1. Introduction

Cross-phase modulation (XPM), as one of the fiber nonlinearities, has been widely studied [1–9]. It appears as one of the most detrimental impairments in high-bit-rate long-haul wavelength-division multiplexing (WDM) transmission systems, especially over low dispersion fiber like nonzero dispersion-shifted fiber (NZDSF) [3]. For On-Off Keying (OOK) WDM systems, XPM-induced impairments can be suppressed by introducing suitable time delays between adjacent channels within each repeater [4], or by using a novel device, obtained by writing a series of narrow-band fiber Bragg gratings [5]. Moreover, dispersion maps have also been brought forward as another attractive method. Ref. [6–9] respectively investigated different dispersion mapping schemes to successfully suppress XPM-induced distortions in 10 Gbit/s OOK systems.

Differential phase-shift keying (DPSK) format has been studied extensively. With better receiver sensitivity and tolerance to fiber nonlinearities than OOK format, DPSK is more suitable for high-speed and long-haul WDM transmission [10]. Nonlinear phase noise, induced by the interaction between optical amplified spontaneous emission (ASE) and fiber Kerr nonlinearities, is one of the fatal impairments in DPSK systems [11–21]. Nonlinear phase noise in single-channel DPSK systems has been widely discussed [11–18]. In DPSK WDM systems, power fluctuations from neighbouring channels can also induce nonlinear phase noise due to XPM [19–21]. Suppression of XPM-induced nonlinear phase noise in DPSK WDM systems by dispersion maps has been mentioned before, but no detailed numerical results have been given, especially for 40 Gbit/s systems. Ref. [20] studied the resonant map with the dispersion at the end of each span fully compensated. Ref. [19] and [21] highlighted incomplete dispersion compensation can suppress XPM-induced nonlinear phase noise. However, Ref. [19] was based on a 10 Gbit/s system and didn’t discuss the dispersion map design. Meanwhile, Ref. [21] didn’t consider the complicated situation with both pre-compensation and residual dispersion per span, in which accumulated dispersion before each span is different.

Most of the previous studies of nonlinear phase noise focused on WDM systems in which all of the channels are modulated with the same DPSK format. However, there is a good reason to investigate the case that different formats are used for each channel. The mixed usage is possible if conventional OOK systems are partially updated to the DPSK format, or the channels from different links with their respective formats are combined in a network node and must co-propagate [22–25]. The reported investigations of multi-format transmissions are mostly 10 Gbit/s or 10 Gbit/s mixed with 40 Gbit/s. To our knowledge, there is lack of studies on 40 Gbit/s multi-format WDM systems. Moreover, there is no comprehensive investigation on both XPM and XPM-induced nonlinear phase noise in DPSK and OOK mixed format systems with different dispersion mapping. Although Ref. [24] discussed XPM-induced impairments of 10 Gbit/s RZ-DPSK signals from adjacent OOK channels and Ref. [26] developed a theoretical model for XPM-induced nonlinear phase noise in DQPSK and OOK mixed WDM transmission, neither of them considered the suppression of that impairment. Much recently, Ref. [27] investigated XPM penalty, induced by neighboring 10 Gbit/s NRZ-OOK, for 42.7 Gbit/s DQPSK signals in a dispersion-managed hybrid DWDM transmission system. However, the adopted pre-compensation is fixed and without optimization.

The paper is organized as follows. In Section 2, we present a numerical study of both XPM effect and XPM-induced nonlinear phase noise in 40 Gbit/s RZ-DPSK WDM systems using NZDSF (typical dispersion 4.5 ps/nm/km). In Section 3, we further investigate XPM-induced impairments from neighboring OOK channels in 40 Gbit/s multi-format (RZ-DPSK and RZ-OOK) NZDSF WDM systems. In Section 4, we discuss the situation with standard single-mode fiber (SSMF, typical dispersion 17 ps/nm/km). Finally, we present our conclusions in Section 5.

2. XPM effect and XPM-induced nonlinear phase noise in 40 Gbit/s RZ-DPSK WDM systems

 

Fig. 1. System configuration in our simulation. For RZ-DPSK WDM system, the three channels all carry RZ-DPSK signals with 33% duty cycle. For multi-format WDM system, the center channel still transmits RZ-DPSK signals, while its two neighbors use 33% RZ-OOK format.

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The transmission system setup in our simulation is shown in Fig. 1. Three channels carrying 40 Gbit/s RZ-DPSK signals with 33% duty cycle are transmitted. The wavelength of the center channel is 1553.60 nm and the channel spacing is 100 GHz. The fiber link consists of 30 spans, each of which includes NZDSF, inline dispersion compensation fiber (DCF) and an erbium-doped fiber amplifier (EDFA). The parameters of the NZDSF are: span length span L_span=100 km, dispersion D=4.5 ps/nm/km, attenuation α=0.21 dB/km and nonlinearity γ=1.32/W/km. The inline DCF, pre- and post-compensation modules are all linear and without loss. The total dispersion of the transmission link is fully compensated at the receiver end. The average launch power is set as ave P_ave=0.9 dBm so that the mean nonlinear phase shift 〈Φ_NL〉=1 rad after N_span=30 spans. Here 〈Φ_NL〉=N_span γP_aveL_eff. L_eff≈1/α is the effective length per span. The noise figure of the inline EDFA is set to be 7.2 dB, resulting in an optical signal-to-noise ratio (OSNR) of ρ_s=25.5 (14 dB) after 30 spans transmission. The OSNR is defined as ρ_s=P_ave/(N_spanS₀B_d). S ₀ is the amplifier noise spectral density in a single polarization. B_d is the optical noise bandwidth which is set the same as the bit-rate of 40 Gbit/s. With only linear Gaussian noise, the error probability can be calculated to 1/2exp(ρ_s)≈4.2×10^-12 [28]. The receiver consists of a second-order Gaussian optical filter with 3 dB bandwidth of 80 GHz, followed by a one-bit delay interferometer, a balanced detector and a fifth-order Bessel electrical low-pass filter with 28 GHz bandwidth. Our numerical simulations are performed with VPItransmissionMaker7.1. We choose the length of a De Bruijn sequence as 4096 bits to contain sufficient bit patterns to capture nonlinear interaction details for the system scenarios in this paper [29].

The performance of RZ-DPSK systems is evaluated by bit-error-rate (BER). We use a semi-stochastic approach [30] for BER estimation to include the influence of signal-ASE and ASE-ASE beating noise. The received sampling signal is optical signal mixed with noise on the constructive port of the detector, while only optical noise on the destructive port. Since they are independent stochastic processes and both fitted by non-central Chi-squared distribution, the moment-generating function (MGF) of the detected electrical signal is [30]

Where the first term corresponds to the constructive port and the second term corresponds to the destructive port. The parameters M, N and E are obtained by fitting the histogram of the detected amplitudes. Thereafter, the BER is calculated from resulting MGF by saddle point approximation method [31]. To reduce the stochastic randomness, the results have been averaged over ten iterations of BER calculations with different noise random-number seeds.

 

Fig. 2. BER as a function of pre-compensation for 40 Gbit/s RZ-DPSK systems. (a) with both XPM effect and XPM-induced nonlinear phase noise; (b) comparing XPM effect with XPM-induced nonlinear phase noise.

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For comparison, we simulate a single-channel system at the wavelength of 1553.60 nm. The results of both single- and three-channel systems are shown in Fig. 2(a). Without pre-compensation (D_pre=0 ps/nm) and residual dispersion per span (D_res=0 ps/nm), the BER of WDM system is distinctly worse more than three orders compared with that of single-channel system. The performance degradation is due to XPM-induced impairments from two adjacent channels. This proves that full dispersion compensation per span can cause constructive addition of XPM-induced impairments span by span, which is consistent with Ref. [19] and [21]. Due to the same reason, for a resonant dispersion map (D_res=0 ps/nm), only optimizing pre-compensation can’t improve system performance effectively. However, when residual dispersion is introduced to each span, e.g. D_res=10 and 20 ps/nm, XPM-induced impairments add destructively span by span. Consequently, the performance of both WDM and single-channel systems is improved with similar optimum pre-compensation. With optimum dispersion mapping, the BER difference between WDM and single-channel system is less than half of one order of magnitude. We conclude that XPM-induced impairments can be effectively suppressed by optimizing both pre-compensation and residual dispersion per span. It should be noted that, not shown in Fig. 2(a), we simulated a five-channel system. The performance of the center channel is very close to that of three-channel system. Therefore, it’s safe to say that the XPM-induced impairments from other WDM channels are also readily suppressed.

In order to distinguish between XPM effect and XPM-induced nonlinear phase noise, we consider the situation with only XPM. Keeping OSNR unchanged, we add ASE noise at the receiver. In doing so, there is no interaction between fiber nonlinearities and ASE noise along the transmission link. Therefore, there is no nonlinear phase noise. Three curve groups are illustrated in Fig. 2(b). The green one is for single-channel RZ-DPSK transmission without nonlinear phase noise. The blue one corresponds for the case of WDM transmission with XPM but without nonlinear phase noise. The red one stands for the situation with both XPM and XPM-induced nonlinear phase noise, which is plotted in Fig. 2(a) and shown here again for comparison. Comparing the red and the blue curves at resonant maps, we find that XPM and XPM-induced nonlinear phase noise have comparable influence on system performance. Without ASE noise, DPSK signals have identical intensity in each time slot. However, due to the encoded phase, DPSK pulses will interfere with each other during dispersive transmission, which leads to intensity fluctuations and results in the system degradation from XPM effect.

3. XPM effect and XPM-induced nonlinear phase noise in 40 Gbit/s multi-format WDM systems

 

Fig. 3. BER as a function of pre-compensation for a 40 Gbit/s RZ-DPSK channel in multi-format WDM systems. (a) with both XPM effect and XPM-induced nonlinear phase noise; (b) comparing XPM effect with XPM-induced nonlinear phase noise. The single-channel RZ-DPSK transmission is shown for comparison.

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Similar investigations are carried out for multi-format WDM systems. The system configuration is shown in Fig. 1. The center channel still carry 40 Gbit/s RZ-DPSK signals, while the two adjacent channels use 40 Gbit/s RZ-OOK formats. The other system parameters are the same as 40 Gbit/s RZ-DPSK WDM systems in Section 2. The results are summarized in Fig. 3. Similar with the case of DPSK WDM systems, Fig. 3(a) shows that optimum dispersion mapping can also suppress XPM-induced impairments in multi-format WDM systems. Comparing Fig. 3(a) with Fig. 2(a), it is clear that without pre-compensation and residual dispersion per span, XPM-induced impairments from adjacent RZ-OOK channels is much more serious than that from adjacent RZ-DPSK channels. Therefore, a large enough inline residual dispersion, e.g. D_res=20 ps/nm, should be used to suppress XPM-induced impairments from neighboring RZ-OOK channels. The random intensity of RZ-OOK bit sequence will induce much more XPM effect to the center channel, which is further proved in Fig. 3(b) by separating XPM effect from XPM-induced nonlinear phase noise. The red and the blue curves at resonant maps in Fig. 3(b) have similar shapes and are very close to each other, except that the red one has a little worse BER. This suggests that XPM effect dominates over XPM-induced nonlinear phase noise in multi-format WDM transmission with resonant maps.

4. Discussions of SSMF

The above investigations focus on low dispersion fiber of NZDSF. In this section, for the sake of comparison, we carry out a similar study for SSMF, with dispersion of 17 ps/nm/km. The system configuration and parameters are the same as that of Fig. 1, except that SSMF is used as the transmission fiber. The results are shown in Fig. 4. Both XPM effect and XPM-induced nonlinear phase noise are considered. Without pre-compensation and residual dispersion per span, the BERs of RZ-DPSK channels in three-channel DPSK and multi-format systems are about one and two orders worse, respectively, than that of single-channel case. Compared with Fig. 2(a) and Fig. 3(a), the performance degradation of SSMF system is much slighter than that of NZDSF case, since high dispersion of SSMF can help to reduce XPM strength [3]. By optimizing both pre-compensation and residual dispersion per span, XPM-induced impairments in SSMF systems can be effectively suppressed.

 

Fig. 4. BER as a function of pre-compensation for a 40 Gbit/s RZ-DPSK channel in DPSK and multi-format WDM systems using SSMF.

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5. Conclusion

We have studied XPM effect and XPM-induced nonlinear phase noise in both RZ-DPSK and multi-format WDM systems operating at 40 Gbit/s. For NZDSF scenarios, if dispersion map is not optimized, RZ-DPSK signals will suffer serious XPM-induced impairments from its neighboring channels through inter-channel nonlinear interaction. These impairments are much more serious when the neighbors are OOK channels, because XPM effect from adjacent OOK channels is much more severe than XPM-induced nonlinear phase noise. For SSMF, XPM-induced impairments are much slighter than that of NZDSF.

Residual dispersion per span will be beneficial to eliminate XPM-induced impairments. By optimizing pre-compensation together with residual dispersion per span, XPM effect and XPM-induced nonlinear phase noise from adjacent RZ-DPSK or RZ-OOK channels can both be suppressed effectively.