## Abstract

In this paper we numerically investigate nonlinear impairments in a WDM system with mixed PM (D)QPSK and OOK channels. First we analyze the dependence of XPM and XPolM on SOP and baud rate in absence of PMD. In this case we find that the nonlinear impairments are highly dependent on relative SOP between the PM (D)QPSK and neighbouring OOK channels. The dependence on relative SOP is more pronounced in differential detection than in coherent detection. However, with increasing values of PMD this dependence decreases, and non-linear tolerance improves.

© 2012 OSA

## 1. Introduction

In order to cope with the increase in demand of high capacity fiber optic transmission links, up gradation of 10 Gbps links with 40/100 Gbps channels has driven the industry to deploy polarization multiplexed quadrature phase-shift keying systems with differential and coherent detection (PM (D)QPSK) as alternatives. However the performance of these PM (D)QPSK channels in pre-existing non-return-to-zero on-off keying (NRZ OOK) based dense wave division multiplexed (DWDM) systems is impacted by nonlinearities induced by cross phase modulation (XPM) and cross polarization modulation (XPolM) [1, 2]. The impact of both XPM and XPolM depends on the baud rate of the PM (D)QPSK channel and relative state of polarization (SOP) to the neighbouring NRZ OOK channels [3, 6]. Here we investigate the dependence of XPM and XPolM on relative SOPs for both coherent and differential detection at three different baud rates of 10 Gbaud, 28 Gbaud and 56 Gbaud. Furthermore, we perform an in depth analysis of the nonlinearities in terms of Stokes vectors of the PM (D)QPSK channel after transmission. We find that the system impact of XPM and XPolM is different for differential and coherent detection. We also observe that polarization mode dispersion (PMD) reduces the dependence of nonlinear impairments on relative SOP of the neighbouring NRZ OOK channels.

## 2. Simulation setup

The setup for simulations using VPI Transmission Maker is shown in Fig. 1
. Two nested Mach-Zehnder modulators (MZMs) in the polarization multiplexed transmitter were driven by De-Bruijn bit sequences (DBBS) of order 13 and 12 in *x* and *y* polarization. Laser sources with 1 MHz linewidth at a wavelength of 1550 nm were used both in the transmitter and as local oscillator. The signal was pulse carved at 50% duty cycle by an extra MZM for generation of return-to-zero signals. The optical fiber link consisted of eight spans of 80 km standard single mode fiber (SSMF) with full inline dispersion compensation. Each span was amplified by ideal (noise free) amplifiers. The attenuation in SSMF was 0.2 dB/km, the dispersion was 17 ps/nm-km and the non-linear coefficient was 1.31 W^{−1}km^{−1}. For dispersion compensating fibers (DCFs) the attenuation coefficient was 0.5 dB/km, the dispersion was −85 ps/nm-km and the non-linear coefficient was 3.5 W^{−1}km^{−1}. Four NRZ OOK channels were driven by different (PRBS) sequences at 10 Gb/s. The separation between the central PM (D)QPSK channel and the NRZ OOK channels was 100 GHz. The NRZ OOK channels were separated by 50 GHz from each other.

At the receiver white Gaussian optical noise is added to the signal (in order to vary the OSNR; performance was then evaluated by calculating the required OSNR to obtain BER = 10^{−3}), before being demultiplexed using a Gaussian filter having a bandwidth of 70 GHz. For both coherent and differential detection, the detected signal was low pass filtered by a fourth order Bessel filter with 3-dB bandwidth equal to 70% of the baud rate. In the coherent detection case, the digital field was reconstructed using the inphase/quadrature components of each polarization (sampled at 2 samples per symbol) and digitally processed by constant modulus algorithm (CMA) based adaptive filters having 7 taps in a butterfly structure [7]. Carrier phase recovery was done by employing “Viterbi-Viterbi” carrier phase estimation [8]. The number of taps for moving average filter in the carrier phase estimation was optimized for every power level and OSNR, by minimizing the average BER for the *x* and *y* polarization components.

## 3. Dependence of non-linear impairments on SOP and baud rate

Simulations were first performed at three baud rates of 10 Gbaud, 28 Gbaud and 56 Gbaud of the PM (D)QPSK channel and two states of polarization of the NRZ OOK channels relative to *x* polarization of the PM (D)QPSK channel at 0°, referred to as case (a), and 45° referred to as case (b). The launched power was increased from linear to non-linear regime and the power into DCFs was 5 dB lower than into SSMFs. The resulting required OSNRs for a bit error rate (BER) of 10^{−3} are shown in Fig. 2
. It can be observed that the back-to-back performance of coherent detection is only marginally better than differential detection (while one could expect a difference of 1 or 2 dB). This is due to the fact that in coherent detection differential coding was used to increase robustness against cycle slips at high power levels [9], which adds a penalty of 0.5 dB to the performance of coherent QPSK transmisssion.

We analyze the results in Fig. 2 with the help of SOPs on the Poincaré Sphere and constellation diagrams of *x* and *y* polarization shown in Fig. 3
. From the transmitter, the central channel has its SOP at either +45°, right hand circular (RHC), −45° or left hand circular (LHC). During transmission, the SOP of the central channel rotates around the total Stokes vector according to the Manakov model [10]. For illustration, the blue dots show the total Stokes vector of the DWDM signal before transmission when all the neighbouring NRZ OOK channels carry a 1 bit, which is the scenario giving maximum polarization rotation. The red dots show the SOP of the received optical signal after transmission.

Points along the meridian shown in magenta on the Poincaré Spheres in Fig. 3 represent signals where the *x* and *y* polarization components have equal amplitudes, but different relative phase. In case (a), it can be observed in Fig. 3 that the central channel signals rotate on that meridian (around the total Stokes vector at LHP). As a result, the *x* and *y* polarization components will accumulate a relative phase shift due to XPM. While the average polarization rotation can be reversed by the butterfly equalizer in coherent detection or an analog polarization controller in differential detection, a pattern-dependent spread along the magenta meridian remains (shown in Poincaré Sphere in 2nd row in Fig. 3). This spread we refer to as XPM induced phase noise and is the dominating effect in case (a). The same phenomenon can also be seen from a different viewpoint, using the constellation diagrams for the *x* and *y* polarization: the XPM-induced phase jitter in *x* is higher than in *y* polarization. The reason is that the XPM induced in *x* polarization by NRZ OOK channels (co-polarized to *x* polarization) is twice as large as XPM in *y* polarization, which is orthogonal to NRZ OOK channels [10].

Points along the green meridian on Poincaré Spheres in Fig. 3 represent signals where the *x* and *y* polarization components have a phase difference of 90° but different amplitudes. In case(b), it can be observed in Fig. 3, 3rd row, that the central channel signals rotate on that meridian (around the total Stokes vector at + 45°). The remaining spread along the green meridian after reversing the average polarization rotation (shown in Poincaré Sphere in 4th row in Fig. 3) leads to polarization crosstalk resulting in additional amplitude jitter. This effect is what we refer to as XPolM [11, 12].This amplitude jitter is also depicted in constellation diagrams (shown in 4th row in Fig. 3) which have a higher amplitude jitter in comparison to constellation diagrams in 2nd row in Fig. 3. From the analysis of constellation diagram in 4th row of Fig. 3 we can also observe that both constellations in *x* and *y* polarization have been impacted by XPM in equal amount, since the NRZ OOK channels are at 45° with respect to both *x* and *y* polarization. The signal in case(b) is therefore impacted by both XPM and XPolM.

To quantify the nonlinear tolerances we compare the non linear threshold (NLT), defined as the launch power for which the required OSNR is increased by 2 dB compared to back-to-back, for the three baud rates and two relative states of polarization in Table 1 .

We can draw several conclusions from the Table. 1. For case (a) the nonlinear threshold increases with baud rate both for differential and coherent detection. This is due to the fact that the nonlinear phase shift introduced by XPM from neighbouring NRZ OOK channels becomes constant over several symbols of the PM (D)QPSK channel when the symbol time becomes relatively shorter than the neighbouring NRZ OOK pulses. This is obviously beneficial in differential detection, because a constant phase shift is effectively canceled. Also for coherent detection, however, carrier phase estimation can mitigate XPM induced phase noise more effectively for higher baud rate, because more slowly varying phase is more easily tracked.

For case (b) and differential detection, the NLT decreases with baud rate, indicating both that XPolM dominates in this case and that the impact of XPolM increases with baud rate. A possible explanation is that the amplitude noise becomes correlated over more symbols with increasing baud rate for XPolM (in the same way as phase noise becomes correlated over more symbols for XPM). Since differential detection compares one noisy symbol to another noisy symbol it may be sensitive to such a correlated amplitude noise. With coherent detection the NLT increases with baud rate also in case (b), which indicates that reduction in XPM is strong enough to impact the system performance even in this case.

Next, the SOP of neighbouring NRZ OOK channels was swept from 0° to 90° with respect to the *x* polarization of central channel to further investigate the dependence of XPM and XPolM on relative SOPs for coherent detection. Figure 4
shows the BER versus NRZ OOK SOPs at 18 dB OSNR for 10 Gbaud, at 19 dB OSNR for 28 Gbaud and at 21 dB OSNR for 56 Gbaud. From the analysis of Fig. 4(a) it is apparent that at 10 Gbaud, BER for *x* and *y* polarization are complimentary of each other and average BER remains fairly constant. Each polarization has maximum BER due to XPM when the NRZ OOK channel SOPs are parallel to it and minimum when SOPs are orthogonal to it. However for 28 Gbaud and 56 Gbaud the average BER as well as the BER of the individual polarizations is maximum at 45° SOP of NRZ OOK channels. This additional penalty is due to XPolM at 45° SOP (which we have discussed above).

In practical DWDM systems the SOPs of the neighbouring NRZ OOK channels are typically random and different for each channel. To investigate how system performance may vary in this case we also performed simulations with 400 different realizations of NRZ OOK signals having random SOPs uniformly distributed over the Poincaré sphere. Statistics were gathered for a coherent PM QPSK system at 28 Gbaud and fixed launched power of 2.2 dBm, and plotted in Fig. 5(a)
, which shows the probability density function (PDF) of the required OSNR for BER of 10^{−3}.

The lowest required OSNR in Fig. 5(a) is 18.4 dB which is the same required OSNR as for case (a) of Fig. 2(b) at 2.2 dBm and the highest required OSNR is 24.1 dB which is the same required OSNR as for case (b) of Fig. 2(b) at 2.2 dBm. The average required OSNR for all SOPs is around 20.1 dB which is closer to the lowest than to the highest required OSNR. The result in Fig. 5(a) indicates that the SOP of all channels need to be quite well aligned in order for the non-linear impact to be highest, and consequently the likelihood of required OSNR to be close to the highest value of 24.1 dB is very low.

## 4. Impact of polarization mode dispersion on non-linear impairments

In the simulations presented above, PMD was neglected. We now run a new set of simulations, in which PMD is included in order to replicate real transmission scenarios. Figure 5(b) shows the probability density functions of required OSNRs for BER of 10^{−3} for different PMD values, at a fixed launched power of 2.2 dBm, for coherent detection. Again, 400 realizations of PMD and random initial SOPs were used. It is apparent that with increasing values of PMD, performance is improved, and the variance of PDFs of required OSNRs decreases, which is in agreement with what has been observed in previous studies [13]. This can be explained by the polarization walk-off effect [14, 15]: PMD causes the SOPs of the different channels to follow different random paths over the Poincaré sphere, so that the SOP rotation induced on the central PM QPSK channel by neighbouring channels through XPM (as argued in Section 3) changes in direction throughout transmission, and does not accumulate as effectively at the end of the link. It is interesting to note from Fig. 5(b) that a PMD above 0.02 ps/km^{1/2} is needed for this process to become noticeable.

## 5. Summary and conclusion

In this paper we numerically investigated the dependence of nonlinear impairments on baud rates and relative SOP between central PM (D)QPSK channel and NRZ OOK channels. We discovered that for systems with very low or no PMD the impact of XPM and XPolM strongly depends on the baud rate and relative SOP between central channel and NRZ OOK channels for both coherent and differential detection. The dependence on relative SOP is more pronounced in differential detection than coherent detection. Secondly we also observed that the dependence of nonlinear impairments on SOPs of neighbouring channels reduces with increasing PMD thus making the system more immune to outage due to high nonlinear penalties.

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