFs) of real and imaginary part of the measured optical noise (markers) with a theoretical Gaussian PDF, shown in Fig. 5(a)
Fig. 5 PDF of real and imaginary components of total noise (markers: measured signals, line: theoretical Gaussian). Insets show an example of measured constellations. (a) PL = −3 dBm (b) PH = + 4 dBm.
for PL and Fig. 5(b) for PH. 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

In this section we show how accurately Eq. (3) models system performance at a fixed distance. We experimentally measure the signal to noise ratio before the decision gate after 15x100km of SSMF propagation as a function of channel power, with and without additional ASE noise loading. From the measurement of the OSNRASE after propagation, together with the back to back characterization of KTRX, we can infer SNRlin. We can therefore obtain the signal to noise ratio due to nonlinearities SNRNL 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.

In Fig. 6(a)
Fig. 6 (a) Measured and modeled total SNR after 15x100km SSMF transmission with and without noise loading versus channel power (b) 1/SNRlin and 1/SNRNL are shown as percentage of the total 1/SNR.
we report SNRtot and its two components SNRlin and SNRNL 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 OSNRASE by roughly 2 dB. Without noise loading, SNRtot has a maximum at the optimal power per channel PNLT~1.5dBm (visualized by a vertical line in Fig. 6). The linear part SNRlin increases as the signal power (and therefore the OSNRASE) grows. The nonlinear signal to noise ratio SNRNL, on the other hand, decreases as the signal power triggers the nonlinear effects. As it can be seen, the measured SNRNL has a slope of roughly-2 dB/dB, therefore confirming its dependence on 1/P2. The solid lines show the results of the model, where Eq. (3) has been used. In Eq. (3), we have used the measured OSNRASE, KTRX as characterized in back to back, and aNL = 3.3e-3 mW−2. The value of aNL has been calculated from the measured value of SNRNL in the case without noise loading, for the power P = 4 dBm/chan, using aNL = 1/(SNRNL∙P2). The same value of aNL is used also in the second series of measurements. The model in both cases fits very well the measured values.

Figure 6(b) shows 1/SNRlin and 1/SNRNL as fraction of 1/SNRtot, in the case without noise loading. Around PNLT, 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]

,3

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).

], as 1/SNRNL accounts for ~33% and 1/SNRlin for ~66% of the total noise-to-signal ratio.

Figure 7(a)
Fig. 7 (a) Q2 factor versus power per channel for 15x100km SSMF transmission, with and without noise loading (Solid lines: model, markers: experiment). (b) Model error vs. power per channel.
shows the measured BER-equivalent Q2 factor for the same system (markers). The modeled values obtained using (3) and (2), using the same aNL value characterized before, are shown as solid lines. Figure 7(b) shows the error of the modeled Q2-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.

We have repeated the same experiment (15x100 km propagation) using nonzero dispersion shifted fiber instead of SSMF. The test channel was at 1546.12 nm, where the local dispersion is ~4 ps/nm/km. Figure 8(a)
Fig. 8 (a) Q2 factor versus power per channel for 15x100km NZDSF transmission, with and without noise loading (Solid lines: model, markers: experiment). (b) Model error vs. power per channel.
shows the modeled and measured Q2 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 Q2 factor with respect to the measured one within ± 0.2 dB.

5. Performance modeling and prediction for variable distance

In this section, we use Eq. (3) to calculate 1/SNRNL for a fixed power and variable number of spans, in order to investigate its dependence on propagation distance. In this configuration 1/SNRNL is proportional to the nonlinear noise variance, so that we can use 1/SNRNL to deduce rules on the accumulation of nonlinearity-induced distortions.

Figure 9(a)
Fig. 9 1/SNRNL calculated from measurements for different channel powers as function of number of spans in dB scale. Solid lines are linear fit. Transmission over (a) SSMF, (b) NZDSF
shows (for SSMF propagation) the values of 1/SNRNL calculated with Eq. (3) from the measured data (electrical SNR, KTX and OSNRASE) 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 ~N1.37, 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 ~N1.33.

Next, we use the new expression of Eq. (4) for aNL in Eq. (3), and with Eq. (2) we predict system performance as function of distance. Results are shown in Fig. 10(a)
Fig. 10 Measured BER-equivalent Q2 factor is shown as function of the number of spans in linear scale. Solid lines are the results of the model. Transmission over (a) SSMF, (b) NZDSF
for SSMF and Fig. 10(b) for NZDSF. In the figures, markers are experimental results of Q2 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 αNL 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

Figure 11(a)
Fig. 11 Numerically calculated variance of nonlinear noise normalized to signal power (a) As function of the number of spans for NZDSF and SSMF, and (b) For a single span as function of chromatic dispersion at the input of the span.
shows the variance of the nonlinear noise, normalized to unit symbol power and calculated over 12.47 GHz, as function of the number of spans for SSMF at 1550 nm and NZDSF at 1550 nm and 1530 nm, assuming −3dBm channel power in all cases. The local dispersion is 17 ps/nm/km, 4.25 ps/nm/km and 2.5 ps/nm/km, respectively. The effective area is 80 μm2 for SSMF, and 72 μm2 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).

For a given fiber type and wavelength, Fig. 10(a) shows that the variance of nonlinear noise versus span count can be approximated with excellent accuracy by a linear fit in dB/dB scale with a slope strictly greater than 1, confirming at least qualitatively the findings of the experimental results presented in section 5. Specifically, we find slopes of 1.18, 1.24 and 1.29dB/dB for SSMF, NZDSF (1550nm) and NZDSF(1530nm). These values are somewhat lower than those found experimentally. The slight discrepancy is assigned to the presence of sources of impairments other than Kerr effect in the experimental setup (polarization dependent loss, imperfect DSP, …)

This faster-than-linear growth of the variance of nonlinear distortions may have two (possibly combined) sources:

  • 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 N2var(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.

In order to test the latter hypothesis, we present in Fig. 11(b) the variance of the nonlinear distortion created by a single span of 100 km of SSMF with 0 dBm channel power as a function of the cumulated chromatic dispersion at the span input. Besides some scatter (due to random realizations of channels sequences, SOP and delays) a clear linear growth of the variance of nonlinear distortions versus chromatic dispersion in dB/dB scale can be observed. Denoting the average slope 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.

In [4

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. 23(11), 742–744 (2011). [CrossRef]

], authors investigate a Nyquist-WDM system, where the channel spacing is equal to the baud rate. They report a linear growth of the nonlinear variance with the number of spans, whereas we report a supra-linear growth. In our system, self phase modulation is known to dominate over cross nonlinear effects [14

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

]. At the Nyquist limit, as in [4

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. 23(11), 742–744 (2011). [CrossRef]

], cross channel effects are stronger, due to the reduced channel spacing. It is therefore reasonable to speculate that intrachannel effects induce a faster growth of the nonlinear variance with respect to interchannel effects, and this will be the object of a separate study.

7. Conclusion

We have experimentally shown that, in the case of a 50-GHz-spaced WDM-PDM-QPSK 100 Gb/s transmission system without optical dispersion compensation, the optical fiber channel can be modeled by an additive Gaussian noise channel with high accuracy. We characterized the noise variance with respect to signal power, ASE level and distance. We proposed a simple empirical model to account for the distance dependence of the nonlinear noise variance, which is shown to be able to predict Q2 factor with accuracy within ± 0.3 dB over a wide range of distances.

Via simulations we have investigated the nature of nonlinear distortions accumulation, showing that not all spans contribute equally to the total nonlinear noise variance, since the nonlinear noise generated in a span is a function of the chromatic dispersion at the span input. Finally, we have shown that a simple model assuming uncorrelated per-span nonlinear distortions whose variance grows as in (5) reasonably justifies (with an error which is within simulation accuracy) the observed variance growth.

Acknowledgments

Authors acknowledge partial funding from French project 100GRIA, and from the Celtic project EO-Net.

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 19(13), 12781–12788 (2011). [CrossRef] [PubMed]

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. 23(11), 742–744 (2011). [CrossRef]

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 16(2), 804–817 (2008). [CrossRef] [PubMed]

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]

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. 37(5), 437–448 (1989). [CrossRef]

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]

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. 22(10), 685–687 (2010). [CrossRef]

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

  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. Express19(13), 12781–12788 (2011). [CrossRef] [PubMed]
  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.23(11), 742–744 (2011). [CrossRef]
  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. Express16(2), 804–817 (2008). [CrossRef] [PubMed]
  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]
  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.37(5), 437–448 (1989). [CrossRef]
  10. J. J. Filliben, “The probability plot correlation coefficient test for normality,” Technometrics17(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]
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