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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 13 — Jun. 18, 2012
  • pp: 14406–14418
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Influence of PMD on fiber nonlinearity compensation using digital back propagation

Guanjun Gao, Xi Chen, and William Shieh  »View Author Affiliations


Optics Express, Vol. 20, Issue 13, pp. 14406-14418 (2012)
http://dx.doi.org/10.1364/OE.20.014406


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Abstract

With ideal nonlinearity compensation using digital back propagation (DBP), the transmission performance of an optical fiber channel has been considered to be limited by nondeterministic nonlinear signal-ASE interaction. In this paper, we conduct theoretical and numerical study on nonlinearity compensation using DBP in the presence of polarization-mode dispersion (PMD). Analytical expressions of transmission performance with DBP are derived and substantiated by numerical simulations for polarization-division-multiplexed systems under the influence of PMD effects. We find that nondeterministic distributed PMD impairs the effectiveness of DBP-based nonlinearity compensation much more than nonlinear signal-ASE interaction, and is therefore the fundamental limitation to single-mode fiber channel capacity.

© 2012 OSA

1. Introduction

Linear fiber impairment has been elegantly resolved by electronic digital signal processing (DSP) in either coherent single-carrier or OFDM systems, and fiber nonlinearity is viewed as the ultimate limiting factor to the fiber channel capacity [1

1. E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008). [CrossRef] [PubMed]

3

3. W. Shieh and C. Athaudage, “coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006). [CrossRef]

]. Many methods have been proposed to mitigate nonlinear impairments, including (i) designing nonlinearity tolerant channels, e.g., optimization of chromatic dispersion maps, application of optical phase conjugation and deployment of large effective area fibers etc. [4

4. A. Chowdhury, G. Raybon, R. J. Essiambre, J. H. Sinsky, A. Adamiecki, J. Leuthold, C. R. Doerr, and S. Chandrasekhar, “Compensation of intrachannel nonlinearities in 40-Gb/s pseudolinear systems using optical-phase conjugation,” J. Lightwave Technol. 23(1), 172–177 (2005). [CrossRef]

,5

5. S. L. Jansen, D. van den Borne, B. Spinnler, S. Calabrò, H. Suche, P. M. Krummrich, W. Sohler, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra-long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24(1), 54–64 (2006). [CrossRef]

], (ii) designing nonlinearity robust modulation formats such as constant envelope with lower PAPR and signal sub-band bandwidth optimization [6

6. W. Shieh and Y. Tang, “Ultrahigh-speed signal transmission over nonlinear and dispersive fiber optic channel: the multicarrier advantage,” IEEE Photon. J. 2(3), 276–283 (2010). [CrossRef]

], and (iii) digital nonlinear impairments compensation at either transmitter or receivers [7

7. X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009). [CrossRef]

]. Among these studies, digital back propagation (DBP) has been proposed to compensate the deterministic fiber nonlinear impairments by digitally back-propagating the received signals which undergo both linear chromatic dispersion and fiber nonlinearity [8

8. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]

17

17. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

]. Various factors that influence the performance of DBP have been investigated including the signal bandwidth, sampling rate, step-size and polarization-mode dispersion [8

8. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]

17

17. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

]. It is assumed that with the exact knowledge of channel information, the deterministic nonlinearity interactions between signals can be completely removed with fine enough back propagation steps and processing power. Subsequently, the fiber capacity will be mainly dependent on the non-deterministic effects including (i) nonlinear interaction between ASE and signals [18

18. R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]

,19

19. 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]

] and (ii) stochastic polarization dependent nonlinearity interaction [16

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

,17

17. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

,20

20. G. Gao, X. Chen, W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation,” OFC’2012, paper OMA3.

]. Without consideration of polarization-mode dispersion (PMD), the nonlinear signal-ASE interaction has been regarded as the fundamental limitation to single-mode fiber system capacity [19

19. 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]

]. However, numerical study shows that PMD will impact the effectiveness of digital back propagation significantly [16

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

,17

17. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

]. In this work, we provide a theoretical study of the influence of PMD on the effectiveness of DBP for polarization-division-multiplexed (PDM) systems. Substantiated by numerical simulations, the theory is used to evaluate the system performance with DBP, capturing the stochastic nature of PMD effect. It is shown that nondeterministic distributed PMD impairs the effectiveness of DBP-based nonlinearity compensation much more than nonlinear signal-ASE interaction, and is therefore the fundamental limitation to the single-mode fiber (SMF) channel capacity.

2. Theoretical derivations of nonlinear transmission performance using DBP

It has been proved that for densely-spaced OFDM systems, all the third-order nonlinear interactions can be considered as four-wave-mixing (FWM) [21

21. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

,22

22. 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]

]. The FWM interaction of subcarriers at frequencies of fifjand fkwould produce a mixing product at frequency offg=fi+fjfk. For PDM systems, the propagation equation for the FWM component at frequencyfg, is given by [23

23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. 3(2), 158–173 (2011). [CrossRef]

]
icg'z+iα2cg'12p(z)σωgcg'+12β2ωg2cg'+γ[(ck'+ci')cj'+(ck'+cj')ci']=0
(1)
where cg'is the FWM component invoked by subcarrier ci,cjand ckafter transmission of M fiber spans, and can be expressed as [22

22. 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]

,23

23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. 3(2), 158–173 (2011). [CrossRef]

]
cg'=γ[(ck+ci)cj+(ck+cj)ci]eαL/2iβgL1eαLjΔβijkLjΔβijk+α1exp(jMΔβ˜ijk)1exp(jΔβ˜ijk)
(2)
where, α,L,ζand β2are fiber loss coefficient, fiber length per span, residual dispersion ratio per span and group-velocity dispersion parameter respectively, superscript ‘ + ’ stands for Hermitian conjugate. γ=8γ0/9is the third-order nonlinear coefficient for fiber with randomly varying birefringence eigen axis. For systems with N subcarriers, the power of FWM noise generated at fg can be obtained as [22

22. 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]

,23

23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. 3(2), 158–173 (2011). [CrossRef]

]
PNL,M=12k=N/2N/2j=N/2N/2Pg,M=12k=N/2N/2j=N/2N/232PiPjPkγ2eαLη1η2η1=|1eαLejΔβijkLjΔβijk+α|21β22(2π)4Δf4j2(kj)2+α2η2=|1exp(jMΔβ˜ijk)1exp(jΔβ˜ijk)|2=sin2(Mj(kj)Δf2(2π)2|β2|2L2ζ2/2)sin2(j(kj)Δf2(2π)2|β2|2L2ζ2/2)
(3)
wherePNL,Mis the average FWM noise power generated at fgand Pi,j,kis the power of interfering subcarrier.η1and η2stand for FWM efficiency for single span and multi-span FWM interference effect [22

22. 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]

24

24. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17(11), 801–803 (1992). [CrossRef] [PubMed]

]. It has been proved that for dispersion uncompensated systems, FWM noise generated in each span are independent and do not interference with each other [22

22. 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]

,23

23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. 3(2), 158–173 (2011). [CrossRef]

]. By converting the summation into integration through f=jΔfand f1=(jk)Δf, the nonlinear noise intensity can be expressed as [22

22. 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]

,23

23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. 3(2), 158–173 (2011). [CrossRef]

]
INL,M=PNL,MΔf=3γ2I34β22(2π)4B/2f1B/2f1B/2B/2η1(f,f1)η2(f,f1)dfdf1=3γ2I3Mln(B/B0)he8πα|β2|he=2(M1+eαLζMMeαLζ)eαLζM(eαLζ1)2+1
(4)
whereBis the signal bandwidth andB0is defined as |β2|/(2π2αB).Iis the signal power density and heis the multi-span FWM interference enhancement factor, which can be approximated as 1 for dispersion uncompensated systems. Then the nonlinear transmission performance such as SNR, maximum SNR, optimum launch power density, spectral efficiency (SE) can be obtained using the closed form solution of Eq. (4) according to [22

22. 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]

,23

23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. 3(2), 158–173 (2011). [CrossRef]

]. Based on the above derivations, we further consider the influence of nondeterministic effects including nonlinear signal-ASE interaction and PMD, and arrive at the closed-form expressions of nonlinear transmission performance considering these effects for dispersion uncompensated PDM systems with DBP.

2.1 Nonlinear Signal-Noise interaction

2.2 PMD-induced uncompensated nonlinear noise

Under influence of PMD, following Eq. (2), the FWM product component invoked by subcarrieri,jandk can be expressed as
cg,M'=γ[(ckp+cip)cjp+(ckp+cjp)cip]eαL/2iβgL1eαLjΔβijkLjΔβijk+α
(12)
where, ci,j,kpis the OFDM subcarriers influenced by PMD after transmission of M spans. It is observed from Eq. (12) that the generated FWM components depend on the polarization states of interfering subcarriers, which change stochastically due to PMD. An intuitive illustration of the PMD influence is as shown in Fig. 2
Fig. 2 Illustration of PMD influence on nonlinear four-wave mixing generation. Sub.: Subcarrier.
.

Then the residual FWM power generated at frequency fg of Mth span averaged over fiber polarization state is given by
Pg,M'=E{|cg,M'|2}=γ2(Δβijk)2+α2E{|(ckp+cip)cjp+(ckp+cjp)cip(ck+ci)cj(ck+cj)ci|2}=3P3γ2(Δβijk)2+α2[134exp(3π3LMDp2f1216)14exp(7LMDp2f1216)]=Pg,M0RPMDRPMD=2[134exp(3π3LMDp2f1216)14exp(7π3LMDp2f1216)]
(15)
wherePg,M0is the generated FWM noise power for systems without DBP-based nonlinear compensation and RPMDis the normalized residual FWM power ratio after DBP. Similar to the derivation of Eq. (4), nonlinear noise density generated at Mth span can be obtained by carrying out the following integrations
IPMDM=3γ2β22(2π)4B0/2B/20RPMD1(f1f)2+fW4dfdf1   =3γ2I34παβ2B0/2B/21f1[134exp(3π3LMDp2f1216)14exp(7LMDp2f1216)]df1   =3γ2I34παβ2ln(B/B0)9γ2I332παβ2[Ei(3π3B2LMDp264)Ei(3π3B0LMDp264)]            3γ2I332παβ2[Ei(7π3B2LMDp264)Ei(7π3B2LMDp264)]
(16)
where Ei is a special function defined as Ei(x)=x(et/t)dt [25

25. A. Milton and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Abramowitz and Stegun, eds. (Dover, 1964).

].

Due to the fact that the FWM noise generated along each span is independent and does not interference with each other for dispersion uncompensated systems, the overall residual FWM noise density is therefore the summation of FWM noise density at each span, expressed as
IPMD=M=1/2Ns1/2IPMDM=3γ2I332παβ2{8Nsln(B/B0)M=1/2Ns1/2[3E1(M)+E2(M)]}E1(M)=Ei(3π3B2LMDp264)Ei(3π3B0LMDp264)E2(M)=Ei(7π3B2LMDp264)Ei(7π3B0LMDp264)
(17)
In Eq. (17), the average DGD at the midpoint of each span is used to approximate the distributed characteristic of PMD effect. Similar to Eq. (11), the system performance parameters under the influence of PMD can be expressed as
 IPMD=(I/I0,PMD)2I,  SNR=I2n0+(I/I0,PMD)2I, IPMDopt=(n0I0,PMD2)1/3 SNRPMDmax=13(I0,PMDn0)2/3, SPMDmax=2log2(1+13(I0,PMDn0)2/3)I0,PMD=1γ32πα|β2|24Nsln(B/B0)M=1/2Ns1/2[9E1(M)+3E2(M)]  
(18)
where subscript ‘PMD’ stands for quantities being influenced by PMD. Finally, the overall nonlinear noise intensity under the influence of both nonlinear signal-ASE interaction and distributed nondeterministic PMD can be given by

INL=INASE+IPMD,    SNR = I2n0+INL
(19)

3. Simulation results and discussions

In order to compensate all the third-order nonlinearities including FWM among different wavelength channels, phase locked LO arrays are required to preserve their phase relations [17

17. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

]. After up sampling and signal field reconstruction, received signals are back propagated with the inverse value of fiber parameters used in the forward transmission. Both the forward and backward propagation are governed by nonlinear Schrödinger vector equation with Manakov-PMD approximation [26

26. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996). [CrossRef]

,27

27. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15(9), 1735–1746 (1997). [CrossRef]

]. The parameters used for simulation are as follows: 21 frequency-continuous channels each covering 25 GHz bandwidth with QPSK modulated subcarriers are transmitted along 10 spans of 100-km uncompensated SSMF fiber link. The forward and backward fiber span parameters are as shown in Table 1

Table 1. Fiber Span Parameters used in Simulation

table-icon
View This Table
. The operation bandwidth of back propagation is the same as forward propagation, much larger than the entire signal bandwidth, performing the ‘full-band’ nonlinearity compensation. The influences of nonlinear signal-ASE interaction and PMD on digital back propagation are simulated separately. Without PMD, the Q factor after nonlinearity compensation is a fixed value. However, under influence of PMD, the Q factor becomes stochastic. Therefore 100 PMD realizations are performed for each specific launch power level to capture this stochastic influence. The statistical distributions of Q factor with PMD parameter of 0.05 and 0.1 ps/kmare shown in Fig. 4
Fig. 4 Statistical distribution of Q factor for (a) launch power of 15 dBm and PMD of 0.05 ps/sqrt(km) (b) launch power of 15 dBm and PMD of 0.1 ps/sqrt(km), (c) launch power of 18 dBm and PMD of 0.05 ps/sqrt(km), and (d) launch power of 18dBm and PMD of 0.1 ps/sqrt(km). Results are obtained over 10 spans of 100 km SSMF links. For each combination of launch power and PMD coefficient, 100 cases are simulated.
.

Comparisons between numerical simulation and theoretical results under both PMD and nonlinear signal-ASE interaction are shown in Fig. 5
Fig. 5 SNR versus launch power under different nonlinear interactions. Linear: linear transmission regime without nonlinear effects. N-ASE: nonlinear signal-ASE interaction impaired DBP systems. PMD: distributed PMD impaired DBP systems. The unit of PMD parameter isps/km. Avg: average SNR obtained from 100 PMD realizations. NBP: no back-propagation nonlinearity compensation. Open symbols are for simulation results and dashed lines for theoretical results. Results are obtained over 10 spans of 100 km SSMF links.
. For distributed PMD impaired DBP systems, average Q factors are presented here and worse performance would also occur due to the stochastic distributions of DGD. From Fig. 5 it is observed our theories match with simulations quite well. The discrepancy of the SNRs between the simulation and theory is less than 1.2 dB. It is also shown that even for systems with a small PMD parameter of 0.05 ps/km, the maximum SNR difference between distributed nonlinear signal-ASE interaction and PMD impaired systems is 3.8 dB, showing much more severe impact of PMD than nonlinear signal-ASE interaction. Compared to systems without nonlinearity compensation, the maximum SNR of DBP systems with PMD parameter of 0.05 and 0.1 ps/kmare improved by 4.3 and 2.7 dB respectively.

By using the derived expressions of Eqs. (11) and (18), we then investigate the information capacity similar to [22

22. 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]

,28

28. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature 411(6841), 1027–1030 (2001). [CrossRef] [PubMed]

], and explore the system performance over the entire 5-THz C band, where the simulations studies would be very likely time prohibitive. The analytical results of nondeterministic PMD impairments are obtained for varying mean PMD values. As shown in Fig. 6(a)
Fig. 6 System performance over 5-THz C band for (a) SNR versus launch power density for 40 spans 100 km SSMF transmissions and (b) spectral efficiency versus number of spans. Linear: linear transmission regime without nonlinear effects. N-ASE: nonlinear signal-ASE limited regimes. PMD: nonlinearity compensation influenced by PMD, the unit of PMD parameter isps/km; NBP: no back-propagation nonlinearity compensation. For PMD limited regimes, mean system performance has been assumed.
, after 40 spans of 100 km SSMF transmissions, PMD has a much more severe influence on the system performance than nonlinear signal-ASE interaction for polarization multiplexing systems with DBP. Even for an extremely small PMD of 0.01 ps/km, the maximum SNR difference between PMD limited and nonlinear signal-ASE interaction limited regimes is more than 2.3 dB. For PMD of 0.05 ps/km, this difference can be as large as 3.6 dB. With a large PMD of 0.8 ps/km, the improvement of nonlinear compensation using DBP becomes negligible. It is worth noting that the results are based on the mean performance and there are many fiber polarization state realizations where the system performance will be worse than these results. In practice, DBP with a course-step size such as 1 step/span would be sufficient to achieve quite good performance, especially for small signal bandwidth, where the polarization states are more likely to be correlated among different frequencies [14

14. L. B. Du, B. Schmidt, and A. Lowery, “Efficient digital backpropagation for PDM-CO-OFDM optical transmission systems,” OFC’ 2010, paper OTuE2.

]. The spectral efficiency that can be achieved is shown in Fig. 6(b), suggesting that even with a small PMD parameter of 0.05ps/km, the fiber channel spectral efficiency and capacity are fundamentally limited by PMD, instead of nonlinear signal-ASE interaction.

4. Conclusion

In this paper, we have derived analytical expressions of nonlinear transmission performance of polarization-division-multiplexed (PDM) systems with digital back propagation, under the influence stochastic PMD effect. These analytical expressions are substantiated by numerical simulations. It is demonstrated that nondeterministic distributed PMD impairs the effectiveness of DBP-based nonlinearity compensation much more than nonlinear signal-ASE interaction, and is therefore the fundamental limitation to single-mode fiber channel capacity. Based on the analytical expressions, the ultimate nonlinear Shannon capacity of single mode fiber systems is then obtained.

Appendix A

DenotingU=(Ux,Uy)T=[(sk+si)sj+(sk+sj)si(ck+ci)cj(ck+cj)ci], and expanding Uinto two polarization components, we obtain

Ux=2sxk*sxisxj+syk*syisxj+syk*syjsxi(2cxk*cxicxj+cyk*cyicxj+cyk*cyjcxi)
(20)

where the superscript ‘*’ stands for scalar conjugate and sx,yi,j,k=cx,yi,j,k+nx,yi,j,kis the OFDM subcarrier cx,yi,j,k, contaminated with ASE noise nx,yi,j,k. Assuming OFDM symbol cx,yi,j,kand noise nx,yi,j,kon different polarizations or different subcarriers are uncorrelated, namely cx1j*cx2k=P2δx1x2δjkand nx1j*nx2k=Pn2δx1x2δjkwhere x1,2stands for x or y polarization component, we obtain the ensemble average power of Uxas

|Ux|2=|2(cxk*+nxk*)(cxi+nxi)(sxj+nxj)+(cyk*+nyk*)(cyi+nyi)(cxj+nxj)                  +(cyk*+nyk*)(cyj+nyj)(cxi+nxi)(2cxk*cxicxj+cyk*cyicxj+cyk*cyjcxi)|2          =94P2Pn+94PPn2+94Pn394P2Pn
(21)

where, PandPnare the power of signal and ASE noise over each subcarrier frequency respectively. By using |Ux|2=|Uy|2, the ensemble average of signal power becomes

|(sk+si)sj+(sk+sj)si(ck+ci)cj(ck+cj)ci|2=|Ux|2+|Uy|2=92P2Pn
(22)

Substituting Eqs. (22) into Eq. (5), we obtain Eq. (6).

Appendix B

DenotingU=(Ux,Uy)T=(ckp+cip)cjp+(ckp+cjp)cip(ck+ci)cj(ck+cj)ci, and expanding Uinto the two polarization components, we obtain

Ux=2(ckxp*cixpcjxpckx*cixcjx)+(ckyp*ciypcjxpcky*ciycjx)+(ckyp*cjypcixpcky*cjycix)
(23)

ck(j)xp=ak(j)ck(j)x+bk(j)ck(j)y,      ck(j)yp=bk(j)*ck(j)x+ak(j)*ck(j)y  
(24)

Then Uxcan be expressed as

Ux=ABA=2ckxp*cixpcjxp+ckyp*ciypcjxp+ckyp*cjypcixp,B=2ckx*cixcjx+cky*ciycjx+cky*cjycix
(25)

where Aand Bare comprised of polarization dependent and independent terms respectively. The average power of Ux is expressed as

E{|Ux|2}=E{|A|2}+E{|B|2}2Re[E{A*B}]
(26)

where ‘Re’ stands for real component of a variable. Taking cias reference subcarrier and assuming its polarization is unchanged along the transmission, we have

cixp*cix=cixpcix*=|cix|2,  ciyp*ciy=ciypciy*=|ciy|2
(27)

We assume that in absence of PMD, the OFDM information symbol ci,j,k;x,yon different polarizations or different subcarriers are uncorrelated, namely cx1j*cx2k=P2δx1x2δjkwith x1,2stands for x or y polarization component. Then from Eqs. (24) and (25), we obtain:

E{|A|2}=E(|B|2)=4|ckx|2|cix|2|cjx|2+|cky|2|ciy|2|cjx|2+|cky|2|cjy|2|cix|2=34P3E{A*B}=E[(2ckxp*cixpcjxp+ckyp*ciypcjxp+ckyp*cjypcixp)*(2ckxp*cixpcjxp+ckyp*ciypcjxp+ckyp*cjypcixp)]  =18P3E{5ak*aj+2bk*bj+2bj*bk+akaj}
(28)

Following the same approach as shown for the statistics of the Jones Matrix [29

29. A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. 20(5), 811–821 (2002). [CrossRef]

,30

30. A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett. 26(10), 675–677 (2001). [CrossRef] [PubMed]

], the correlation coefficients of Jones Matrix elements at subcarrier jand kcan be proved as:

E{ak*aj}=12[exp(Δτ2¯Δϖ28)+exp(7Δτ2¯Δϖ224)]E{bk*bj}=E{bkbj*}=12[exp(Δτ2¯Δϖ28)exp(7Δτ2¯Δϖ224)]E{akaj}=exp(7Δτ2¯Δϖ224)
(29)

where Δτ2¯=3πΔτ¯2/8with Δτ¯ denoting the average DGD and Δϖ=2π(fjfk)is the angle frequency difference between fjand fk. We have omitted the relatively lengthy steps to derive Eq. (29). We also note that the statistics of Jones matrixMk(j)defined in this paper is different than that of conventional Jones matrix in [29

29. A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. 20(5), 811–821 (2002). [CrossRef]

,30

30. A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett. 26(10), 675–677 (2001). [CrossRef] [PubMed]

] because we enforceMk(j)as a 2x2 identity matrix for the ith subcarrier (the reference subcarrier). Substituting Eqs. (28) and (29) into Eq. (26), we obtain

E{|Ux|2}=P38[129exp(Δτ2¯Δϖ28)3exp(7Δτ2¯Δϖ224)]
(30)

Thus the ensemble average power of Uis obtained as

E{|U|2}=E{|Ux|2+|Uy|2}=2E{|Ux|2}         =14P3[129exp(Δτ2¯Δϖ28)3exp(7Δτ2¯Δϖ224)]
(31)

References and links

1.

E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008). [CrossRef] [PubMed]

2.

S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express 15(5), 2120–2126 (2007). [CrossRef] [PubMed]

3.

W. Shieh and C. Athaudage, “coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006). [CrossRef]

4.

A. Chowdhury, G. Raybon, R. J. Essiambre, J. H. Sinsky, A. Adamiecki, J. Leuthold, C. R. Doerr, and S. Chandrasekhar, “Compensation of intrachannel nonlinearities in 40-Gb/s pseudolinear systems using optical-phase conjugation,” J. Lightwave Technol. 23(1), 172–177 (2005). [CrossRef]

5.

S. L. Jansen, D. van den Borne, B. Spinnler, S. Calabrò, H. Suche, P. M. Krummrich, W. Sohler, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra-long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24(1), 54–64 (2006). [CrossRef]

6.

W. Shieh and Y. Tang, “Ultrahigh-speed signal transmission over nonlinear and dispersive fiber optic channel: the multicarrier advantage,” IEEE Photon. J. 2(3), 276–283 (2010). [CrossRef]

7.

X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009). [CrossRef]

8.

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]

9.

E. F. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express 16(20), 16124–16137 (2008). [CrossRef] [PubMed]

10.

E. F. Mateo and G. Li, “Compensation of interchannel nonlinearities using enhanced coupled equations for digital backward propagation,” Appl. Opt. 48(25), F6–F10 (2009). [CrossRef] [PubMed]

11.

E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010). [CrossRef] [PubMed]

12.

E. F. Mateo, X. Zhou, and G. Li, “Improved digital backward propagation for the compensation of inter-channel nonlinear effects in polarization-multiplexed WDM systems,” Opt. Express 19(2), 570–583 (2011). [CrossRef] [PubMed]

13.

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010). [CrossRef] [PubMed]

14.

L. B. Du, B. Schmidt, and A. Lowery, “Efficient digital backpropagation for PDM-CO-OFDM optical transmission systems,” OFC’ 2010, paper OTuE2.

15.

E. Ip and J. M. Kahn, “Nonlinear impairment compensation using backpropagation,” in Optical Fiber, New Developments, C. Lethien, Ed., In-Tech, Vienna Austria, December (2009).

16.

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

17.

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

18.

R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]

19.

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]

20.

G. Gao, X. Chen, W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation,” OFC’2012, paper OMA3.

21.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

22.

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]

23.

W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J. 3(2), 158–173 (2011). [CrossRef]

24.

K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17(11), 801–803 (1992). [CrossRef] [PubMed]

25.

A. Milton and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Abramowitz and Stegun, eds. (Dover, 1964).

26.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996). [CrossRef]

27.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15(9), 1735–1746 (1997). [CrossRef]

28.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature 411(6841), 1027–1030 (2001). [CrossRef] [PubMed]

29.

A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. 20(5), 811–821 (2002). [CrossRef]

30.

A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett. 26(10), 675–677 (2001). [CrossRef] [PubMed]

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

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: April 16, 2012
Revised Manuscript: May 14, 2012
Manuscript Accepted: May 14, 2012
Published: June 13, 2012

Citation
Guanjun Gao, Xi Chen, and William Shieh, "Influence of PMD on fiber nonlinearity compensation using digital back propagation," Opt. Express 20, 14406-14418 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14406


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References

  1. E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express16(2), 753–791 (2008). [CrossRef] [PubMed]
  2. S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express15(5), 2120–2126 (2007). [CrossRef] [PubMed]
  3. W. Shieh and C. Athaudage, “coherent optical orthogonal frequency division multiplexing,” Electron. Lett.42(10), 587–589 (2006). [CrossRef]
  4. A. Chowdhury, G. Raybon, R. J. Essiambre, J. H. Sinsky, A. Adamiecki, J. Leuthold, C. R. Doerr, and S. Chandrasekhar, “Compensation of intrachannel nonlinearities in 40-Gb/s pseudolinear systems using optical-phase conjugation,” J. Lightwave Technol.23(1), 172–177 (2005). [CrossRef]
  5. S. L. Jansen, D. van den Borne, B. Spinnler, S. Calabrò, H. Suche, P. M. Krummrich, W. Sohler, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra-long-haul phase-shift-keyed transmission,” J. Lightwave Technol.24(1), 54–64 (2006). [CrossRef]
  6. W. Shieh and Y. Tang, “Ultrahigh-speed signal transmission over nonlinear and dispersive fiber optic channel: the multicarrier advantage,” IEEE Photon. J.2(3), 276–283 (2010). [CrossRef]
  7. X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol.27(16), 3632–3640 (2009). [CrossRef]
  8. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol.26(20), 3416–3425 (2008). [CrossRef]
  9. E. F. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express16(20), 16124–16137 (2008). [CrossRef] [PubMed]
  10. E. F. Mateo and G. Li, “Compensation of interchannel nonlinearities using enhanced coupled equations for digital backward propagation,” Appl. Opt.48(25), F6–F10 (2009). [CrossRef] [PubMed]
  11. E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express18(14), 15144–15154 (2010). [CrossRef] [PubMed]
  12. E. F. Mateo, X. Zhou, and G. Li, “Improved digital backward propagation for the compensation of inter-channel nonlinear effects in polarization-multiplexed WDM systems,” Opt. Express19(2), 570–583 (2011). [CrossRef] [PubMed]
  13. L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express18(16), 17075–17088 (2010). [CrossRef] [PubMed]
  14. L. B. Du, B. Schmidt, and A. Lowery, “Efficient digital backpropagation for PDM-CO-OFDM optical transmission systems,” OFC’ 2010, paper OTuE2.
  15. E. Ip and J. M. Kahn, “Nonlinear impairment compensation using backpropagation,” in Optical Fiber, New Developments, C. Lethien, Ed., In-Tech, Vienna Austria, December (2009).
  16. E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol.28(6), 939–951 (2010). [CrossRef]
  17. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J.2(5), 816–832 (2010). [CrossRef]
  18. R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28(4), 662–701 (2010). [CrossRef]
  19. 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]
  20. G. Gao, X. Chen, W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation,” OFC’2012, paper OMA3.
  21. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008). [CrossRef] [PubMed]
  22. 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]
  23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J.3(2), 158–173 (2011). [CrossRef]
  24. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett.17(11), 801–803 (1992). [CrossRef] [PubMed]
  25. A. Milton and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Abramowitz and Stegun, eds. (Dover, 1964).
  26. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996). [CrossRef]
  27. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15(9), 1735–1746 (1997). [CrossRef]
  28. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature411(6841), 1027–1030 (2001). [CrossRef] [PubMed]
  29. A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol.20(5), 811–821 (2002). [CrossRef]
  30. A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett.26(10), 675–677 (2001). [CrossRef] [PubMed]

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