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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 27 — Dec. 17, 2012
  • pp: 28779–28785
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Analytical results on back propagation nonlinear compensator with coherent detection

Takahito Tanimura, Markus Nölle, Johannes Karl Fischer, and Colja Schubert  »View Author Affiliations


Optics Express, Vol. 20, Issue 27, pp. 28779-28785 (2012)
http://dx.doi.org/10.1364/OE.20.028779


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Abstract

We derive analytic formulas for the improvement in effective optical signal-to-noise ratio brought by a digital nonlinear compensator for dispersion uncompensated links. By assuming Gaussian distributed nonlinear noise, we are able to take both nonlinear signal-to-signal and nonlinear signal-to-noise interactions into account. In the limit of weak nonlinear signal-to-noise interactions, we derive an upper boundary of the OSNR improvement. This upper boundary only depends on fiber parameters as well as on the total bandwidth of the considered wavelength-division multiplexing (WDM) signal and the bandwidth available for back propagation. We discuss the dependency of the upper boundary on different fiber types and also the OSNR improvement in practical system conditions. Furthermore, the analytical formulas are validated by numerical simulations.

© 2012 OSA

1. Introduction

Digital equalization for fiber nonlinearity is an indispensable technology to extend the transmission reach of high spectral efficiency (SE) transmission systems, which combine coherently detected higher order modulation formats with wavelength division multiplexing (WDM). In particular, digital signal processing (DSP) using back propagation nonlinear compensation (NLC) has been applied [1

1. T. Tanimura, T. Hoshida, S. Oda, T. Tanaka, C. Ohsima, and J. C. Rasmussen, “Systematic analysis on multi-segment dual-polarisation nonlinear compensation in 112 Gb/s DP-QPSK coherent receiver,” in Tech. Digest of European Conference on Optical Communication,2009, paper 9.4.5.

4

4. D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010). [CrossRef]

] to compensate for nonlinearity due to the Kerr effect except for limitations due to inter-channel nonlinearity, amplified spontaneous emission (ASE) noise [5

5. D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (2011). [CrossRef] [PubMed]

,6

6. L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The limits of digital backpropagation in nonlinear coherent fiber-optic links,” in Tech. Digest of European Conference on Optical Communication,2012, P4.14.

], and polarization-mode dispersion (PMD) [7

7. G. Gao, X. Chen, and W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation in the presence of PMD,” in Tech. Digest of Optical Fiber Communication Conference,2012, paper OM3A.5.

,8

8. T. Tanimura, S. Oda, T. Hoshida, L. Li, Z. Tao, and J. C. Rasmussen, “Experimental characterization of nonlinearity mitigation by digital back propagation and nonlinear polarization crosstalk canceller under high PMD condition,” in Tech. Digest of Optical Fiber Communication Conference,2011, JWA020.

]. The performance improvements by NLC combined with single- and/or multi-channel coherent reception have been demonstrated with both experiments and numerical simulations [1

1. T. Tanimura, T. Hoshida, S. Oda, T. Tanaka, C. Ohsima, and J. C. Rasmussen, “Systematic analysis on multi-segment dual-polarisation nonlinear compensation in 112 Gb/s DP-QPSK coherent receiver,” in Tech. Digest of European Conference on Optical Communication,2009, paper 9.4.5.

4

4. D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010). [CrossRef]

] and discussed [5

5. D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (2011). [CrossRef] [PubMed]

8

8. T. Tanimura, S. Oda, T. Hoshida, L. Li, Z. Tao, and J. C. Rasmussen, “Experimental characterization of nonlinearity mitigation by digital back propagation and nonlinear polarization crosstalk canceller under high PMD condition,” in Tech. Digest of Optical Fiber Communication Conference,2011, JWA020.

]. However, so far no simple analytical formula has been derived, which is able to describe the OSNR improvement brought by NLC in WDM links. The main issue is that the NLC performance depends not only on the considered NLC scheme itself but also on transmission system parameters, e.g. the fiber launch power, the fiber type and the dispersion map. Although numerical simulations based on the well-known Manakov equation can be used for evaluation of the NLC performance, the approach of numerical simulation would be time-consuming and the results strongly depend on certain system setup.

In this paper, we further extend this analysis and theoretically derive a simple formula of effective OSNR improvement brought by NLC taking both signal-to-signal and signal-to-noise interference into account. The theoretical predictions are verified with numerical simulations and show good agreement with the simulation results if the fiber launch power is so high that the system requires NLC. In the limit of weak signal-to-noise interference, we end up with a simple formula for the OSNR improvement brought by NLC as an upper boundary [14

14. T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator in coherent transmission systems,” in Tech. Digest of European Conference on Optical Communication,2012, P3.11.

]. It should be noted that this simple formula is independent of fiber span length, number of spans, symbol rate, and channel spacing of the WDM channels, and the OSNR improvement therefore can be calculated with the help of the fiber parameters, the bandwidth used for NLC, and the total optical bandwidth occupied by all WDM channels. We discuss the dependency of the upper boundary on different fiber types and also the OSNR improvement in practical system conditions.

2. Modeling of a back propagation NLC with transmission links

The nonlinear noise power caused by signal-to-ASE interference in the m-th span is proportional toGASE(m)GTx,ch2=(mPASE/(BnNs))(PTx,ch/(RsΔf))2, where GASE(m) is the power spectral density of the ASE noise at the m-th span and GTx,ch is the power spectral density of the signal [5

5. D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (2011). [CrossRef] [PubMed]

,6

6. L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The limits of digital backpropagation in nonlinear coherent fiber-optic links,” in Tech. Digest of European Conference on Optical Communication,2012, P4.14.

]. In the summation of the nonlinear noise power of each span, we assume the span contributions to be uncorrelated random variables. Then the nonlinear noise power due to signal-to-ASE interference after Ns-spans transmission is given by [10

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

]
PNL,sigASE=(1+Ns)γ22πα|β2|RsΔfln(BWDM)PTx,ch2PASE.
(5)
Equation (5) shows that the nonlinear noise power generated by nonlinear signal-to-ASE interactions grows quadratically with the power per channel PTx,ch [6

6. L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The limits of digital backpropagation in nonlinear coherent fiber-optic links,” in Tech. Digest of European Conference on Optical Communication,2012, P4.14.

]. Note that these formulas so far do not take into account that nonlinear impairments are mitigated by using back propagation NLC.

Considering NLC with a limited bandwidth BNLC, the nonlinear noise generated by signal-to-signal interactions within this bandwidth can be removed, while the nonlinear noise power generated from outside this bandwidth and the noise power generated by signal-to-ASE interactions remain as residual noise. Substituting the integration range of f in Eq. (4) to be (BNLC/2, BWDM/2) with the condition BWDM > BNLC > Bo, the residual nonlinear noise power after NLC can be expressed as:
PNL,sigsig,w/NLC=Nsγ2πα|β2|RsΔf2ln(BWDMBNLC)PTx,ch3Bn,
(6)
where BNLC is the optical bandwidth used for NLC, typically corresponding to the electrical bandwidth of a single coherent receiver. The OSNR improvement generated by a back propagation NLC is defined as the effective OSNR-difference between the achievable maximum effective OSNR with and without the use of NLC, respectively.

In the limit of weak signal-to-ASE interference (PNL,sigASE0), a maximum value of effective OSNR without nonlinear compensation is given by
OSNRmax,w/oNLC=23PASE2/3(2πα|β2|Δf2Rsγ2NsBnln(BWDM/Bo))1/3
(7)
from Eqs. (1), (2), and (4). A maximum value of effective OSNR with nonlinear compensation can be similarly obtained from Eqs. (1), (2), and (6):
OSNRmax,w/NLC=23PASE2/3(2πα|β2|Δf2Rsγ2NsBnln(BWDM/BNLC))1/3.
(8)
The maximum values of the OSNR are obtained by optimizing with respect to the value of PTx,ch. From Eqs. (7) and (8), the effective OSNR improvement in [dB] by using NLC without signal-to-ASE interference can be derived as [14

14. T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator in coherent transmission systems,” in Tech. Digest of European Conference on Optical Communication,2012, P3.11.

]:
OSNRimprovementbyNLC=103log10(ln(BWDM)ln(Bo)ln(BWDM)ln(BNLC)).
(9)
Equation (9) gives an upper boundary for the performance improvement by a back propagation NLC. It should be noted that the upper boundary is independent of fiber span length, number of spans, symbol rate, and channel spacing of the DWDM channels within the assumptions in [10

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

].

3. System setup for numerical simulation

To compare the results of the numerical simulations with the theoretical results, a back-to-back model [15

15. E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Tech. Digest of European Conference on Optical Communication,2011, paper We.7.B.2.

] was used. The effective OSNR by theory can be converted to a corresponding Q-factor. Assuming DP-QPSK modulation, the Q-factor is given by Q = k OSNR Bn/Rs where k is a parameter in the back-to-back model [15

15. E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Tech. Digest of European Conference on Optical Communication,2011, paper We.7.B.2.

] (ηXT in [15

15. E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Tech. Digest of European Conference on Optical Communication,2011, paper We.7.B.2.

] was set to zero.) The k, which introduces a rigid translation of the BER vs. OSNR curve, can be obtained from back-to-back simulations. In this setup, the k-parameter in the back-to-back model was evaluated to be 0.88 and this value was used in following analysis. The net Q-gain due to NLC is defined as the Q-difference between the achievable maximum Q-factors with and without the use of NLC, respectively.

4. Results and discussion

First, we compared the agreement between Q-factors obtained by theory ('signal-signal model' and 'signal-signal + signal-ASE model') and numerical simulation with a single channel NLC (BNLC = 35 GHz) after transmission over 120 spans SMF. Figure 3
Fig. 3 Q-factor after transmission as a function of the fiber launched power. NLC using only a single channel (BNLC = 35 GHz).
shows Q-factors after transmission for theory and simulation. We observed, at least in the linear regime, a good agreement between the analytical 'signal-signal model' and the simulation, where the Q-factor increases initially with launched power, reaches a peak value, and then decreases with a further increase in power because of the onset of the nonlinear effects. However, the difference between the results for the analytical 'signal-signal model' and the simulation is larger for higher launch powers. This is due to the nonlinear interaction between signal and ASE, which are not included in the analytical formula. With modification of the model to include signal-ASE interference, we observed an improved agreement between Q-factors derived by theory and simulation. The differences between Q-factor predicted by 'signal-signal + signal-ASE' model and numerical simulation were within 0.5 dB regardless of NLC. These results support that the derived equations can be used to evaluate the performance of NLC.

Next, we investigated the dependency of the improvement caused by the NLC on the bandwidth used for the nonlinear compensation. Figure 4
Fig. 4 Net Q-gain brought by NLC as a function of the bandwidth used for NLC with BWDM = 11 × 37.5GHz.
shows the simulated and theoretical Q-gains as a function of bandwidth used for NLC BNLC after transmission over 120 spans SMF. The total bandwidth is BWDM = 11 × 37.5 GHz. The net Q-gain brought by the NLC shows consistent improvement over the entire range of BNLC from 20 to 200 GHz for both simulation and theory. It can be seen that the difference between the predicted Q-gain by the 'signal-signal model' and the 'signal-signal + signal-ASE model' was increasing with bandwidth used for NLC. This is due to the residual signal-to-ASE interference, which is not included in the 'signal-signal model'. The differences between Q-gain predicted by 'signal-signal + signal-ASE model' and numerical simulation were within only 0.15 dB. The optimum fiber input power with NLC PTX,ch,opt,w/NLC to achieve maximum values of OSNR monotonically increases with the bandwidth used for NLC. In particular, employing 'signal-to-signal model' ('signal-to-signal + signal-to-ASE model)', the values of PTX,ch,opt,w/NLC are calculated as 0 (−0.3), 0.6 (0.2), 1.1 (0.5), and 1.5 (0.9) dBm per channel at the bandwidth used for NLC of 50, 100, 150, and 200 GHz, respectively. Note that the optimum powers are reduced when also accounting for nonlinear signal-to-ASE interactions.

Finally, we consider the OSNR improvement by using the NLC model including nonlinear signal-to-ASE interactions. In this case, the OSNR improvement by NLC depends not only on BNLC, BWDM, and Bo but also on other system parameters. To highlight the dependence on system parameters, we swept the number of transmission spans Ns from 20 to 150. Figure 5(b) illustrates the influence of the number of spans (Ns) on the OSNR improvement by NLC. To maximize values of the OSNR, PTx,ch were optimized at each condition of Ns and BNLC in Fig. 5(b). The other simulation parameters were fixed at the values described in the caption of Fig. 5(b). We observed that increasing the number of spans degrades the achievable OSNR improvement by NLC through nonlinear signal-to-ASE interactions. After 20-spans transmission, the optimum fiber launch power was increased from −1.4 dBm to 0.5 dBm per channel for increasing bandwidth BNLC from 20 GHz to 1 THz even in the presence of nonlinear signal-to-ASE interactions. The increase of optimum fiber launch power enabled by a higher bandwidth BNLC allows achieving a higher Q-factor. On the other hand, the optimum launch power is reduced with increasing number of spans Ns through nonlinear signal-to-ASE interactions. For example, the optimum fiber launch power was decreased from 0.5 dBm to 0 dBm per channel when increasing the number of spans of Ns from 20 to 150 spans and keeping the bandwidth BNLC constant at 1 THz.

5. Conclusions

We have derived analytical formulas for the OSNR improvement brought by a digital back propagation nonlinear compensator for dispersion uncompensated transmission links. These formulas are based on an analytical model, which assumes Gaussian distributed nonlinear noise, and is able to take both nonlinear signal-to-signal and signal-to-ASE interactions into account. The obtained analytical results were verified by numerical simulations of DP-QPSK WDM transmission. In the limit of weak nonlinear signal-to-ASE interactions, we derived an upper boundary for the improvement brought by NLC that only depends on the fiber parameters, the overall WDM bandwidth and the bandwidth available for back-propagation. For a particular fixed transmission system, the improvement monotonically increases with the bandwidth used for NLC.

References and links

1.

T. Tanimura, T. Hoshida, S. Oda, T. Tanaka, C. Ohsima, and J. C. Rasmussen, “Systematic analysis on multi-segment dual-polarisation nonlinear compensation in 112 Gb/s DP-QPSK coherent receiver,” in Tech. Digest of European Conference on Optical Communication,2009, paper 9.4.5.

2.

S. Oda, T. Tanimura, T. Hoshida, C. Ohshima, H. Nakashima, Z. Tao, and J. C. Rasmussen, “112 Gb/s DP-QPSK transmission using a novel nonlinear compensator in digital coherent receiver,” in Tech. Digest of Optical Fiber Communication Conference,2009, paper OThR6.

3.

E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol. 28(6), 939–951 (2010). [CrossRef]

4.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010). [CrossRef]

5.

D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (2011). [CrossRef] [PubMed]

6.

L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The limits of digital backpropagation in nonlinear coherent fiber-optic links,” in Tech. Digest of European Conference on Optical Communication,2012, P4.14.

7.

G. Gao, X. Chen, and W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation in the presence of PMD,” in Tech. Digest of Optical Fiber Communication Conference,2012, paper OM3A.5.

8.

T. Tanimura, S. Oda, T. Hoshida, L. Li, Z. Tao, and J. C. Rasmussen, “Experimental characterization of nonlinearity mitigation by digital back propagation and nonlinear polarization crosstalk canceller under high PMD condition,” in Tech. Digest of Optical Fiber Communication Conference,2011, JWA020.

9.

A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” in Tech. Digest of European Conference on Optical Communication,1993, paper MoC2.4.

10.

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

11.

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23(11), 742–744 (2011). [CrossRef]

12.

G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express 19(26), B440–B451 (2011). [CrossRef]

13.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30(10), 1524–1539 (2012). [CrossRef]

14.

T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator in coherent transmission systems,” in Tech. Digest of European Conference on Optical Communication,2012, P3.11.

15.

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Tech. Digest of European Conference on Optical Communication,2011, paper We.7.B.2.

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2330) Fiber optics and optical communications : Fiber optics communications

ToC Category:
Subsystems for Optical Networks

History
Original Manuscript: October 2, 2012
Revised Manuscript: November 15, 2012
Manuscript Accepted: November 16, 2012
Published: December 12, 2012

Virtual Issues
European Conference on Optical Communication 2012 (2012) Optics Express

Citation
Takahito Tanimura, Markus Nölle, Johannes Karl Fischer, and Colja Schubert, "Analytical results on back propagation nonlinear compensator with coherent detection," Opt. Express 20, 28779-28785 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28779


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References

  1. T. Tanimura, T. Hoshida, S. Oda, T. Tanaka, C. Ohsima, and J. C. Rasmussen, “Systematic analysis on multi-segment dual-polarisation nonlinear compensation in 112 Gb/s DP-QPSK coherent receiver,” in Tech. Digest of European Conference on Optical Communication,2009, paper 9.4.5.
  2. S. Oda, T. Tanimura, T. Hoshida, C. Ohshima, H. Nakashima, Z. Tao, and J. C. Rasmussen, “112 Gb/s DP-QPSK transmission using a novel nonlinear compensator in digital coherent receiver,” in Tech. Digest of Optical Fiber Communication Conference,2009, paper OThR6.
  3. E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol.28(6), 939–951 (2010). [CrossRef]
  4. D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010). [CrossRef]
  5. D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express19(4), 3449–3454 (2011). [CrossRef] [PubMed]
  6. L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The limits of digital backpropagation in nonlinear coherent fiber-optic links,” in Tech. Digest of European Conference on Optical Communication,2012, P4.14.
  7. G. Gao, X. Chen, and W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation in the presence of PMD,” in Tech. Digest of Optical Fiber Communication Conference,2012, paper OM3A.5.
  8. T. Tanimura, S. Oda, T. Hoshida, L. Li, Z. Tao, and J. C. Rasmussen, “Experimental characterization of nonlinearity mitigation by digital back propagation and nonlinear polarization crosstalk canceller under high PMD condition,” in Tech. Digest of Optical Fiber Communication Conference,2011, JWA020.
  9. A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” in Tech. Digest of European Conference on Optical Communication,1993, paper MoC2.4.
  10. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010). [CrossRef] [PubMed]
  11. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011). [CrossRef]
  12. G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express19(26), B440–B451 (2011). [CrossRef]
  13. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012). [CrossRef]
  14. T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator in coherent transmission systems,” in Tech. Digest of European Conference on Optical Communication,2012, P3.11.
  15. E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Tech. Digest of European Conference on Optical Communication,2011, paper We.7.B.2.

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