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Abstract
The inter-core crosstalk (IC-XT) of multi-core fiber (MCF) limits the capacity of space division multiplexing system (SDM) fundamentally. We develop a closed-form expression of the magnitude of IC-XT for various types of signals, which can well explain the mechanism of different fluctuation behaviors of the real-time short-term average crosstalk (STAXT) and bit error ratio (BER) for the optical signals with and without strong optical carrier. The experimental verifications with the real-time measurement of the BER and outage probability in a 7 × 10-Gb/s SDM system agree well with the proposed theory and confirm that the unmodulated optical carrier plays a substantial role in fluctuation of BER. The range of fluctuation can be reduced by 3 orders of magnitude for the optical signal without optical carrier. We also investigate the effect of IC-XT in a long-haul transmission system based on a recirculating 7-core fiber loop and develop a frequency-domain IC-XT measurement technique. Longer transmission distance is shown to have a narrower BER fluctuation range, since IC-XT is no longer the only dominant factor on transmission performance.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Multi-core fiber (MCF) associated with space division multiplexing (SDM) technology is regarded as one of the most promising solutions to overcome the capacity limit of the current optical communication system built on traditional single-core fibers [1,2]. Recently, MCF is reshaping the layouts of state-of-the art fiber-optic system from short reach to long haul, with its unique features of multiple parallel channels with similar delays. It can not only improve the capacity density of an extremely limited panel space inside the datacenter [3,4], but also reduce the energy consumption by the shared amplification among all cores for submarine cable system [5]. Despite these advantages, however, MCF suffers from a fresh fundamental issue unseen in single-core fibers, i.e., the inter-core crosstalk (IC-XT), which prevents the utilization of each core as a complete individual spatial channel to increase the capacity linearly.
The IC-XT in each core originates from the random coupling from other cores that share the same cladding. It occurs at randomly appearing discrete phase matching points (PMPs) along the fiber. The reduced core pitch of MCF for higher core density magnifies this crosstalk, making it a vital issue even in weakly-coupled MCFs [6,7]. Careful fiber designs such as heterogeneous cores [8] or trench-assisted cores [9] can alleviate the challenge, but cannot eliminate the IC-XT completely. Moreover, the IC-XT is stochastic due to longitudinally varying perturbations of the deployed MCF, such as the bend, twist, and structure fluctuation [10,11]. Eventually, IC-XT becomes a fundamental performance limiting factor of the MCF-based optical transmission system. For example, the maximum achieved signal-to-noise ratio (SNR) will be limited when the accumulated IC-XT cannot be neglected.
Recent investigations also link the performance fluctuation of the optical signals with considerable carrier (hereafter referred as carrier-supported signal) with the stochastic nature of the IC-XT. This is particularly critical in the intensity-modulation direct-detection (IM/DD) system, which dominates the short-reach scenarios. As reported in [11], the short-term average crosstalk (STAXT) of an on-off keying (OOK) signal exhibits strong fluctuation with >10-dB dynamic range, resulting in up to 5 orders of magnitude of fluctuations in the bit error ratio (BER) [12] and the risk of transmission outage [13–15]. The reports on the suppressed STAXT fluctuation with carrier-less signal (hereafter referred as carrier-free signal) also confirm the relation between the optical carrier and performance instability from another perspective [11]. The different behaviors on the fluctuations can be roughly explained by the reduced coherence with broadband modulation [10,11], but it lacks exact mathematic models of the variance between carrier-supported and carrier-free signals. There are few reports on the transmission stability based on the carrier-free signal [16], and none report of a unified performance stability comparison between the carrier-supported signal and the carrier-free signal. Another unexplored issue is the stability behaviour of IC-XT in the long-haul transmission scenarios.
In this work, we further develop the theoretical model for the IC-XT induced by the carrier-supported signal. Our model gives the closed form expression of the variance of IC-XT caused by the carrier-supported signal, carrier-free signal, and continuous wave (CW), respectively. We also make a comprehensive comparison of the real-time BER and outage probability of the signals with carrier (OOK) and without carrier (differential phase shift keying, DPSK) in a 41-km 7-core MCF. The results agree with our theory very well. Finally, we investigate the IC-XT performance under 820-km MCF transmission system based on a parallel 7-core MCF loop system. A novel frequency-domain technique is also developed to measure the IC-XT in each loop accurately. As the transmission distance increases, impairments such as amplified spontaneous emission (ASE), polarization-mode dispersion (PMD), and chromatic dispersion (CD) increase, IC-XT is no longer the only dominant factor on transmission performance, thus the BER gradually stabilizes.
2. Theoretical model of IC-XT
Although the statistical distributions of the IC-XT of CW light and carrier-free signals have been well studied theoretically in earlier works like [6,9,10,11], we further improve the modelling of IC-XT by also considering the theoretical analysis of carrier-supported signals. The three cases of IC-XT, including the single wavelength CW light, carrier-supported signal and carrier-free signal, are systematically analyzed and compared, in which we deduce their means and variances.
According to the mode coupling theory, the IC-XT of weakly-coupled MCF comes from the discrete PMP, which occurs randomly along the MCF due to longitudinally varying perturbations. The total IC-XT can be approximated as the linear summation of contributions from all PMPs and thus expressed by the following mode coupling equation for a CW light [6]
where Am→n is the normalized complex amplitude of the IC-XT in core n from core m with N PMPs, φk is the random phase difference at the k-th PMP which is uniformly distributed in [0, 2π], and Knm is the discrete coupling coefficient from core m to core n, whose absolute value is determined by [9] where κ, β, R, Dnm, and γ represent the mode-coupling coefficient, the propagation constant, the bending radius, the distance between core n and core m, and the twist rate, respectively.The total power of IC-XT of a CW light is denoted by a time-varying random variable and its probability distribution function conforms to a chi-square distribution (according to the central limit theorem). Strong fluctuations of the STAXT of a CW light was measured to a range of ∼20 dB [10].
The statistical properties of IC-XT of modulated wideband signals, however, are different from those of a CW light. This is mainly because that the modulation phase shift introduced by the difference between the propagation constants of the two cores (the inter-core skew, snm) changes the phase matching state of the PMPs. Thus, we can obtain the crosstalk transfer function (XTTF) for a certain length of MCF by considering the wavelength dependence to Eq. (1) [10] as
Thus, the IC-XT power induced by these three signals can be expressed as
Here we split the complex integral into an accumulation of contributions from M discrete narrow spectral lines, where Ω0 and Ωl are the contribution of carrier and narrow spectral lines at frequency of ωl, respectively, and given by
Furthermore, considering statistical independence, we could obtain the following results
Therefore, the average power of IC-XT induced by different kinds of optical signal with the same power are equal, given as
Given that it is unrealistic to deduce the variance of the IC-XT analytically from Eq. (8–10), numerical simulations by Monte Carlo method can be performed to obtain the statistical characteristics of the random variables in our model [11]. From Fig. 3 in [11], we can find that the variances of each Ωl are almost equal, but all covariances between different Ωl are almost negligible. This is valid when inter-core skew reaches several nanoseconds, which is a typical situation for MCF having a length of tens of kilometers. Meanwhile, the decorrelation bandwidth is calculated to be ∼100 MHz under this condition.
By omitting the covariance part between different Ωl, we make the following simplifications for the variance of IC-XT caused by the three kinds of signals as
Since the variance of Ωl and Ω0 are almost equal, the variance of IC-XT power caused by wideband carrier-free signal is M times smaller than that caused by CW light with the same optical power. Here, the maximum value of M is approximately equal to the ratio of the modulation bandwidth (tens of GHz) to the decorrelation bandwidth of IC-XT (typical 0.1 GHz-1 GHz). Thus, M is in the order of tens to hundreds. As a result, the fluctuations of STAXT caused by carrier-free signal are greatly suppressed by more than 10 dB.
For carrier-supported signal, the variance of IC-XT can be divided into separate contributions from the CW carrier and the “carrier-free” modulated signal, where the ratio of two parts is decided by the CSPR. Similar to the carrier-free signal, the variance from the wideband signal part is reduced more than M times so as to be neglected. Meanwhile, the variance from the carrier part is also smaller than that of CW light. Even assuming the ideal minimum value of c = 1 (infinite extinction ratio) for the carrier-supported signal, it is only 4 times smaller than the CW case, which is still a pretty large value.
According to the above theoretical analysis, we can conclude that the STAXT fluctuation caused by carrier-supported signal is partly suppressed but still in the same order of magnitude with the CW light case, which is much more severe than the carrier-free signal case.
3. Experimental setup of 10-Gb/s OOK/DPSK SDM transmission based on a 41 km 7-core fiber
We first conduct experiment to evaluate the stochastic behavior of IC-XT and its impact in an MCF-based system with the experimental setup as shown in Fig. 1. Here, we choose OOK and DPSK for comparison as they are the most representative modulation formats in carrier-supported signal and carrier-free signal respectively. In the transmitter, 10-Gb/s pseudo-random bit sequence (PRBS) with a word length of 215−1 is generated by a pulse pattern generator (PPG) and utilized to modulate the CW light centered at 1550.12 nm via a Mach-Zehnder modulator (MZM). The MZM is biased at the quadrature point or null point with the swing of driving signal of Vπ or 2Vπ for OOK and DPSK signal, respectively. The generated signal is first boosted by a high-power erbium-doped fiber amplifier (EDFA) and divided into two channels. One of the two channels is used as the test channel while the other is further split into six identical copies, launched into the other six cores of the 41-km long 7-core MCF together to serve as crosstalk channels. By exciting the six surrounding cores simultaneously, the decorrelation time of STAXT is greatly reduced, allowing us to use a shorter measurement time to characterize the STAXT completely [17,18]. The signal-to-crosstalk power ratio can be controlled by the two inserted variable optical attenuators (VOAs) so that the IC-XT levels can be set freely. Before launching into MCF, the optical signal in the test channel first passes an acousto-optic modulator (AOM) as a switch for STAXT measurement and a polarization scrambler (PS) to perturb the polarization of the signal light.
Fig. 1. Schematic diagram of the IC-XT measurement and BER performance evaluation experimental setup and the cross section of 7-core fiber. PPG, pulse pattern generator; MZM, Mach-Zehnder modulator; EDFA, erbium-doped fiber amplifier; PS, polarization scrambler; OF, optical filter; VOA, variable optical attenuator; PM, power meter; CDR, clock and data recovery module; BERT, bit error ratio tester.
The cross-section of the 41-km 7-core fiber used in our experiment is shown in the inset of Fig. 1. The cladding diameter is 150 µm and the core pitch is 42 µm. The dispersion parameter of each core is measured to be ∼20.8 ps/nm/km at 1550 nm by a commercial chromatic dispersion analyzer. The measured results of insertion loss (IL) and average IC-XT for the 7 cores together with fan-in/fan-out device are listed in Table 1. It is shown that the IL of each core is ∼11 dB, except that core 3 has a larger loss of 12.6 dB. The IC-XT between adjacent cores and non-adjacent cores are around -40 dB and less than -50 dB, respectively. Considering this small IC-XT level, the test channel is loaded to the center core (core 1), and six crosstalk channels are loaded to the six outer cores (core 2-7) to get a large overall IC-XT to investigate the IC-XT induced BER performance deterioration.
Table 1. IL&IC-XT Measurement Result of 41-km 7-Core Fiber with Fan-in/fan-out (In dB)
At receiver side, a power meter (PM) is utilized to measure the overall IC-XT of core 1. Here, the IC-XT is defined as the ratio of the crosstalk power which couples from other cores to the core of interest to the signal power. Thus, measuring the IC-XT needs to get the signal power and the crosstalk power separately. To focus on the pure effect of IC-XT, we use dispersion compensation fiber (DCF) to remove the effect of CD in our experiment. Similarly, we utilize a pre-amplifier to assure the received optical power well above the receiver sensitivity to minimize the effects of electrical noise of photodetector (PD). Besides, for DPSK signal, a one-bit delay line interferometer is used to demodulate the differential phase information to intensity signal for PD detection. Meanwhile, for a fair comparison, a single-ended PD is used both for OOK and DPSK. The decision is performed in a clock and data recovery (CDR) module. The data and regenerated clock are sent to a BER tester (BERT) for real-time BER measurement. A personal computer (PC) is connected to the PM and the BERT to record the real-time STAXT and BER at regular intervals.
4. Experimental comparison of BER performance stability between OOK signal and DPSK signal
We performed real-time BER measurements for both modulation formats over 30 minutes at different average IC-XT levels from -26 dB to -13 dB, and the results are shown in Fig. 2. The BER variation due to the time-varying polarization mismatch between signal and crosstalk [12] is smoothed via a random polarization scrambling with a period of 1 ms, which is much smaller than the BER measurement interval of 2 s. It is worth noting that the transmission system can achieve error-free transmission when the crosstalk channels are turned off, which guarantees that the system performance is limited by the IC-XT dominantly. As expected, the OOK signal exhibits severe BER fluctuations with the range of several orders of magnitude at all average IC-XT levels due to the severe fluctuations of STAXT determined by the strong optical carrier. In contrast, for DPSK signal, since there is no single-frequency optical carrier and the signal bandwidth of 10 GHz is much larger than the decorrelation bandwidth, the STAXT fluctuations are well suppressed. Correspondingly, the BER performance is relatively stable, with gentle fluctuations of much less than 1 order of magnitude.
Fig. 2. Real-time BER performance of 10-Gbit/s OOK signal and 10-Gbit/s DPSK signal respectively in 30 minutes under different average IC-XT levels.
In order to analyze the statistical characteristics of the BER performance for the two modulation formats more clearly, we plotted the line chart with error bar shown in Fig. 3, based on results in Fig. 2. The blue and red solid curves are the average BER, while the endpoints of the error bars are the extreme values of BER fluctuations and the shaded color bands are the 95% confidence interval. It is shown that the BER fluctuation range of DPSK signal is several times smaller than that of OOK at all IC-XT levels. In addition, average BER of DPSK deteriorates more slowly than OOK with the increase of IC-XT. When the average IC-XT is small enough (< -19 dB), the average BER performance of the OOK signal is better than DPSK, otherwise it is worse. This could be attributed to the difference in the mechanism of how the IC-XT affects the BER for the carrier-free signal and carrier-supported signal. For the carrier-free signal, the crosstalk serves as the distortion to the signal with the same bandwidth, which can be modeled as additive Gaussian white noise [6]. For the carrier-supported signal, however, the crosstalk term is dominated by the unmodulated carrier, which exhibits more of a fluctuating power but less in-band distortions. However, due to serious fluctuations, the worst case of the short-term BER of the OOK is always worse than that of DPSK. Finally, we can confirm that the BER stability of DPSK signal is much better than that of OOK signal in MCF enabled SDM transmission.
Fig. 3. Average BER together with the BER fluctuation bounds of 10-Gbit/s OOK signal/10-Gbit/s DPSK signal in 30 minutes as a function of average IC-XT level.
The advantage of stable BER performance for practical transmission links is that the outage probability caused by BER fluctuations can be greatly reduced under the same average IC-XT level. For example, Fig. 4 gives the long-time BER measurement for OOK signal and DPSK signal during 5 hours. The average IC-XT is set to -17 dB for both modulation formats and the BER sampling interval is 5 s which means 3600 samples are recorded in total. It is shown that 122 BER samples of OOK signal exceed the KP4 FEC limit of 2.1 × 10−4, which means the outage probability of OOK signal is around 3.4% with -17 dB average IC-XT. In contrary, all BER samples of DPSK signal is below the KP4 FEC4 limit. In other words, under the same outage probability requirement, the DPSK signal can endure higher IC-XT level.
Fig. 4. Long time BER performance and outage probability of OOK signal/DPSK signal with the average IC-XT of -17 dB and the KP4 FEC limit of 2.1 × 10−4.
5. 7-core fiber recirculating loop based long-haul IC-XT and BER performance test
Furthermore, we study the IC-XT and BER performance in long-haul transmission with an MCF-based recirculating loop as shown in Fig. 5. The transmitter and receiver part of the experimental setup is almost the same as those in Fig. 1, except that a low-speed PD (∼100 MHz bandwidth) together with an oscilloscope are introduced to monitor the loop power status in receiver side. The recirculating loop consists of a spool of 41-km-long 7-core fiber, DCF, EDFA, waveshaper, polarization scrambler and an AOM as loop switch. The waveshaper works as an optical filter to suppress the out-of-band ASE noise. Here, the period of polarization scrambling needs to be synchronized with the loop period to ensure that polarization-dependent effects such as polarization-dependent loss (PDL) or PMD do not accumulate anomalously with loop turns. In addition, a delay signal generator is required to produce synchronous control signals for AOMs, polarization scrambler and the BERT.
Fig. 5. Schematic diagram of the 7-core fiber loop based long-haul IC-XT measurement and BER performance evaluation experimental setup. DG, delay generator; WS, waveshaper; OSA, optical spectrum analyzer.
It is worth mentioning that the IC-XT measurement and BER test cannot be carried out simultaneously in this loop system. When performing IC-XT measurements, it is impossible to turn off the signal path and measure the crosstalk power by a power meter as we used above. Here, we propose an IC-XT measurement scheme by introducing a separate crosstalk channel with a different wavelength and measuring its power via an optical spectrum analyzer (OSA). The data-carrying signal occupies the optical carrier centered at 1550.12 nm, while the crosstalk channel is centered at 1551.72 nm. A wavelength division multiplexing (WDM) de-multiplexer is then utilized to separate the signal and crosstalk, and they are injected into core 1 and core 2-7, respectively. To accommodate the crosstalk signal inside the loop, a passband centered at 1551.72 nm needs to be added into the profile of waveshaper. Finally, we use a third AOM with synchronized driving signal as a timing mask to select the required number of turns for the IC-XT measurement in OSA.
In the experiment, the launching power of each core is set as 0 dBm. Meanwhile, we select DPSK format for both signal and crosstalk in the measurement and average five times for each independent measurement, in order to reduce the impact of STAXT fluctuations on the measurement results. As shown in the insets of Fig. 6, stronger signal peak at 1550.12 nm and a weaker crosstalk peak at 1551.72 nm can be clearly seen in the optical spectrum, where the difference between these two peaks is the accumulated IC-XT level. Meanwhile, as the transmission distance increases, the signal peak remains the same level, while the crosstalk peak gradually increases, that is, the average IC-XT gradually increases. The measured accumulated IC-XTs as a function of transmission distance are given in Fig. 6. Obviously, the experimentally measured IC-XT growth curve is in good agreement with the theoretical linear growth curve, which indicates that the relay optical amplification in long-haul transmission has no significant impact on the accumulation of IC-XT.
Fig. 6. Experimental measurement results and theoretical curves of accumulated IC-XT level as a function of transmission distance.
We further perform long-term real-time BER measurement. The BERs of OOK/DPSK after 10/15/20 turns (410 km/615 km/820 km transmission) are measured continuously for 30 minutes, and the results are shown in Fig. 7. The average BERs after transmission increase along with the increase of distance due to the gradual accumulation of IC-XT. Compared with the results as shown in Fig. 3, the measured BER here is slightly higher for the same average IC-XT level, which can be attributed to the accumulated ASE noise, PMD and residual CD in the long-haul transmission. For example, the average BER of DPSK is deteriorated from 1 × 10−6 to 5 × 10−6 with IC-XT level of 20 dB after 820-km transmission.
Fig. 7. Real-time BER performance of 10-Gbit/s OOK signal/10-Gbit/s DPSK signal in 30 minutes after 10/15/20 laps.
In terms of BER fluctuations, DPSK signals are still much more stable than OOK signals after long-haul transmission. However, compared with the short-distance results in Fig. 2 and Fig. 3, it is apparent that the BER fluctuation range is reduced after long-haul transmission, which is even obvious on the lower bound of OOK signals. For example, the lower bound of the instantaneous BER fluctuation of OOK signal after 20 laps (∼5 × 10−7) is significantly compressed compared with the BER results at same average IC-XT level (-20 dB) in Fig. 3 (∼5 × 10−9). This reduction in the BER fluctuation range is also caused by other transmission impairments (ASE noise, PMD and residual CD). These impairments accumulate with the number of loops and result in a “noise floor” even in the absence of IC-XT, thus, the instantaneous BER no longer exhibits extremely low values. In addition, it should be noted that the upper bound of BER fluctuation does not change significantly, which means the IC-XT is the dominant role when the instantaneous IC-XT is at an extremely large value. On the other hand, for DPSK signal, since its STAXT is inherently very stable, the BER fluctuation range will not be significantly compressed like OOK signal. Therefore, the stability advantage of the carrier-free signal will gradually decrease as the transmission distance increases.
6. Conclusion
In this paper, we have developed a complete mathematic model for the respective variance of IC-XT induced by the carrier-supported and carrier-free signals, which gives a closed form expression for the respective IC-XT level of these two signals and continuous wave. Then a comprehensive comparison of the real-time BER and outage probability of these two types of signals with OOK and DPSK in a 41-km 7-core MCF is experimentally performed, and the results show that carrier-free signal outperforms carrier-supported signal on transmission stability. Furthermore, the parallel loop based long-haul transmission results show that the BER fluctuations will become gentler as the transmission distance increases, indicating that these fluctuations caused by IC-XT might become trivial in ultra-long-haul MCF transmission links.
Funding
National Key Research and Development Program of China (2019YFB2204004); National Natural Science Foundation of China (62101049, 62105032).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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