Modified phase-conjugate twin wave schemes for fiber nonlinearity mitigation

Yukui Yu; Jian Zhao

doi:10.1364/OE.23.030399

1. Introduction

Signal distortions generated by the fiber Kerr nonlinearity impose a significant limitation on the achievable capacity and transmission reach in optical communications [1]. Various investigations have been performed to mitigate the fiber nonlinearity effect, such as digital back propagation [2–7], nonlinear equalizers [8,9], perturbation-based methods [10–14], optical phase conjugate [15–18], and nonlinear Fourier transform [19–22]. However, these solutions exhibit either high complexity or reduced network flexibility. Recently, a new scheme, named phase-conjugate twin wave (PCTW) technique, was proposed [23,24]. A pair of phase-conjugated optical signals (defined as E_x and E_y = E_x*) co-propagates in the fiber at two orthogonal polarizations. The nonlinear distortions on these two signals are anti-correlated and thus can be mitigated by superimposing the twin signals at the receiver. This principle was also extended to space-division-multiplexing systems in multi-core fibers [25], and orthogonal frequency division multiplexing (OFDM) systems [26]. Although this method is simple and effective, it reduces the spectral efficiency (SE) by a factor of two. A dual-PCTW scheme with quadrature pulse shaping [27] was proposed to address the halved-SE problem and the concept was also applied to OFDM systems [28]. However, this method places more stringent requirement on the specifications of transceivers, by sending 9-quadrature amplitude modulation (9-QAM) signals into the fiber to realize the SE of the quadrature phase-shift keying (QPSK) format. The increase of the signal level also hinders the scalability of this method to higher-level formats such as 8-QAM.

In this paper, we propose a novel modified PCTW (M-PCTW) scheme to enhance the SE while maintaining the performance benefits of the conventional PCTW method. In contrast to the dual PCTW method, our technology does not have to increase the level of the transmitted signals. The principle is based on the finding that when one of the phase-conjugated twin signals is modulated with additional bits E_a (i.e. E_y = E_x*∙E_a, where E_a is a phase-modulated field with unit amplitude), at the receiver, once E_a can be correctly decoded, E_x can still suppress the nonlinear distortions in a similar manner as in the conventional PCTW method. The information carried in E_a increases the SE and is also found as the dominant source limiting the overall system performance. We further introduce redundancy via error correction code to reduce the errors in the additional bits E_a. By doing so, the SE is significantly improved while the performance can approach that of the conventional PCTW method. Simulations of a 25-Gbaud polarization-division-multiplexed (PDM) QPSK system over 15,200 km were performed and it is shown that the normalized SE (NSE) can be enhanced from 50% to >80% with negligible performance degradation compared to the conventional PCTW method. We also show that this scheme can be applied to higher-level formats and the NSE of a 25-Gbaud 8-QAM system over 6,400 km can be increased from 50% to 75%.

2. Principle

2.1. Conventional PCTW scheme

Figure 1 shows the principle of the conventional PCTW scheme. In this method, the x- and y-polarization signals at the transmitter are phase conjugated. Assuming the signals of these two polarizations at the distance L are E_x(L,t) and E_y(L,t), respectively, we have E_y(0,t) = (E_x(0,t))* with (∙)* referring as the complex conjugation operation. These two signals are launched into the fiber, and the first-order nonlinear distortions on the two polarization signals in the frequency domain, δE_x(L,ω) and δE_y(L,ω), are:

\begin{matrix} δ E_{x (or y)} (L, ω) = i \frac{9}{8} γ P_{0} L_{e f f} \int_{- \infty}^{+ \infty} d ω_{1} \int_{- \infty}^{+ \infty} d ω_{2} η (ω_{1} ω_{2}) \\ \times [E_{x (or y)} (0, ω + ω_{1}) E_{x (or y)} (0, ω + ω_{2}) E_{x (or y)}^{*} (0, ω + ω_{1} + ω_{2}) \\ + E_{y (or x)} (0, ω + ω_{1}) E_{x (or y)} (0, ω + ω_{2}) E_{y (or x)}^{*} (0, ω + ω_{1} + ω_{2})] \end{matrix}

where

E_{x (or y)} (0, ω) = \int_{- \infty}^{+ \infty} E_{x (or y)} (0, t) e^{- i ω t} d t / \sqrt{2 π}

L_eff, γ, and η(ω₁ω₂) are the effective length, the fiber nonlinear Kerr coefficient, and the dimensionless nonlinear transfer function, respectively. For a symmetric dispersion and power map along the link, η(ω₁ω₂) is real-valued. Given E_y(0,t) = (E_x(0,t))* or E_y(0,ω) = (E_x(0,-ω))*, we can derive from Eq. (1):

δ E_{y} (L, ω) = - δ E_{x} {(L, - ω)}^{*}, δ E_{y} (L, t) = - δ E_{x} {(L, t)}^{*}

Consequently, at the receiver, the nonlinear distortions can be cancelled and the original signal can be recovered by using:

E_{r} (t) = E_{x} (L, t) + E_{y} {(L, t)}^{*} = 2 E_{x} (0, t)

This method is effective but reduces the SE by a factor of two.

Fig. 1 Principle of the conventional PCTW scheme.

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2.2. Proposed M-PCTW schemes

In this subsection, we will propose an improved scheme, named M-PCTW-II to improve the SE while maintaining the performance benefits of the conventional PCTW method. To reach M-PCTW-II, we also introduce two intermediate schemes, named generalized PCTW (G-PCTW) and M-PCTW-I, whose principle is illustrated in Figs. 2 and 3, respectively.

Fig. 2 The principle of the G-PCTW scheme.

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Fig. 3 Principle of the M-PCTW-I scheme.

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In G-PCTW, we set E_y(0,t) = E_a × (E_x(0,t))*, where E_a is a constant field with unit amplitude, the nonlinear distortions can be derived from Eq. (1) as:

δ E_{y} (L, ω) = - δ E_{x} {(L, - ω)}^{*} \times E_{a}, δ E_{y} (L, t) = - δ E_{x} {(L, t)}^{*} \times E_{a}

At the receiver, the original optical signal is recovered by using:

E_{r} (t) = E_{x} (L, t) + {(E_{y} (L, t) / E_{a})}^{*} = [E_{x} (0, t) + δ E_{x} (L, t)] + [E_{x} (0, t) - δ E_{x} (L, t)] = 2 E_{x} (0, t)

Equation (6) implies that when a constant field with unit amplitude is multiplied to E_y(0,t), the fiber nonlinear distortion can still be cancelled. However, E_a does not carry any additional information, and one may think if it is possible to also encode E_a to increase the SE. To investigate this, we firstly prove the following relationship in the Appendix:

δ E_{x} (L, t) \times E_{y}^{*} (0, t) = - δ E_{y}^{*} (L, t) \times E_{x} (0, t)

Equation (7) is satisfied when self/cross phase modulation (SPM/XPM) and cross polarization modulation (XPolM) dominate the nonlinear effects. Equation (7) also assumes that E_a has the unit amplitude. However, no limitation is placed on E_x and so high-level formats can be applied to E_x, as will be shown in the simulations. Based on Eq. (7), we set E_y(0,t) = E_a × (E_x(0,t))*, where E_a, instead of a constant, is an encoded field with unit amplitude. At the receiver, if E_a is correctly decoded, E_x can still be recovered without nonlinear distortions:

E_{r} (t) = E_{x} (L, t) + {(E_{y} (L, t) / E_{a})}^{*} = 2 E_{x} (0, t) + δ E_{x} (L, t) + δ E_{y}^{*} (L, t) / E_{a}^{*} = 2 E_{x} (0, t)

Figure 3 depicts the principle of M-PCTW-I. Additional information is encoded in E_a before it multiplies to E_x(0,t)*. Figure 4 shows an example when E_x is a QPSK signal with the symbol period of T. In this example, E_a is encoded as either 1 or −1 in each T so that 1/T additional bits are encoded in E_a and the total transmission rate increases from 2/T to 3/T compared to the conventional PCTW. Similarly, when E_a is encoded as 1, −1, j or –j, the transmission rate is 4/T, the same as that of the PDM QPSK signal. Without the loss of generality, we define the i^th signal levels in E_a as E_a_,_i, 1 ≤ i ≤ N, where log₂N is the number of bits encoded in E_a, m_a. At the receiver, after coherent detection to recover E_x and E_y, the signals are split into N paths, with the i^th path performing the operation:

Fig. 4 An example to illustrate the encoding of E_y(0,t).

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E_{r, i} (t) = E_{x} (L, t) + {(E_{y} (L, t) / E_{a, i})}^{*}, 1 \leq i \leq N

Figure 5 depicts the constellation of E_r_,_i when E_x is a QPSK signal and E_a is encoded as (a) 1 or −1; (b) 1, −1, j or –j. Table 1 analyses the outputs of different paths for the case of Fig. 5(a). It can be seen that the constellation of each path in the cases of Figs. 5(a) and 5(b) is no longer the same as that of E_x, and has five and nine points, respectively. In the case of Fig. 5(a) (or Table 1), when E_a = 1, the output of path 1, E_r_,1, is equal to the transmitted QPSK signal E_x while E_r_,2 = 0. Conversely, when E_a = −1, E_r_,2 gives the QPSK signal E_x while E_r_,1 = 0. A selection module compares the outputs of these two paths and selects the larger one. This selection process operates every symbol period T and decides either 1 or −1 is transmitted in E_a. The selected path is then sent to the next de-mapping module to decide E_x. In Fig. 5(a), we mark the points, which represent that when these points occur at the output of a path, this path is selected to decode E_x and the index of this path can be used to decode E_a. Similarly, we also mark the points in Fig. 5(b) and it is seen that in this case, a path can also be selected if the absolute value of the output of this path is larger than that of the other three paths.

Fig. 5 Constellation diagram of a path at the receiver, E_r,i, when E_x is a QPSK signal and E_a is encoded as (a) 1 or −1; (b) 1, −1, j or –j. Solid dots are the constellation points and the marked dots represent that when these points occur at the output of a path, this path is selected.

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Table 1. Outputs of the two receiver paths when E_x is a QPSK signal and E_a is 1 or −1

View Table

Note that although the outputs of the paths in Fig. 3 show a five- or nine-point constellation, the transmitted signals in the fiber, E_x and E_y, are still QPSK signals. The algorithms in the coherent receiver to recover E_x and E_y are also standard ones for the QPSK format. This is in contrast to the prior art [27,28], where 9-QAM signals are transmitted over the fiber and detected at the receiver to achieve the SE of the QPSK format. In addition, the setup in Fig. 3 requires N multiplications, additions, and comparisons, whose complexity is negligible compared to that for coherent detection, and so this scheme is simple.

M-PCTW-I can improve the SE. However, Eq. (8) is derived based on the assumption that E_a is correct. In fact, E_a is the dominant source limiting the system performance because as we can see in Fig. 3, the decoding of E_x is dependent on the correct selection of paths and improper decoding of E_a also increases the errors in E_x. Therefore, the correctness of E_a is more important than that of E_x. We thus propose M-PCTW-II to add redundancy in E_a to mitigate this limiting source, as shown in Fig. 6. In practice, this redundancy can be realized in different ways, e.g. by using forward error correction (FEC) or reducing the symbol rate of E_a from 1/T to 1/(2T). The net SE is also variable by adjusting the overhead of the FEC:

N o r m a l i z e d SE (NSE) = (m_{a} / (1 + r) + m_{x}) / (2 m_{x})

where m_a and m_x are the number of bits encoded in E_a and E_x respectively. r represents the FEC overhead in E_a. The SE is normalized with respect to the conventional PDM signal. For example, when E_x is a QPSK signal, E_a is a BPSK signal, and the overhead of FEC for E_a is 25%, the NSEs for the conventional PCTW, M-PCTW-I and M-PCTW-II, are 50%, 75%, and 70%, respectively. By using FEC, the performance of E_a can be greatly improved, enabling the overall performance approaching that of the conventional PCTW method. In this paper, without the loss of generality, we use the BCH code in E_a.

Fig. 6 Principle of the M-PCTW-II scheme.

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Note that the FEC here is applied to illustrate the concept. In practice, this FEC for E_a can reduce the bit error rate (BER) below a level (e.g. 10⁻⁴) and another FEC is then used to further reduce the BER of both E_a and E_x to 10⁻¹⁵. Alternatively, a strong FEC can be used to directly reduce the BER of E_a to 10⁻¹⁵, and another FEC is then applied to E_x based on the correct E_a. The complexity depends on the type of FEC, the overhead and the gain.

3. Simulation setup and results

We simulated 25-Gbaud PDM QPSK and 8-QAM systems to verify the advantage of the proposed scheme, where the 8-QAM format was in the form as in [29]. A pseudo-random bit sequence with a length of 2²³-1 was used to generate binary bits, which were mapped to PDM QPSK/8-QAM signals to implement different schemes (conventional PDM-QPSK/8-QAM, PCTW, G-PCTW, M-PCTW-I, and M-PCTW-II). In M-PCTW-II, BCH code was used as the FEC. The fiber link consisted of spans of 80-km standard single-mode fiber with a loss of 0.2 dB/km, a nonlinear coefficient of 1.31/W/km, and a dispersion coefficient of 17 ps/km/nm. At the end of each span, an optical amplifier with 16-dB gain and 4.5-dB noise figure was used to compensate the fiber loss. The total simulated length in the QPSK and 8-QAM systems was 15,200 km and 6,400 km, respectively. No in-line dispersion compensation was used. Unless otherwise stated, a symmetric dispersion map was adopted by adding ideal pre- and post-dispersion compensation after the transmitter and before the receiver. At the receiver, the signal was coherently detected. Laser linewidth was set to be zero, but Viterbi-Viterbi and blind phase search algorithms were still used to compensate the nonlinearity induced phase rotation for QPSK and 8-QAM signals, respectively. The simulated number of symbols was 2¹⁷. The simulation was iterated two times independently with different PRBS patterns and the BER was averaged to obtain the Q factor.

We firstly focus on the QPSK system. Figure 7(a) shows the Q factor versus signal launch power for the conventional PDM QPSK at different baud rate, 25-Gbaud PCTW, and one case of 25-Gbaud G-PCTW. The field E_a, as defined in section 2.2, is a constant field in G-PCTW and defined as exp(jφ). It is seen that 25-Gbaud conventional QPSK system shows the worst transmission performance. Reducing the baud rate in this system can improve the performance in the linear region (launch power < −2 dBm), but exhibits only slight improvement in the optimal Q factor due to the fiber nonlinearity. As expected, PCTW improves the optimal Q factor significantly (~3 dB at −1-dBm launch power). On the other hand, G-PCTW has a similar Q factor versus launch power curve as the PCTW method. Figure 7(b) depicts the optimal Q factor versus φ for G-PCTW. The figure shows that the performance is insensitive to φ. This implies that multiplying the constant field exp(jφ) to E_x* in the y polarization can still maintain the capability of E_x in fiber nonlinearity mitigation as in PCTW. Figure 8 shows the constellation diagrams of (a) conventional PDM QPSK (b) PCTW and (c) G-PCTW. The rectangular points represent the constellation of incorrectly decoded data. It is confirmed that both PCTW and G-PCTW have clear constellations, and the number of errors is much less than that in the conventional QPSK system.

Fig. 7 (a) Q factor versus the signal launch power at 15,200 km for 25-, 18.75- and 12.5-Gbaud conventional PDM QPSK, 25-Gbaud PCTW, and 25-Gbaud G-PCTW. (b) Optimal Q factor of G-PCTW versus φ. In (a) and (b), E_a is a constant field exp(jφ) in G-PCTW.

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Fig. 8 Constellation diagrams of (a) 25-Gbaud conventional PDM QPSK (b) 25-Gbaud PCTW and (c) 25-Gbaud G-PCTW. The signal launch power is −1 dBm. Rectangular points represent the incorrectly decoded data.

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Although G-PCTW maintains the performance of the conventional PCTW method, it does not offer any enhancement in the SE. We thus modulate E_a to add additional information. Figure 9 shows the decoding setup of the M-PCTW-I scheme when (a) E_a = 1 or −1; (b) E_a = 1, −1, j or –j. After coherent detection and analogue-to-digital conversion, the x- and y-polarization signals are split into two and four paths for Figs. 9(a) and 9(b), respectively. The selection module compares the intensity of those paths on a symbol-by-symbol basis, and selects the larger (or the largest) one. This selection process implicitly decodes the information in E_a. The output of the selected paths is then used to decode E_x.

Fig. 9 Decoding setup of M-PCTW-I when (a) one bit and (b) two bits are modulated in E_a.

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Figure 10 shows the Q factor versus signal launch power for the conventional PCTW, E_x in M-PCTW-I when the data in E_a are assumed to be correctly decoded, and M-PCTW-I where E_a is decoded by selecting the larger (or the largest). Figure 10(a) and 10(b) represent the cases that one bit and two bits are modulated in E_a, respectively. For E_x in M-PCTW-I when E_a is correct, the BER is obtained by dividing the errors in E_x by the transmitted bits in E_x (i.e. the number of symbols × 2). This case is impractical but important to reveal the implications of the system. In practice, a FEC can be applied to E_a, as studied later, to ensure no errors in E_a and to isolate the BER of E_x. The figure shows that once E_a is correctly decoded, the performance of E_x in M-PCTW-I is similar to that in the conventional PCTW method, although its conjugate at the y polarization is modulated with additional bits E_a. The slight degradation in the Q factor at the optimal signal power might arise from the assumption of Eq. (7) which only considers SPM/XPM and XPolM effects. On the other hand, the performance of M-PCTW-I is degraded when the decoding errors in E_a are considered. When one bit is modulated in E_a, the NSE increases from 50% to 75% but the optimal Q factor is reduced by ~1.5 dB compared to the PCTW method. Note that the performance in this case is still ~1.5 dB better than the conventional PDM QPSK signal. In contrast, when the NSE increases to 100% by modulating two bits in E_a, the performance is further degraded, implying that E_a is the dominant error source.

Fig. 10 Q factor versus the signal launch power when (a) one bit and (b) two bits are modulated in E_a. Squares: the BER is obtained by dividing the errors only in E_x by the total transmitted bits in E_x (the number of symbols × 2).

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To understand the reasons for this performance degradation, Figs. 11 and 12 depict the constellation diagrams of the output of a path at the receiver, the output of the selection module, and the output of the selection module when E_a is correctly decoded. The rectangular points represent the constellation of the incorrectly decoded data. As described previously, the output of each path in M-PCTW-I shows either a five- or a nine-point constellation. Ideally, when the path that has the largest intensity is selected, the selection module gives the QPSK constellation of E_x. However, in practice, if E_a is wrongly decoded due to nonlinearity-induced distortion and/or the noise, E_x would also be decoded incorrectly although the constellation at the output of the selection module may still be clear. It implies that the decoding of E_a is more important than that of E_x. In Figs. 11(c) and 12(c), we can see that once E_a is correctly decoded, the number of errors in E_x is significantly reduced to a value similarly to in the PCTW method. These results further confirm two important implications: 1) M-PCTW-I can still suppress nonlinear distortions for E_x in a similar manner as the conventional PCTW method. 2) E_a is the dominant source limiting the system performance.

Fig. 11 Constellation diagrams of (a) the output of a path; (b) the output of the selection module; (c) the output of the selection module when the decoded E_a is correct (i.e. the selected paths are correct). One bit is modulated in E_a and the signal launch power is −1 dBm.

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Fig. 12 Constellation diagrams of (a) the output of a path; (b) the output of the selection module; (c) the output of the selection module when the decoded E_a is correct (i.e. the selected paths are correct). Two bits are modulated in E_a and the signal launch power is −1 dBm.

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The first implication is still valid when an arbitrary phase rotation is applied to E_a. Figure 13(a) shows the optimal Q factor versus φ for PCTW and E_x in M-PCTW-I when E_a is correctly decoded. In the figure, the phase of E_a is rotated with an arbitrary constant: for the one-bit case (circles), E_a is modulated with exp(jφ) or exp(j∙(π + φ)); for the two-bit case (squares), E_a is modulated with exp(jφ), exp(j∙(π/2 + φ)), exp(j∙(π + φ)), or exp(j∙(3π/2 + φ)). It is seen that similar to the G-PCTW, applying a constant phase rotation to the modulated field E_a in M-PCTW-I does not influence the performance of E_x.

Fig. 13 (a) The optimal Q factor versus the rotated phase for PCTW and E_x in M-PCTW-I when E_a is correctly decoded. (b) The optimal Q factor versus the percentage of dispersion pre-compensation for PCTW and E_x in M-PCTW-I when E_a is correctly decoded.

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In previous results, a symmetric dispersion map is adopted. Pre- and post-compensated dispersion are both 50% of the total accumulated dispersion in the fiber link. This dispersion map results in the optimal performance in the conventional PCTW. Figure 13(b) shows the optimal Q factor versus the percentage of pre-compensated dispersion for both PCTW and E_x in M-PCTW-I assuming that E_a is correctly decoded. It can be seen that in PCTW, 50% of dispersion pre-compensation is indeed optimal and there is around 1-dB penalty when the dispersion is 100% pre- or post-compensated. On the other hand, E_x in the M-PCTW-I also shows the best performance for ~50% of dispersion pre-compensation. However, the Q factor in this case is less sensitive to the percentage of dispersion pre-compensation and the fluctuation in the optimal Q factor is around 0.5 dB. In particular, it is seen that under 0% and 100% pre-compensation, E_x in M-PCTW-I exhibits slightly better performance than PCTW. Additional results show that this difference is augmented for higher signal powers. This might be because four wave mixing (FWM) cannot be mitigated in these cases and has higher influence on PCTW due to strong correlation between x and y polarizations in this system.

Figure 13 investigates the first implication in the M-PCTW-I method. However, from Fig. 10, it can be seen that the overall system performance is still degraded due to E_a. In the following, without the loss of generality, we introduce redundancy in E_a via the BCH code to improve the performance while maintaining the advantage of high SE. Figure 14 depicts the Q factor versus the signal launch power when (a) one bit and (b) two bits are modulated in E_a. In Fig. 14(a), BCH codes (1023, 893) and (1023, 798) are applied to E_a and the NSEs are 72% and 70%, respectively. In Fig. 14(b), BCH codes (1023, 618) and (1023, 513) are used, which correspond to the NSE of 80% and 75%, respectively. In both cases, it can be seen that by utilizing the redundancy to reduce the errors in E_a, the optimal Q factor is significantly improved and approaches that of the conventional PCTW method even when the NSE is very close to M-PCTW-I. The residual penalty of M-PCTW-II at lower (<-4 dBm) and higher (1 dBm) power region is due to imperfect correction of errors in E_a using these BCH codes, which in turn limits the overall performance.

Fig. 14 Q factor versus the signal launch power for different schemes when (a) one bit and (b) two bits are modulated in E_a. Diamonds and pluses: the BER is obtained by dividing all errors by the total net transmitted bits excluding the BCH overhead.

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Figure 15 shows the Q-factor improvement of the conventional PCTW and the proposed M-PCTW-II at the optimal signal launch power with respect to the conventional 25-Gbaud PDM QPSK signal. One bit and two bits are added in E_a in Figs. 15(a) and 15(b), respectively. BCH code (1023, q) is applied to obtain different NSE. In Fig. 15(a), the cases of 63%, 68%, 70%, 72%, and 75% NSE correspond to q values of 513, 718, 798, 893, and 1023, respectively. In Fig. 15(b), q values of 218, 413, 463, 513, 573, 618, 668, 718, and 768 are applied to realize the NSE of 60%, 70%, 73%, 75%, 78%, 80%, 83%, 85%, and 88%, respectively. It is seen that in both figures M-PCTW-II can significantly improve the NSE while maintaining the performance benefits of the conventional PCTW method. In M-PCTW-II, some points with higher NSE exhibits even better performance than that with 50% NSE due to the denominator in the calculation of BER (the total net transmitted bits excluding the BCH overhead). For example, the denominator is the number of symbols × 2 (or 3) for 50% (or 75%) NSE. When one bit is modulated in E_a, >3-dB Q-factor improvement is observed for NSE values up to 72%. Note that the maximal NSE (without BCH) in this case is 75%. By adding two bits in E_a, >3-dB Q-factor improvement can be achieved when NSE increases to 80%. This verifies the advantage of the proposed method over the conventional PCTW.

Fig. 15 Q-factor improvement versus the NSE when (a) one bit and (b) two bits are modulated in E_a. Different NSEs are obtained by using the BCH code in E_a with different overhead.

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The concept of the proposed scheme can be extended to higher-level formats. Figure 16 shows the Q-factor improvement of the conventional PCTW and the proposed M-PCTW-II at 6,400 km when E_x is an 8-QAM signal and (a) one or (b) two bits are added in E_a. Note that E_a needs to have unit amplitude and so three added bits are not applicable in 8-QAM. The maximal NSE for Figs. 16(a) and 16(b) is 67% and 83%, respectively. The improvement is with respect to the 25-Gbaud PDM 8-QAM signal and BCH code (1023, q) is used to obtain different NSE. In Fig. 16(a), the cases of 54%, 58%, 61%, 62%, 63%, 64%, 65% and 67% NSE correspond to q of 218, 503, 678, 768, 798, 828, 923, and 1023, respectively. In Fig. 16(b), q values of 218, 503, 678, 768, 798, 828, 858, 923, and 1023 are applied to realize the NSE of 57%, 66%, 72%, 75%, 76%, 77%, 78%, 80%, and 83%, respectively. The figure shows that the proposed scheme can enhance the NSE of the 8-QAM system from 50% to 65% and 75% in Figs. 16(a) and 16(b), respectively, while maintaining 3-dB Q-factor improvement. This is due to the fact that Eq. (7) does not place any restriction on the format of E_x (or E_y) once E_a has the unit amplitude. On the other hand, it is seen that the performance of M-PCTW-II at 50% NSE degrades (~0.5 dB) with respect to the conventional PCTW. This might be because the effect of FWM increases in higher-level formats while Eq. (7) assumes that SPM/XPM and XPolM dominate the fiber nonlinearity.

Fig. 16 Q-factor improvement versus the NSE when (a) one bit and (b) two bits are modulated in E_a. E_x is an 8-QAM signal and the fiber length is 6,400 km

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4. Conclusions

We have proposed a novel M-PCTW-II method, which can significantly improve the NSE while preserving the benefits of the conventional PCTW method in fiber nonlinearity mitigation. We have shown that by modulating one of the conjugated signals with additional bits E_a, the information encoded in the conjugated variants E_x can still suppress the nonlinear distortions in a similar manner as in the conventional PCTW method, while that encoded in the additional bits E_a is the dominant source limiting the system performance. We further introduce redundancy to the additional bits via FEC to overcome this performance bottleneck, enabling the overall system performance close to that of the conventional PCTW method while significantly enhancing the SE. Simulations of a 25-Gbaud PDM QPSK system over 15,200 km were performed to verify the findings, and the results show that the proposed scheme can increase the NSE from 50% to 80% while maintaining the performance benefits of the conventional PCTW method. The concept of the proposed method can also be extended to the 8-QAM system, where the simulation over 6,400-km single-mode fiber shows that the NSE can be enhanced from 50% to 75%.

Appendix

In this Appendix, we will prove Eq. (7). We firstly write δE_x(L,t) and δE_y(L,t) as:

\begin{array}{l} δ E_{x (or y)} (L, t) = \int_{- \infty}^{+ \infty} δ E_{x (o r y)} (L, ω) \cdot \exp (- 2 π i ω t) d ω \\ = i \frac{9}{8} γ P_{0} L_{e f f} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} η (ω_{1} ω_{2}) \times [E_{x (or y)} (0, ω + ω_{1}) E_{x (or y)} (0, ω + ω_{2}) E_{x (or y)}^{*} (0, ω + ω_{1} + ω_{2}) \\ + E_{y (or x)} (0, ω + ω_{1}) E_{x (or y)} (0, ω + ω_{2}) E_{y (or x)}^{*} (0, ω + ω_{1} + ω_{2})] \cdot \exp (- 2 π i ω t) d ω \cdot d ω_{1} \cdot d ω_{2} \end{array}

From Eq. (11), the left-hand side of Eq. (7) can be obtained as:

\begin{array}{l} δ E_{x} (L, t) \times E_{y}^{*} (0, t) = E_{y}^{*} (0, t) \times i \frac{9}{8} γ P_{0} L_{e f f} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} η (ω_{1} ω_{2}) \cdot \exp (- 2 π i ω t) d ω \cdot d ω_{1} \cdot d ω_{2} \\ \times [E_{x} (0, ω + ω_{1}) E_{x} (0, ω + ω_{2}) E_{x}^{*} (0, ω + ω_{1} + ω_{2}) \\ + E_{y} (0, ω + ω_{1}) E_{x} (0, ω + ω_{2}) E_{y}^{*} (0, ω + ω_{1} + ω_{2})] \end{array}

Similarly, the right-hand side of Eq. (7) is derived as:

\begin{array}{l} δ E_{y}^{*} (L, t) \times E_{x} (0, t) = - E_{x} (0, t) \times i \frac{9}{8} γ P_{0} L_{e f f} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} η^{*} (ω_{1} ω_{2}) \cdot \exp (2 π i ω t) d ω \cdot d ω_{1} \cdot d ω_{2} \\ \times [E_{y}^{*} (0, ω + ω_{1}) E_{y}^{*} (0, ω + ω_{2}) E_{y} (0, ω + ω_{1} + ω_{2}) \\ + E_{x}^{*} (0, ω + ω_{1}) E_{y}^{*} (0, ω + ω_{2}) E_{x} (0, ω + ω_{1} + ω_{2})] \end{array}

In this paper, we assume that SPM/XPM and XPolM dominate the nonlinear effect. This is valid when the phase-matching bandwidth is much less than the signal bandwidth and η(ω₁ω₂) is close to a delta function. For example, for 15,200 km and a dispersion coefficient of 17 ps/nm/km, the phase-matching bandwidth is less than 1 GHz [30]. Under this assumption, either ω₁ or ω₂ is zero with η(ω₁ω₂) = η. For ω₁ = 0, Eq. (12) can be written as:

\begin{array}{l} δ E_{x} (L, t) \times E_{y}^{*} (0, t) = E_{y}^{*} (0, t) \times i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \exp (- 2 π i ω t) d ω \cdot d ω_{2} \\ \times [E_{x} (0, ω) E_{x} (0, ω + ω_{2}) E_{x}^{*} (0, ω + ω_{2}) + E_{y} (0, ω) E_{x} (0, ω + ω_{2}) E_{y}^{*} (0, ω + ω_{2})] \\ = E_{y}^{*} (0, t) \times i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \exp (- 2 π i ω_{p} t) d ω_{p} \cdot d ω_{p, 2} \\ \times [E_{x} (0, ω_{p}) {| E_{x} (0, ω_{p, 2}) |}^{2} + E_{y} (0, ω_{p}) E_{x} (0, ω_{p, 2}) E_{y}^{*} (0, ω_{p 2})] \\ = i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} d ω_{p, 2} \times [E_{x} (0, t) E_{y}^{*} (0, t) \cdot {| E_{x} (0, ω_{p, 2}) |}^{2} + {| E_{y} (0, t) |}^{2} \cdot E_{x} (0, ω_{p, 2}) E_{y}^{*} (0, ω_{p 2})] \end{array}

where ω_p = ω and ω_p_,2 = ω + ω₂. On the other hand, when ω₁ = 0, Eq. (13) is derived as:

\begin{array}{l} δ E_{y}^{*} (L, t) \times E_{x} (0, t) = - E_{x} (0, t) \times i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \exp (2 π i ω t) d ω \cdot d ω_{2} \\ \times [E_{y}^{*} (0, ω) E_{y}^{*} (0, ω + ω_{2}) E_{y} (0, ω + ω_{2}) + E_{x}^{*} (0, ω) E_{y}^{*} (0, ω + ω_{2}) E_{x} (0, ω + ω_{2})] \\ = - E_{x} (0, t) \times i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \exp (2 π i ω_{p} t) d ω_{p} \cdot d ω_{p, 2} \\ \times [E_{y}^{*} (0, ω_{p}) {| E_{y} (0, ω_{p, 2}) |}^{2} + E_{x}^{*} (0, ω_{p}) E_{y}^{*} (0, ω_{p, 2}) E_{x} (0, ω_{p, 2})] \\ = - i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} d ω_{p, 2} \times [E_{x} (0, t) E_{y}^{*} (0, t) \cdot {| E_{y} (0, ω_{p, 2}) |}^{2} + {| E_{x} (0, t) |}^{2} \cdot E_{x} (0, ω_{p, 2}) E_{y}^{*} (0, ω_{p, 2})] \end{array}

when E_y(0,t) = E_a × (E_x(0,t))*, where E_a has the unit amplitude. It is proved from Eqs. (14) and (15) that Eq. (7) is satisfied. Note that there is no limitation on E_y(0,t) or E_x(0,t). Therefore, high-level formats can be applied to E_x. On the other hand, for ω₂ = 0, Eq. (12) is derived as:

\begin{array}{l} δ E_{x} (L, t) \times E_{y}^{*} (0, t) = E_{y}^{*} (0, t) \times i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \exp (- 2 π i ω t) d ω \cdot d ω_{1} \\ \times [E_{x} (0, ω + ω_{1}) E_{x} (0, ω) E_{x}^{*} (0, ω + ω_{1}) + E_{y} (0, ω + ω_{1}) E_{x} (0, ω) E_{y}^{*} (0, ω + ω_{1})] \\ = E_{y}^{*} (0, t) \times i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \exp (- 2 π i ω_{p} t) d ω_{p} \cdot d ω_{p, 1} \\ \times [E_{x} (0, ω_{p}) {| E_{x} (0, ω_{p, 1}) |}^{2} + E_{x} (0, ω_{p}) {| E_{y} (0, ω_{p, 1}) |}^{2}] \\ = i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} d ω_{p, 2} \times [E_{x} (0, t) E_{y}^{*} (0, t) \cdot {| E_{x} (0, ω_{p, 1}) |}^{2} + E_{x} (0, t) E_{y}^{*} (0, t) \cdot {| E_{y} (0, ω_{p, 1}) |}^{2}] \end{array}

where ω_p = ω and ω_p_,1 = ω + ω₁. Similarly, Eq. (13) is derived as:

\begin{array}{l} δ E_{y}^{*} (L, t) \times E_{x} (0, t) = - E_{x} (0, t) \times i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \exp (2 π i ω t) d ω \cdot d ω_{1} \\ \times [E_{y}^{*} (0, ω + ω_{1}) E_{y}^{*} (0, ω) E_{y} (0, ω + ω_{1}) + E_{x}^{*} (0, ω + ω_{1}) E_{y}^{*} (0, ω) E_{x} (0, ω + ω_{1})] \\ = - E_{x} (0, t) \times i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \exp (2 π i ω_{p} t) d ω_{p} \cdot d ω_{p, 1} \\ \times [E_{y}^{*} (0, ω_{p}) {| E_{y} (0, ω_{p, 1}) |}^{2} + E_{y}^{*} (0, ω_{p}) {| E_{x} (0, ω_{p, 1}) |}^{2}] \\ = - i \frac{9}{8} γ P_{0} L_{e f f} η \int_{- \infty}^{+ \infty} d ω_{p, 2} \times [E_{x} (0, t) E_{y}^{*} (0, t) \cdot {| E_{y} (0, ω_{p, 1}) |}^{2} + E_{x} (0, t) E_{y}^{*} (0, t) \cdot {| E_{x} (0, ω_{p, 1}) |}^{2}] \end{array}

Comparison between Eqs. (16) and (17) shows that Eq. (7) is still satisfied.

Acknowledgments

This work was supported by the Science Foundation Ireland under grant number 11/SIRG/I2124 and China Scholarship Council.

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	E_a = 1				E_a = −1
	E_x = j∙π/4	E_x = j∙3π/4	E_x = -j∙π/4	E_x = -j∙3π/4	E_x = j∙π/4	E_x = j∙3π/4	E_x = -j∙π/4	E_x = -j∙3π/4
Path 1	E_x = j∙π/4	E_x = j∙3π/4	E_x = -j∙π/4	E_x = -j∙3π/4	0	0	0	0
Path 2	0	0	0	0	E_x = j∙π/4	E_x = j∙3π/4	E_x = -j∙π/4	E_x = -j∙3π/4

Modified phase-conjugate twin wave schemes for fiber nonlinearity mitigation

Abstract

1. Introduction

2. Principle

2.1. Conventional PCTW scheme

2.2. Proposed M-PCTW schemes

3. Simulation setup and results

4. Conclusions

Appendix

Acknowledgments

References and links

Cited By

Figures (16)

Tables (1)

Equations (17)

Optics Express