Influence of PMD on fiber nonlinearity compensation using digital back propagation

Guanjun Gao; Xi Chen; William Shieh

doi:10.1364/OE.20.014406

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 [1–3]. 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 [6], and (iii) digital nonlinear impairments compensation at either transmitter or receivers [7]. 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 [19]. 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 _{$f_{i}, f_{j}$}and_{$f_{k}$}would produce a mixing product at frequency of_{$f_{g} = f_{i} + f_{j} - f_{k}$}. For PDM systems, the propagation equation for the FWM component at frequency_{$f_{g}$}, is given by [23]

i \frac{\partial c_{g}^{'}}{\partial z} + i \frac{α}{2} c_{g}^{'} - \frac{1}{2} \vec{p} (z) \cdot \vec{σ} ω_{g} c_{g}^{'} + \frac{1}{2} β_{2} ω_{g}^{2} c_{g}^{'} + γ [(c_{k}^{' +} c_{i}^{'}) c_{j}^{'} + (c_{k}^{' +} c_{j}^{'}) c_{i}^{'}] = 0

where _{_{$c_{g}^{'}$}}is the FWM component invoked by subcarrier _{$c_{i}$},_{_{$c_{j}$}}and _{$c_{k}$}after transmission of M fiber spans, and can be expressed as [22,23]

c_{g}^{'} = γ [(c_{k}^{+} c_{i}) c_{j} + (c_{k}^{+} c_{j}) c_{i}] e^{- α L / 2 - i β_{g} L} \frac{1 - e^{- α L - j Δ β_{i j k} L}}{j Δ β_{i j k} + α} \frac{1 - \exp (j M \cdot Δ {\tilde{β}}_{i j k})}{1 - \exp (j Δ {\tilde{β}}_{i j k})}

where, _$α$,_$L$,_$ζ$and _{$β_{2}$}are fiber loss coefficient, fiber length per span, residual dispersion ratio per span and group-velocity dispersion parameter respectively, superscript ‘ + ’ stands for Hermitian conjugate. _{$γ = 8 γ_{0} / 9$}is 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 _{$f_{g}$} can be obtained as [22,23]

\begin{array}{l} P_{N L, M} = \frac{1}{2} \sum_{k = - N / 2}^{N / 2} \sum_{j = - N / 2}^{N / 2} 〈 P_{g, M} 〉 = \frac{1}{2} \sum_{k = - N / 2}^{N / 2} \sum_{j = - N / 2}^{N / 2} \frac{3}{2} P_{i} P_{j} P_{k} γ^{2} e^{- α L} η_{1} η_{2} \\ η_{1} = {| \frac{1 - e^{- α L} e^{- j Δ β_{i j k} L}}{j Δ β_{i j k} + α} |}^{2} \approx \frac{1}{β_{2}^{2} {(2 π)}^{4} Δ f^{4} j^{2} {(k - j)}^{2} + α^{2}} \\ η_{2} = {| \frac{1 - \exp (j M Δ {\tilde{β}}_{i j k})}{1 - \exp (j Δ {\tilde{β}}_{i j k})} |}^{2} = \frac{\sin^{2} (M j (k - j) Δ f^{2} {(2 π)}^{2} {| β_{2} |}^{2} L^{2} ζ^{2} / 2)}{\sin^{2} (j (k - j) Δ f^{2} {(2 π)}^{2} {| β_{2} |}^{2} L^{2} ζ^{2} / 2)} \end{array}

where_{$P_{N L, M}$}is the average FWM noise power generated at _{$f_{g}$}and _{$P_{i, j, k}$}is the power of interfering subcarrier._{$η_{1}$}and _{$η_{2}$}stand for FWM efficiency for single span and multi-span FWM interference effect [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 _{$f = j Δ f$}and _{$f_{1} = (j - k) Δ f$}, the nonlinear noise intensity can be expressed as [22,23]

\begin{array}{l} I_{N L, M} = \frac{P_{N L, M}}{Δ f} = \frac{3 γ^{2} I^{3}}{4 β_{2}^{2} {(2 π)}^{4}} \int_{- B / 2 - f_{1}}^{B / 2 - f_{1}} \int_{B / 2}^{B / 2} η_{1} (f, f_{1}) η_{2} (f, f_{1}) d f d f_{1} = \frac{3 γ^{2} I^{3} M \ln (B / B_{0}) h_{e}}{8 π α | β_{2} |} \\ h_{e} = \frac{2 (M - 1 + e^{- α L ζ M} - M e^{- α L ζ}) e^{- α L ζ}}{M {(e^{- α L ζ} - 1)}^{2}} + 1 \end{array}

where_$B$is the signal bandwidth and_{$B_{0}$}is defined as _{$| β_{2} | / (2 π^{2} α B)$}._$I$is the signal power density and _{$h_{e}$}is 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 [22,23]. 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 _{_{$N_{s}$}}spans (‘A’ to ‘C’) and digitally back propagated through _{$N_{s}$}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 _{$N_{s}$}spans of digital back propagation. For instance, considering the ASE noise (_{$n_{0}$}) generated at M_th fiber link (Location ‘B’), the signal and the noise _{$n_{0}$}are transmitted from M_th span (‘B’) to the receiver (‘C’) after _{$N_{s} - M$}spans forward transmission. With exactly _{$N_{s} - M$}spans of back propagation (from ‘C'’ to ‘B'’), the nonlinear interaction between _{$n_{0}$}and signal can be removed. However, after back propagated by further _$M$spans (from ‘B'’ to ‘A'’), extra nonlinear signal noise interaction is generated, equivalent to the nonlinear signal-ASE interaction of _$M$spans. The uncompensated nonlinear components generated by M_th span EDFA can be expressed as

\begin{array}{l} c_{g, M}^{'} = γ [(s_{k}^{+} s_{i}) s_{j} + (s_{k}^{+} s_{j}) s_{i} - (c_{k}^{+} c_{i}) c_{j} - (c_{k}^{+} c_{j}) c_{i}] e^{- α L / 2} e^{- j β_{g} L} \cdot \\ \frac{1 - e^{- α L} e^{- j Δ β_{i j k} L}}{j Δ β_{i j k} + α} \frac{1 - \exp (j M \cdot Δ {\tilde{β}}_{i j k})}{1 - \exp (j Δ {\tilde{β}}_{i j k})} \end{array}

where, _{$s_{i, j, k} = c_{i, j, k} + n_{i, j, k}$}is the subcarrier signal contaminated with ASE noise introduced at the M_th amplifier, _{$n_{i, j, k}$}denoting the ASE noise added to subcarrier _{$i, j or k$}, and _{$c_{i, j, k}$}is the information symbol for subcarrier _{$i, j$}or _$k$after the first _$M$spans without added ASE noises. We now carry out analysis using the same procedures as in [22,23], 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.

Fig. 1 Illustration of nonlinear signal-ASE interaction for transmission systems using DBP. SSMF: standard single-mode fiber, EDFA: erbium doped fiber amplifier.

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As shown in Appendix A, the power of nonlinear signal-ASE beating component at subcarrier _{$f_{g}$}can be expressed as

P_{g, M}^{'} = {| c_{g, M}^{'} |}^{2} = \frac{9}{2} P^{2} P_{n} γ^{2} η_{1} η_{2}

where, _$P$and _{$P_{n}$}denote power of OFDM signal and ASE noise at each subcarrier. Following the similar derivations to Eqs. (3) and (4) [22,23], the nonlinear signal-ASE beating density caused by the noise originated from the M_th amplifier can be expressed as

I_{n o i s e, M} = \frac{9 γ^{2} M I^{2} I_{n} \ln ({B / B}_{0}) h_{e, M}}{8 π α | β_{2} |} = \frac{9 γ^{2} M I^{2} n_{0} \ln ({B / B}_{0}) h_{e, M}}{4 π α | β_{2} |}

where, _{$I_{n} = 2 n_{0} = M e^{α L} h \cdot υ \cdot N F$}is the optical ASE noise density accounting for both polarizations, _$h$ is the Planck constant, _$υ$is the light frequency, and _{$N F$} is the noise figure for each amplifier. _$B$is entire signal bandwidth, and _{$B_{0}$} and _{$h_{e}$}are expressed as [22,23]

B_{0} = \frac{α}{2 π^{2} B | β_{2} |}, h_{e, M} = \frac{2 (M - 1 + e^{- α L ζ M} - M e^{- α L ζ}) e^{- α L ζ}}{M {(e^{- α L ζ} - 1)}^{2}} + 1

It is noted that _{$h_{e, M}$}is the multi-span FWM interference enhancement factor and can be approximated as 1 for dispersion uncompensated systems. Then the overall nonlinear signal-noise beating density caused by all the amplifiers along the transmission links can be expressed as

I_{N - A S E} = \sum_{M = 1}^{N_{s}} I_{n o i s e, M} = \sum_{M = 1}^{N_{s}} \frac{9 M γ^{2} I^{2} n_{0} \ln ({B / B}_{0}) h_{e, M}}{4 π α | β_{2} |}

Approximating _{$h_{e}$}as 1 for dispersion uncompensated systems, _{$I_{N - A S E}$}can be further simplified as following

I_{N - A S E} = \sum_{M = 1}^{N_{s}} I_{n o i s e, M} = \frac{9 γ^{2} I^{2} n_{0} \ln ({B / B}_{0}) N_{s} (N_{s} + 1)}{8 π α | β_{2} |}

By introducing a fiber characteristic dependent factor_{$I_{0, N - A S E}$}, the concise expressions of nonlinear transmission performance under the influence of nonlinear signal-ASE interaction can be obtained as

\begin{array}{l} I_{N - A S E} = \frac{I^{2}}{I_{0, N - A S E}}, I_{0, N - A S E} = \frac{4 π α | β_{2} |}{\sum_{M = 1}^{N_{s}} 9 M γ^{2} n_{0} \ln ({B / B}_{0}) h_{e, M}}, SNR = \frac{I}{2 n_{0} + (I^{2} / I_{0, N - A S E})} \\ I_{N - A S E}^{o p t} = \sqrt{2 n_{0} I_{0, N - A S E}}, {SNR}_{N - A S E}^{\max} = \sqrt{I_{0, N - A S E} / 8 n_{0}}, S_{N - A S E}^{\max} = 2 \log_{2} (1 + \sqrt{I_{0, N - A S E} / 8 n_{0}}) \end{array}

where, _$SNR$is the signal-to-noise ratio, _{$I_{N - A S E}^{o p t}$},_{${SNR}_{N - A S E}^{\max}$}and _{$S_{N - A S E}^{\max}$}denote optimal launch power density, maximum SNR and maximum spectral efficiency, respectively.

2.2 PMD-induced uncompensated nonlinear noise

Under influence of PMD, following Eq. (2), the FWM product component invoked by subcarrier_$i$,_$j$and_$k$ can be expressed as

c_{g, M}^{'} = γ [(c_{k}^{p +} c_{i}^{p}) c_{j}^{p} + (c_{k}^{p +} c_{j}^{p}) c_{i}^{p}] e^{- α L / 2 - i β_{g} L} \frac{1 - e^{- α L - j Δ β_{i j k} L}}{j Δ β_{i j k} + α}

where, _{$c_{i, j, k}^{p}$}is 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.

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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 [20]. After digital back propagation, the residual FWM component generated along M_th fiber span can be expressed as following

c_{g, M}^{'} = γ [(c_{k}^{p +} c_{i}^{p}) c_{j}^{p} + (c_{k}^{p +} c_{j}^{p}) c_{i}^{p} - (c_{k}^{+} c_{i}) c_{j} - (c_{k}^{+} c_{j}) c_{i}] e^{- α L / 2} e^{- j β_{g} L} \frac{1 - e^{- α L} e^{- j Δ β_{i j k} L}}{j Δ β_{i j k} + α}

where, _{$((c_{k}^{+} c_{i}) c_{j} + (c_{k}^{+} c_{j}) c_{i})$} denotes the deterministic signals that can be removed by DBP if there is no PMD. Following the derivations in Appendix B, we obtain

\begin{array}{l} E {〈 {| (c_{k}^{p +} c_{i}^{p}) c_{j}^{p} + (c_{k}^{p +} c_{j}^{p}) c_{i}^{p} - (c_{k}^{+} c_{i}) c_{j} - (c_{k}^{+} c_{j}) c_{i} |}^{2} 〉} \\ = \frac{1}{4} P^{3} [12 - 9 \exp (- \frac{\bar{Δ τ_{m}^{2}} Δ ϖ^{2}}{8}) - 3 \exp (- \frac{7 \bar{Δ τ_{m}^{2}} Δ ϖ^{2}}{24})] \\ = 3 P^{3} [1 - \frac{3}{4} \exp (- \frac{3 π^{3} L \cdot M \cdot D_{p}^{2} f_{1}^{2}}{16}) - \frac{1}{4} \exp (- \frac{7 π^{3} L \cdot M \cdot D_{p}^{2} f_{1}^{2}}{16})] \end{array}

where ‘_{$E {}$}’ stands for the ensemble average over fiber polarization state, to be distinguished with ‘_{$〈〉$}’ which is the ensemble average over transmitted subcarrier constellations of the two polarizations. _{$\bar{Δ τ_{m}} \equiv E {Δ τ_{m}} = \sqrt{L \cdot M} D_{P}$}is the average DGD after _$M$spans, _{$D_{P}$}denotes fiber PMD parameter and _{$\bar{Δ τ_{m}^{2}} = 3 π \cdot {\bar{Δ τ_{m}}}^{2} / 8$} is used assuming Maxwellian distribution of _{$Δ τ_{m}$}. We have used over bar as a shorthand for _{$E {}$}. _{$Δ ϖ = 2 π f_{1}$}is the angle frequency difference and _{$f_{1}$}denotes the frequency difference between _{$f_{j}$}and _{$f_{k}$}.

Then the residual FWM power generated at frequency _{$f_{g}$} of M_th span averaged over fiber polarization state is given by

\begin{array}{l} P_{g, M}^{'} = E {〈 {| c_{g, M}^{'} |}^{2} 〉} = \frac{γ^{2}}{{(Δ β_{i j k})}^{2} + α^{2}} E {〈 {| (c_{k}^{p +} c_{i}^{p}) c_{j}^{p} + (c_{k}^{p +} c_{j}^{p}) c_{i}^{p} - (c_{k}^{+} c_{i}) c_{j} - (c_{k}^{+} c_{j}) c_{i} |}^{2} 〉} \\ = \frac{3 P^{3} γ^{2}}{{(Δ β_{i j k})}^{2} + α^{2}} [1 - \frac{3}{4} \exp (- \frac{3 π^{3} L \cdot M \cdot D_{p}^{2} f_{1}^{2}}{16}) - \frac{1}{4} \exp (- \frac{7 L \cdot M \cdot D_{p}^{2} f_{1}^{2}}{16})] = P_{g, M}^{0} R_{P M D} \\ R_{P M D} = 2 [1 - \frac{3}{4} \exp (- \frac{3 π^{3} L \cdot M \cdot D_{p}^{2} f_{1}^{2}}{16}) - \frac{1}{4} \exp (- \frac{7 π^{3} L \cdot M \cdot D_{p}^{2} f_{1}^{2}}{16})] \end{array}

where_{$P_{g, M}^{0}$}is the generated FWM noise power for systems without DBP-based nonlinear compensation and _{$R_{P M D}$}is the normalized residual FWM power ratio after DBP. Similar to the derivation of Eq. (4), nonlinear noise density generated at M_th span can be obtained by carrying out the following integrations

\begin{array}{l} I_{P M D}^{M} = \frac{3 γ^{2}}{β_{2}^{2} {(2 π)}^{4}} \int_{B_{0} / 2}^{B / 2} \int_{0}^{\infty} R_{P M D} \frac{1}{{(f_{1} f)}^{2} + f_{W}^{4}} d f d f_{1} \\ = \frac{3 γ^{2} I^{3}}{4 π α β_{2}} \int_{B_{0} / 2}^{B / 2} \frac{1}{f_{1}} [1 - \frac{3}{4} \exp (- \frac{3 π^{3} L \cdot M \cdot D_{p}^{2} f_{1}^{2}}{16}) - \frac{1}{4} \exp (- \frac{7 L \cdot M \cdot D_{p}^{2} f_{1}^{2}}{16})] d f_{1} \\ = \frac{3 γ^{2} I^{3}}{4 π α β_{2}} \ln (B / B_{0}) - \frac{9 γ^{2} I^{3}}{32 π α β_{2}} [E i (- \frac{3 π^{3} B^{2} L \cdot M \cdot D_{p}^{2}}{64}) - E i (- \frac{3 π^{3} B_{0} L \cdot M \cdot D_{p}^{2}}{64})] \\ - \frac{3 γ^{2} I^{3}}{32 π α β_{2}} [E i (- \frac{7 π^{3} B^{2} L \cdot M \cdot D_{p}^{2}}{64}) - E i (- \frac{7 π^{3} B^{2} L \cdot M \cdot D_{p}^{2}}{64})] \end{array}

where _{$E i$} is a special function defined as _{$E i (x) = \int_{- \infty}^{x} (e^{t} / t) d t$} [25].

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

\begin{array}{l} I_{P M D} = \sum_{M = 1 / 2}^{N_{s} - 1 / 2} I_{P M D}^{M} = \frac{3 γ^{2} I^{3}}{32 π α β_{2}} {8 N_{s} \ln (B / B_{0}) - \sum_{M = 1 / 2}^{N_{s} - 1 / 2} [3 E_{1} (M) + E_{2} (M)]} \\ E_{1} (M) = E i (- \frac{3 π^{3} B^{2} L \cdot M \cdot D_{p}^{2}}{64}) - E i (- \frac{3 π^{3} B_{0} L \cdot M \cdot D_{p}^{2}}{64}) \\ E_{2} (M) = E i (- \frac{7 π^{3} B^{2} L \cdot M \cdot D_{p}^{2}}{64}) - E i (- \frac{7 π^{3} B_{0} L \cdot M \cdot D_{p}^{2}}{64}) \end{array}

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

\begin{array}{l} I_{P M D} = {(I / I_{0, P M D})}^{2} I, SNR= \frac{I}{2 n_{0} + {(I / I_{0, P M D})}^{2} I}, I_{P M D}^{o p t} = {(n_{0} I_{0, P M D}^{2})}^{1 / 3} \\ {SNR}_{P M D}^{\max} = \frac{1}{3} {(\frac{I_{0, P M D}}{n_{0}})}^{2 / 3}, S_{P M D}^{\max} = 2 \log_{2} (1 + \frac{1}{3} {(\frac{I_{0, P M D}}{n_{0}})}^{2 / 3}) \\ I_{0, P M D} = \frac{1}{γ} \sqrt{\frac{32 π α | β_{2} |}{24 N_{s} \ln (B / B_{0}) - \sum_{M = 1 / 2}^{N_{s} - 1 / 2} [9 E_{1} (M) + 3 E_{2} (M)]}} \end{array}

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

I_{N L} = I_{N - A S E} + I_{P M D}, SNR = \frac{I}{2 n_{0} + I_{N L}}

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 .

Fig. 3 Simulation setup for polarization-division-multiplexed CO-OFDM system with digital back propagation. LOs: local oscillators, PBC/S: polarization beam combiner/splitter.

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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]. 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 _{_{$p s / \sqrt{k m}$}}are shown in Fig. 4 .

Table 1. Fiber Span Parameters used in Simulation

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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.

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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 _{$p s / \sqrt{k m}$}, 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 _{$p s / \sqrt{k m}$}are improved by 4.3 and 2.7 dB respectively.

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 is_{_{$p s / \sqrt{k m}$}}. 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.

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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 _{$p s / \sqrt{k m}$}, 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 _{$p s / \sqrt{k m}$}, this difference can be as large as 3.6 dB. With a large PMD of 0.8 _{$p s / \sqrt{k m}$}, 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]. The spectral efficiency that can be achieved is shown in Fig. 6(b), suggesting that even with a small PMD parameter of 0.05_{$p s / \sqrt{k m}$}, the fiber channel spectral efficiency and capacity are fundamentally limited by PMD, instead of nonlinear signal-ASE interaction.

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 is_{$p s / \sqrt{k m}$}; NBP: no back-propagation nonlinearity compensation. For PMD limited regimes, mean system performance has been assumed.

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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

Denoting_{$U = {(U_{x}, U_{y})}^{T} = [(s_{k}^{+} s_{i}) s_{j} + (s_{k}^{+} s_{j}) s_{i} - (c_{k}^{+} c_{i}) c_{j} - (c_{k}^{+} c_{j}) c_{i}]$}, and expanding _$U$into two polarization components, we obtain

U_{x} = 2 s_{x}^{k *} s_{x}^{i} s_{x}^{j} + s_{y}^{k *} s_{y}^{i} s_{x}^{j} + s_{y}^{k *} s_{y}^{j} s_{x}^{i} - (2 c_{x}^{k *} c_{x}^{i} c_{x}^{j} + c_{y}^{k *} c_{y}^{i} c_{x}^{j} + c_{y}^{k *} c_{y}^{j} c_{x}^{i})

where the superscript ‘*’ stands for scalar conjugate and _{$s_{x, y}^{i, j, k} = c_{x, y}^{i, j, k} + n_{x, y}^{i, j, k}$}is the OFDM subcarrier _{$c_{x, y}^{i, j, k}$}, contaminated with ASE noise _{$n_{x, y}^{i, j, k}$}. Assuming OFDM symbol _{$c_{x, y}^{i, j, k}$}and noise _{$n_{x, y}^{i, j, k}$}on different polarizations or different subcarriers are uncorrelated, namely _{$〈 c_{x_{1}}^{j *} c_{x_{2}}^{k} 〉 = \frac{P}{2} δ_{x_{1} x_{2}} δ_{j k}$}and _{$〈 n_{x_{1}}^{j *} n_{x_{2}}^{k} 〉 = \frac{P_{n}}{2} δ_{x_{1} x_{2}} δ_{j k}$}where _{$x_{1, 2}$}stands for x or y polarization component, we obtain the ensemble average power of _{$U_{x}$}as

\begin{array}{l} 〈 {| U_{x} |}^{2} 〉 = 〈 | 2 (c_{x}^{k *} + n_{x}^{k *}) (c_{x}^{i} + n_{x}^{i}) (s_{x}^{j} + n_{x}^{j}) + (c_{y}^{k *} + n_{y}^{k *}) (c_{y}^{i} + n_{y}^{i}) (c_{x}^{j} + n_{x}^{j}) \\ {+ (c_{y}^{k *} + n_{y}^{k *}) (c_{y}^{j} + n_{y}^{j}) (c_{x}^{i} + n_{x}^{i}) - (2 c_{x}^{k *} c_{x}^{i} c_{x}^{j} + c_{y}^{k *} c_{y}^{i} c_{x}^{j} + c_{y}^{k *} c_{y}^{j} c_{x}^{i}) |}^{2} 〉 \\ = \frac{9}{4} P^{2} P_{n} + \frac{9}{4} P P_{n}^{2} + \frac{9}{4} P_{n}^{3} \approx \frac{9}{4} P^{2} P_{n} \end{array}

where, _$P$and_{$P_{n}$}are the power of signal and ASE noise over each subcarrier frequency respectively. By using _{$〈 {| U_{x} |}^{2} 〉 = 〈 {| U_{y} |}^{2} 〉$}, the ensemble average of signal power becomes

〈 {| (s_{k}^{+} s_{i}) s_{j} + (s_{k}^{+} s_{j}) s_{i} - (c_{k}^{+} c_{i}) c_{j} - (c_{k}^{+} c_{j}) c_{i} |}^{2} 〉 = 〈 {| U_{x} |}^{2} 〉 + 〈 {| U_{y} |}^{2} 〉 = \frac{9}{2} P^{2} P_{n}

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

Appendix B

Denoting_{$U = {(U_{x}, U_{y})}^{T} = (c_{k}^{p +} c_{i}^{p}) c_{j}^{p} + (c_{k}^{p +} c_{j}^{p}) c_{i}^{p} - (c_{k}^{+} c_{i}) c_{j} - (c_{k}^{+} c_{j}) c_{i}$}, and expanding _$U$into the two polarization components, we obtain

U_{x} = 2 (c_{k x}^{p *} c_{i x}^{p} c_{j x}^{p} - c_{k x}^{*} c_{i x} c_{j x}) + (c_{k y}^{p *} c_{i y}^{p} c_{j x}^{p} - c_{k y}^{*} c_{i y} c_{j x}) + (c_{k y}^{p *} c_{j y}^{p} c_{i x}^{p} - c_{k y}^{*} c_{j y} c_{i x})

where _{$c_{i, j, k}$}and _{$c_{i, j, k}^{p}$}denote the OFDM information symbol for subcarrier _$i$, _$j$ and _$k$without and with the influence of PMD respectively. Defining Jones matrix at subcarrier _$k$ or_$j$ as _{$M_{k (j)} = [\begin{matrix} a_{k (j)} & b_{k (j)} \\ - b_{k (j)}^{*} & a_{k (j)}^{*} \end{matrix}]$}, _{$c_{k (j)}^{p}$}can be related to _{$c_{k (j)}$}by _{$c_{k (j)}^{p} = M_{k (j)} c_{k (j)}$}, namely,

c_{k (j) x}^{p} = a_{k (j)} c_{k (j) x} + b_{k (j)} c_{k (j) y}, c_{k (j) y}^{p} = - b_{k (j)}^{*} c_{k (j) x} + a_{k (j)}^{*} c_{k (j) y}

Then _{$U_{x}$}can be expressed as

\begin{array}{l} U_{x} = A - B \\ A = 2 c_{k x}^{p *} c_{i x}^{p} c_{j x}^{p} + c_{k y}^{p *} c_{i y}^{p} c_{j x}^{p} + c_{k y}^{p *} c_{j y}^{p} c_{i x}^{p}, B = 2 c_{k x}^{*} c_{i x} c_{j x} + c_{k y}^{*} c_{i y} c_{j x} + c_{k y}^{*} c_{j y} c_{i x} \end{array}

where _$A$and _$B$are comprised of polarization dependent and independent terms respectively. The average power of _{$U_{x}$} is expressed as

E {〈 {| U_{x} |}^{2} 〉} = E {〈 {| A |}^{2} 〉} + E {〈 {| B |}^{2} 〉} - 2 Re [E {〈 A^{*} B 〉}]

where ‘Re’ stands for real component of a variable. Taking _{$c_{i}$}as reference subcarrier and assuming its polarization is unchanged along the transmission, we have

c_{i x}^{p *} c_{i x} = c_{i x}^{p} c_{i x}^{*} = {| c_{i x} |}^{2}, c_{i y}^{p *} c_{i y} = c_{i y}^{p} c_{i y}^{*} = {| c_{i y} |}^{2}

We assume that in absence of PMD, the OFDM information symbol _{$c_{i, j, k; x, y}$}on different polarizations or different subcarriers are uncorrelated, namely _{$〈 c_{x_{1}}^{j *} c_{x_{2}}^{k} 〉 = \frac{P}{2} δ_{x_{1} x_{2}} δ_{j k}$}with _{$x_{1, 2}$}stands for x or y polarization component. Then from Eqs. (24) and (25), we obtain:

\begin{array}{l} E {〈 {| A |}^{2} 〉} = E (〈 {| B |}^{2} 〉) = 〈 4 {| c_{k x} |}^{2} {| c_{i x} |}^{2} {| c_{j x} |}^{2} + {| c_{k y} |}^{2} {| c_{i y} |}^{2} {| c_{j x} |}^{2} + {| c_{k y} |}^{2} {| c_{j y} |}^{2} {| c_{i x} |}^{2} 〉 = \frac{3}{4} P^{3} \\ E {〈 A^{*} B 〉} = E [〈 {(2 c_{k x}^{p *} c_{i x}^{p} c_{j x}^{p} + c_{k y}^{p *} c_{i y}^{p} c_{j x}^{p} + c_{k y}^{p *} c_{j y}^{p} c_{i x}^{p})}^{*} (2 c_{k x}^{p *} c_{i x}^{p} c_{j x}^{p} + c_{k y}^{p *} c_{i y}^{p} c_{j x}^{p} + c_{k y}^{p *} c_{j y}^{p} c_{i x}^{p}) 〉] \\ = \frac{1}{8} P^{3} E {5 a_{k}^{*} a_{j} + 2 b_{k}^{*} b_{j} + 2 b_{j}^{*} b_{k} + a_{k} a_{j}} \end{array}

Following the same approach as shown for the statistics of the Jones Matrix [29,30], the correlation coefficients of Jones Matrix elements at subcarrier _$j$and _$k$can be proved as:

\begin{array}{l} E {a_{k}^{*} a_{j}} = \frac{1}{2} [\exp (- \frac{\bar{Δ τ^{2}} \cdot Δ ϖ^{2}}{8}) + \exp (- \frac{7 \bar{Δ τ^{2}} \cdot Δ ϖ^{2}}{24})] \\ E {b_{k}^{*} b_{j}} = E {b_{k} b_{j}^{*}} = \frac{1}{2} [\exp (- \frac{\bar{Δ τ^{2}} \cdot Δ ϖ^{2}}{8}) - \exp (- \frac{7 \bar{Δ τ^{2}} \cdot Δ ϖ^{2}}{24})] \\ E {a_{k} a_{j}} = \exp (- \frac{7 \bar{Δ τ^{2}} \cdot Δ ϖ^{2}}{24}) \end{array}

where _{$\bar{Δ τ^{2}} = 3 π {\bar{Δ τ}}^{2} / 8$}with _{$\bar{Δ τ}$} denoting the average DGD and _{$Δ ϖ = 2 π (f_{j} - f_{k})$}is the angle frequency difference between _{$f_{j}$}and _{$f_{k}$}. We have omitted the relatively lengthy steps to derive Eq. (29). We also note that the statistics of Jones matrix_{$M_{k (j)}$}defined in this paper is different than that of conventional Jones matrix in [29,30] because we enforce_{$M_{k (j)}$}as a 2x2 identity matrix for the _{$i_{t h}$} subcarrier (the reference subcarrier). Substituting Eqs. (28) and (29) into Eq. (26), we obtain

E {〈 {| U_{x} |}^{2} 〉} = \frac{P_{}^{3}}{8} [12 - 9 \exp (- \frac{\bar{Δ τ^{2}} Δ ϖ^{2}}{8}) - 3 \exp (- \frac{7 \bar{Δ τ^{2}} Δ ϖ^{2}}{24})]

Thus the ensemble average power of _$U$is obtained as

\begin{array}{l} E {〈 {| U |}^{2} 〉} = E {〈 {| U_{x} |}^{2} 〉 + 〈 {| U_{y} |}^{2} 〉} = 2 E {〈 {| U_{x} |}^{2} 〉} \\ = \frac{1}{4} P^{3} [12 - 9 \exp (- \frac{\bar{Δ τ^{2}} Δ ϖ^{2}}{8}) - 3 \exp (- \frac{7 \bar{Δ τ^{2}} Δ ϖ^{2}}{24})] \end{array}

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Span length	100 km	PMD parameter	0.1 ps/sqrt(km)
Fiber loss	0.2 dB/km	Number of spans	10
Dispersion parameter	16 ps/nm/km	Maximum step size	1 km
Nonlinearity parameter	0.00133 /w/km	Maximum nonlinear rotation	0.003 rad
Noise figure	6 dB	Simulation bandwidth	800 GHz

Influence of PMD on fiber nonlinearity compensation using digital back propagation

Abstract

1. Introduction

2. Theoretical derivations of nonlinear transmission performance using DBP

2.1 Nonlinear Signal-Noise interaction

2.2 PMD-induced uncompensated nonlinear noise

3. Simulation results and discussions

4. Conclusion

Appendix A

Appendix B

References and links

Cited By

Figures (6)

Tables (1)

Equations (31)

Optics Express