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 Optical Society of America
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 [1–3]. 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. [4,5], (ii) designing nonlinearity robust modulation formats such as constant envelope with lower PAPR and signal sub-band bandwidth optimization , and (iii) digital nonlinear impairments compensation at either transmitter or receivers . 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 [8–17]. Various factors that influence the performance of DBP have been investigated including the signal bandwidth, sampling rate, step-size and polarization-mode dispersion [8–17]. 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 [18,19] and (ii) stochastic polarization dependent nonlinearity interaction [16,17,20]. 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 . However, numerical study shows that PMD will impact the effectiveness of digital back propagation significantly [16,17]. 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) [21,22]. The FWM interaction of subcarriers at frequencies of andwould produce a mixing product at frequency of. For PDM systems, the propagation equation for the FWM component at frequency, is given by 22,23]22,23]22–24]. It has been proved that for dispersion uncompensated systems, FWM noise generated in each span are independent and do not interference with each other [22,23]. By converting the summation into integration through and , the nonlinear noise intensity can be expressed as [22,23]Eq. (4) according to [22,23]. 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
For systems with ideal nonlinear compensation using DBP as shown in Fig. 1 , signals are transmitted along spans (‘A’ to ‘C’) and digitally back propagated through virtual fiber spans (‘C'’ to ‘A'’) with inverse value of fiber parameters used in forward propagation, removing any of the deterministic nonlinear effect. However, with ASE noise added at each amplifier in the forward propagation, the distributive nonlinear signal-ASE interaction cannot be compensated after spans of digital back propagation. For instance, considering the ASE noise () generated at Mth fiber link (Location ‘B’), the signal and the noise are transmitted from Mth span (‘B’) to the receiver (‘C’) after spans forward transmission. With exactly spans of back propagation (from ‘C'’ to ‘B'’), the nonlinear interaction between and signal can be removed. However, after back propagated by further spans (from ‘B'’ to ‘A'’), extra nonlinear signal noise interaction is generated, equivalent to the nonlinear signal-ASE interaction of spans. The uncompensated nonlinear components generated by Mth span EDFA can be expressed as22,23], arriving at the similar expressions of Eq. (4), but considering the influence of nonlinear signal-ASE beating interactions in addition to nonlinear signal-signal beating interactions.
As shown in Appendix A, the power of nonlinear signal-ASE beating component at subcarrier can be expressed asEqs. (3) and (4) [22,23], the nonlinear signal-ASE beating density caused by the noise originated from the Mth amplifier can be expressed as22,23]
2.2 PMD-induced uncompensated nonlinear noise
Under influence of PMD, following Eq. (2), the FWM product component invoked by subcarrier,and can be expressed asEq. (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 .
Under influence of PMD, the polarization states of subcarriers with frequency difference beyond the PMD correlation bandwidth will experience different evolutions, which are stochastic and cannot be accurately estimated during the intermediate steps of back propagation. Therefore the nonlinearity compensation cannot be performed in perfection any more. Next we derive the analytical expressions of PMD influence on DBP addressing the stochastic nature of signal polarization states, extending our previous analysis using the correlations of PMD vectors . After digital back propagation, the residual FWM component generated along Mth fiber span can be expressed as following
Then the residual FWM power generated at frequency of Mth span averaged over fiber polarization state is given byEq. (4), nonlinear noise density generated at Mth span can be obtained by carrying out the following integrations25].
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 asEq. (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
3. Simulation results and discussions
We conduct numerical simulation to verify the close-form expressions for uncompensated nonlinearity noise under the influence of PMD and nonlinear signal-ASE interactions as discussed in the previous section. The schematic of simulation setup for the densely-spaced PDM-OFDM systems with DBP is shown in Fig. 3 .
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 . 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 [26,27]. 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 . 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 are shown in Fig. 4 .
Comparisons between numerical simulation and theoretical results under both PMD and nonlinear signal-ASE interaction are shown in Fig. 5 . 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 , 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 are 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 [22,28], 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) , 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 , 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 , this difference can be as large as 3.6 dB. With a large PMD of 0.8 , 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 . The spectral efficiency that can be achieved is shown in Fig. 6(b), suggesting that even with a small PMD parameter of 0.05, the fiber channel spectral efficiency and capacity are fundamentally limited by PMD, instead of nonlinear signal-ASE interaction.
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.
Denoting, and expanding into two polarization components, we obtain
where the superscript ‘*’ stands for scalar conjugate and is the OFDM subcarrier , contaminated with ASE noise . Assuming OFDM symbol and noise on different polarizations or different subcarriers are uncorrelated, namely and where stands for x or y polarization component, we obtain the ensemble average power of as
where, andare the power of signal and ASE noise over each subcarrier frequency respectively. By using , the ensemble average of signal power becomes
Denoting, and expanding into the two polarization components, we obtain
where and denote the OFDM information symbol for subcarrier , and without and with the influence of PMD respectively. Defining Jones matrix at subcarrier or as , can be related to by , namely,
Then can be expressed as
where and are comprised of polarization dependent and independent terms respectively. The average power of is expressed as
where ‘Re’ stands for real component of a variable. Taking as reference subcarrier and assuming its polarization is unchanged along the transmission, we have
We assume that in absence of PMD, the OFDM information symbol on different polarizations or different subcarriers are uncorrelated, namely with stands for x or y polarization component. Then from Eqs. (24) and (25), we obtain:
where with denoting the average DGD and is the angle frequency difference between and . We have omitted the relatively lengthy steps to derive Eq. (29). We also note that the statistics of Jones matrixdefined in this paper is different than that of conventional Jones matrix in [29,30] because we enforceas a 2x2 identity matrix for the subcarrier (the reference subcarrier). Substituting Eqs. (28) and (29) into Eq. (26), we obtain
Thus the ensemble average power of is obtained as
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