1. Introduction

Spatial division multiplexing (SDM) is gradually becoming considered as an inevitable evolution in optical communications systems, in order to handle increasingly growing bandwidth demands [1]. In this context, multiple candidates have been proposed as transmission medium for future SDM optical communications. These candidates include bundles of single-core fibers, few-mode fibers (FMFs), multi-core fibers (MCFs) with cores supporting single or few-mode transmission as well as coupled or uncoupled regimes [2].

In this paper, we categorize SDM systems according to the coupling between spatial channels occurring along the transmission path [2]. A coupled transmission regime is associated with the use of fibers along which the lightwaves on different spatial channels are strongly mixed such as FMFs or coupled-core MCFs (CC-MCFs). These systems typically require the use of SDM multiple-input multiple output subsystems (SDM-MIMO) to undo the crosstalk between spatial channels. Uncoupled SDM systems are the alternative, where crosstalk between spatial channels is deliberately minimized, as in homogeneous or heterogeneous uncoupled single-mode MCFs (SM-MCFs). These systems typically do not use SDM-MIMO and crosstalk between spatial channels is handled as an additional transmission penalty [2,3]. Extensive research on the impact of crosstalk on SM-MCFs [3,4] and it’s dynamics [5] has been performed in recent years, showing that crosstalk may ultimately limit their performance [6]. This has driven research into techniques such as the use of heterogeneous SM-MCFs, where the refractive indexes of adjacent cores are designed to be distinct from each other to reduce inter-core interactions [7]. Another possible approach is the use of direction interleaving, allowing a crosstalk reduction by more than 10 dB per 100 km [8]. Nevertheless, the increasing demand for spatial spectral efficiency has led to the development of SM-MCFs with increasingly higher core counts subsequently leading to stronger crosstalk impact [9,10].

In this context, this group has recently proposed and demonstrated the use of SDM-MIMO to undo crosstalk after long distance transmission on a homogeneous SM-MCF [11]. The proposed method exploited two aspects of transmission with SDM-MIMO in SM-MCFs that have been previouslly identified in [12]. Firstly, the required memory length of the equalizers scales linearly with the differences in propagation delays between interacting cores, referred to here as inter-core skew (ICS). The ICS may reach several tens of nanoseconds per span [13] in homogeneous SM-MCFs and hundreds of nanoseconds per span [7] in heterogeneous MCFs. As such, the prohibitively large memory length requirements of such SDM-MIMOs may be considered a major limitation for digital compensation of crosstalk in SM-MCF transmission links. However, [11] has introduced the use of periodic ICS compensation to reduce the required SDM-MIMO memory length, showing that it could potentially reach those used in coupled SDM systems.

The second point proposed by [12] is that the complexity of equalization in SM-MCFs scales with the number of cores rather than the square of the number of modes, as in the case of FMFs or CC-MCFs. This can be intuitively understood by considering that the crosstalk interactions affecting a given core occur mainly with it’s nearest neighbors. Therefore, one may neglect interactions with other cores, reducing the complexity of the required SDM-MIMO [12]. This point has become increasingly relevant as recent demonstrations using high mode-count FMFs have required the use of very complex SDM-MIMO sub-systems [14]. The work in [11] has shown that most of the crosstalk penalty in a homogeneous SM-MCF long distance transmission link can be eliminated by a reduced order SDM-MIMO.

This paper extends [11] by including a detailed description of the ICS compensation technique. We evaluate the validity limits of this technique through numerical simulation, showing that the memory length of the system can be maintained independent of the transmission distance as long as the accumulated crosstalk is below a threshold of approximately −11 dB. We also include an experimental evaluation of the use of ICS compensation to maintain the required memory length of the system. It is shown that the memory length of our system is approximately 20 ns and this value is maintained up to its maximum reachable transmission distance of approximately 10000 km. We have also reprocessed our previous data to assume and SDM-MIMO with the minimum required memory length (20 ns) to demonstrate partial undoing of the crosstalk penalty with PDM-QPSK signals. Finally, we extend the experimental demonstration of the proposed technique to PDM-16QAM and PDM-64QAM signals, achieving reach gains of 14% in both cases.

This paper is structured as follows. Section 2 presents the principle behind the use of SDM-MIMO systems in uncoupled-core SM-MCFs as well as the use of ICS compensation to render the required memory length constant with transmission distance. Section 3 contains an experimental evaluation of the proposed method as well as it’s demonstration using PDM-QPSK, PDM-16QAM and PDM-64QAM signals. Final conclusions are presented in section 4.

2. Reducing the memory length of transmission systems using MCFs

In this section, we describe the ICS compensation approach in further detail. Figure 1(a) shows the impact of residual ICS on the temporal duration of the impulse response of a long distance transmission link composed by multiple spans of a 2-core SM-MCF considering only the effect of group velocity. In this simple example, we can define the SM-MCF transfer matrix as having four coefficients, h_i,j with i and j referring to cores 1 or 2. Without transmission, the impulse responses relating the signals at the output of each core with themselves at the fiber input, h_1,1 and h_2,2, are short time-aligned impulses. Also, the impulse responses relating to the interaction between different cores, h_1,2 and h_2,1, are null, indicating the absence of crosstalk. After transmission through the first span, the impulse responses for the signals in the cores become delayed from one another due to the ICS of that span, T_ICS,1. In addition, the nearly uniform occurrence of crosstalk along the SM-MCF [4] yields rectangular shapes for h_1,2 and h_2,1 with a temporal width corresponding to T_ICS,1 and an area depending on the average crosstalk in that span. Note that the rectangular shape is merely a rough approximation. In reality, the impulse response would be composed of a finite sum of impulses with random amplitudes and phases, corresponding to the crosstalk contributions of the phase matching points along the fiber [4]. The sum of their magnitudes and temporal extent would correspond to the average crosstalk and the ICS, respectively. We are also neglecting the crosstalk power that partially couples back to its originating core. This secondary phenomenon will be referred to in this work as second order crosstalk and is generally neglected in uncoupled-core MCF systems due to its low magnitude. However, it will be shown later in this section that it constitutes the main physical limitation to the effectiveness of ICS compensation. Finally, we are neglecting dynamic fluctuations of the ICS and average crosstalk power. The former have been shown in [13] to be substantially smaller than the average ICS. The latter should change at a very slow rate, compared with the equalizers’ update rate [4].

 

Fig. 1 Evolution of the system impulse response along a long distance MCF transmission link (a) without ICS compensation and (b) with periodic ICS compensation.

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At this point, the impulse response of the transmission system has a memory length corresponding to the ICS of the first span. As the number of spans increases, the propagation delay between cores will increase linearly with the transmission distance, if the ICS is similar on all spans. As such, the memory length of the system increases linearly with the transmission distance as ${\sum }_i{T}_{\text{ICS},i}$. Several techniques may be used to reduce the memory length or the transmission system, such as scrambling cores on different spans or using cores with opposing ICS. Nevertheless, these techniques have limited effectiveness to reduce the required memory length of the system or require specially designed fibers to compensate skew, which may be impractical to produce.

Here, we propose the use of optical delay lines (ODLs) to periodically compensate the ICS, as shown in Fig. 1(b). The ODLs must be chosen in a way that the propagation delays of all the cores become the same. As a result, the impulse responses h_1,1 and h_2,2 are aligned after the ODLs and the crosstalk contributions in h_1,2 and h_2,1 are shifted to match the temporal positions where the crosstalk contributions of the subsequent span will occur. If ODLs are placed after each span in the link, the crosstalk contributions of all spans will be superimposed. In these conditions, the memory length of the system becomes max {T_ICS,i}, where max{·} stands for the maximum value. In practice, this implies that the memory length of the system will corresponds to the maximum ICS of any span in the link. It also implies that the required memory length becomes independent of the transmission distance. The validity of these assumptions will be addressed in the following.

Figure 2 shows an evaluation of the accumulated crosstalk and required memory length as a function of the span count on a simulated two-core SM-MCF link with identical spans. We have considered crosstalk per span values of −28 dB, −31 dB and −32 dB. For the sake of convenience, we express the ICS values per span in terms of equalizer taps and we assumed the values of 500, 1000 and 1500 taps per span. For each span count, we computed the system impulse response and estimated the required memory length to encompass an arbitrarily chosen 99.93% of the signal energy. Figure 2(a) shows examples of the system impulse response when considering an ICS of 1000 taps after 25, 50 and 100 spans. For the sake of convenience, we present the sum of the magnitudes of h_1,1, h_1,2, h_2,1 and h_2,2 instead of the individual impulse responses, which is sufficient to evaluate the required memory length. Figure 2(a) shows the effect of the flat crosstalk contribution on the impulse response increasing in magnitude linearly with the transmission distance but maintaining its temporal extent. It also shows the impact of second order crosstalk growing substantially faster than the crosstalk. Note that in our example the second order crosstalk is easily distinguishable from the main crosstalk due to the ICS compensation, which partially separates both components temporally. The second order crosstalk grows at twice the rate of the main crosstalk component because it originates from the growing main crosstalk component. One may expect that the contribution of the second order crosstalk to the system memory length is negligible up to the point where its magnitude is comparable to that of the crosstalk. This is shown in Figure 2(b) and (c). As the number of spans increases, the total accumulated crosstalk increases in a nearly linear manner but the required memory length remains constant. When the total accumulated crosstalk reaches approximately −11 dB, we can no longer assume that it increases linearly and the second order crosstalk can no longer be neglected. However, we note that the threshold of −11 dB depends strongly on our previous assumption that the required memory length would need to encompass 99.93% of the signal energy. From this point, the second order crosstalk will increase the required memory length of the system with the square root of the transmission distance. Empirically from Fig. 2 we can estimate the maximum span count for constant required memory length as:

 

Fig. 2 Simulation evaluation of the dependence of the required memory length for SDM-MIMO on the accumulated crosstalk, the ICS per span and the number of spans, when using ICS compensation. a) Sum of the magnitudes of the simulated impulse responses with 1000 taps of ICS after 25, 50 and 100 spans. b) Accumulated crosstalk as a function of the number of spans for different values of crosstalk per span. c) Required SDM-MIMO memory length in taps, as a function of the span count for different values of ICS per span (expressed in taps) and different values of crosstalk per span. Note that the points where the required memory length stops being constant correspond to an accumulated crosstalk of approximately −11 dB.

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In this section, we have described the effectiveness of the ICS compensation method to reduce the memory length of a SM-MCF system and evaluated its validity limits. However, we stress that the application of this technique may be limited to the case of homogeneous SM-MCFs, where residual ICS values of a few tens of nanoseconds per span lead to already very demanding system memory lengths. In the case of heterogeneous SM-MCFs with ICS values intentionally set at several hundred nanoseconds per span, the corresponding memory length of the system may prevent its inversion with SDM-MIMO equalizers. For this reason, we may consider that the use of high-order MIMO to compensate the crosstalk in SM-MCF transmission systems is likely to be limited to the case of homogeneous SM-MCFs, which is perhaps the one that would most benefit from digital crosstalk compensation.

3. Demonstration of a long distance MCF link supported by SDM-MIMO

3.1. Experimental setup

Figure 3 shows a simplified diagram of the experimental setup. The light from an 100 kHz linewidth external cavity laser (ECL) operating at 1552 nm was split by a 1×3 power splitter and sent to 3 dual-parallel dual-polarization Mach-Zehnder modulators (DP-MZM) after appropriate polarization alignment using polarization controllers (PCs). The modulators were driven by 12 arbitrary waveform generators (AWG) operating at 60 GS/s. The AWGs produced pre-equalized signals to compensate for the frequency limitations of the transmitters. Each AWG produced a distinct pseudo-random binary signal with a length of 2¹⁵−1 symbols, such that the output of each modulator was a PDM-QPSK signal. Later in this experiment, the AWG signals were modified to produce also PDM-16QAM or PDM-64QAM signals. The power of the three signals was adjusted using erbium-doped fiber amplifiers (EDFAs) followed by variable optical attenuators (VOAs). Transmission was emulated using 3 synchronized recirculating loops running through 3 cores of a 53.7 km 7-core MCF. After the transmission fiber, the signals were amplified and filtered by band-pass filters (BPFs) to limit the ASE power. The BPFs were followed by polarization scramblers (PS) synchronized with the loop, to reduce the impact of polarization dependent loss, and variable optical delay lines (VODLs) consisting of free-space optical delays along with fixed optical patch cords. The VODLs were used to statically align the average recirculation times of the loops. Note that a real implementation would generally dispense the use of variable devices. Static ODLs would likely be sufficient to compensate the average ICS with the dynamic ICS accounted for by an increase of the memory length of the SDM-MIMO. In this experiment, the required delays were measured by using reference signals identical to the 3 cores and measuring the arrival times with an oscilloscope. The required delays were less than 28 ns. Acousto-optic modulators (AOM) were used to control the recirculating signals.

 

Fig. 3 Experimental setup.

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Inset A of Fig. 3 shows a profile of the MCF. It had a cladding and core diameters of 160 μm and 8 μm, respectively. Each core had a step index profile and the core pitch was 44 μm, yielding crosstalk values of approximately −42 dB. Reference [11] presents a full table of the average crosstalk values in this fiber. For this experiment, we used 3 adjacent cores 1, 2 and 3 for the recirculating loops. The remaining cores 4 to 7 carried dummy signals to generate additional crosstalk. The latter were produced by extracting a sample of the signal from core 1 at the input of each recirculation and amplifying it. The amplified sample was then split using a 1×4 splitter and each replica was delayed using optical patch cords before being input into the corresponding core. The patch cord delays where all longer than 100 ns and different by more than 5 ns, decorrelating the replicas with respect to core 1 and amongst themselves.

The receiver consisted of 3 coherent receivers (CORX). Each CORX was composed by a polarization-diverse 90 degree hybrids, where the incoming signals were combined with light from a 100 kHz linewidth ECL used as local oscillator, followed by 4 balanced photodetectors (PDs). The 3 groups of PDs had electrical bandwidths of 37 GHz, 25 GHz and 20 GHz for channels 1, 2 and 3, respectively. The electrical signals were digitized by a 12-channel real-time oscilloscope operating at 80 GS/s and processed offline using MATLAB and C. The DSP consisted of resampling to 2 samples per symbol and normalization stages, followed by the SDM-MIMO. The latter took the form of a 6×6 butterfly structure with 2049-tap equalizers. The equalizer taps were computed using a data-aided least-mean squares algorithm (DA-LMS).

3.2. Experimentally measured SDM-MIMO impulse responses

Figure 4(a) shows the evolution of the impulse responses of the SDM-MIMO equalizers obtained for transmission distances from 1074 km (20 recirculations) to 10203 km (190 recirculations). To facilitate the interpretation of the impulse response, we have represented the sum of the magnitudes of the 36 impulse responses of the 6×6 MIMO. It is clear that the relevant contributions of signal and crosstalk fall within an 800 tap window, regardless of the transmission distance. Nevertheless, the superposition of the signal and crosstalk contributions renders them difficult to distinguish. As such, we have computed a variant of the impulse response where the signals from different cores are misaligned in a way that separates the contributions of crosstalk as shown in Fig. 4(b). Note that this misalignment was used only for the sake of assisting interpretation and was not used further in this work. It is shown that the crosstalk contributions between cores 1 and cores 2 or 3 extend over a range of approximately 800 taps, which correspond to an ICS of 20 ns. The crosstalk contributions between cores 2 and 3 extend over a range of less than 300 taps, corresponding to an ICS of 7.5 ns. Despite the differences in temporal extent, all the crosstalk contributions have similar power. In addition, the crosstalk contributions shown in Fig. 4(b) are not flat, as initially depicted in Section 2. This seems to result mainly from imperfections of the ICS compensation. We did not observe significant variations of the required equalizer length between measurements, which suggests that the impact of the randomly varying dynamic ICS was not significant in this experiment. These observations indicate, that the crosstalk between these cores may partially be undone by an SDM-MIMO equalizers with a memory length of 800 taps or 20 ns, as described in the following section. Finally, the impact of the crosstalk originating from cores 4 to 7 is not distinguishable in Fig. 4 because those signals are not included in the SDM-MIMO. This crosstalk may be expected to degrade mainly the signal in core 1, since it has 4 crosstalk-inducing neighbors. In contrast, the signals in cores 2 or 3 are only impaired by one crosstalk-inducing neighbor.

 

Fig. 4 Impulse responses of the SDM-MIMO equalizers for various transmission distances. a) Original equalizer impulse response and b) impulse response with delayed contributions to highlight the different crosstalk contributions.

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3.3. Transmission performance using SDM-MIMO and high-order modulation formats

To evaluate the transmission performance with and without the use of an SDM-MIMO, we used the experimental setup shown in Fig. 3. However, the SDM-MIMO equalizers were modified to support only 800 taps as this was shown previously to provide sufficient memory to encompass the impulse response of the system. In addition, after convergence using the DA-LMS algorithm, the equalizer update mechanisms were switched to a decision-directed mode. Carrier recovery was performed within the equalizer loop using data-aided or decision directed adaptations of the method proposed in [15]. Finally, we were able to switch the SDM-MIMO from a full 6×6 mode, where the interactions between the signals propagated through different cores were taken into account, to a core-independent mode, where the interactions between the signals on different cores were ignored. The later mode was equivalent to using 3 independent receivers, each using an equivalent 2×2 MIMO. This allowed comparing the use of the 6×6 MIMO with the independent detection of the transmitted signals, as is usually performed with uncoupled core MCFs [3].

Figure 5 shows the Q-factor dependence on the transmission distance for PDM-QPSK, PDM-16QAM and PDM-64QAM with and without the 6×6 SDM-MIMO. For all transmission distances, we used optimum launch powers of −4 dBm for PDM-QPSK and PDM-16QAM and −2 dBm for PDM-64QAM. The use of the SDM-MIMO has lead to Q-factor gains for all considered modulation formats. However, we note that those gains were not constant with the transmission distance. Instead they seemed to increase, stabilizing after a certain reach. The transmission reaches for maximum Q-factor gain were approximately 3800 km with 0.6 dB gain for PDM-QPSK, 1800 km with 0.4 dB gain for PDM-16QAM, and 300 km with 0.2 dB gain for PDM-64QAM. Assuming a minimum acceptable Q-factor of 5.7 dB, commonly used as a threshold for soft decision forward error correction with an overhead of 20% [16], the use of the SDM-MIMO provided reach increases of 9% for PDM-QPSK, and 14% for PDM-16QAM and PDM-64QAM. These translate into transmission reaches of 9780 km, 2631 km, and 503 km, respectively.

 

Fig. 5 Q-Factor dependence on the transmission distance using PDM-QPSK, PDM-16QAM and PDM-64QAM using optimum launch powers with and without MIMO demultiplexing.

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3.4. Discussion

Our experimental results have shown that an SDM-MIMO with 20 ns of memory length combined with periodic ICS compensation was sufficient for the transmission of PDM-QPSK signals over single-mode homogeneous MCFs with a reach of 9780 km. The requirement of 20 ns of memory length in our system is comparable to previously reported experiments using few-mode fibers with long-haul transmission distances [2]. We may highlight the work reported in [17], where a 20 ns SDM-MIMO was used to achieve a transmission reach of 3500 km. Recently, [18] have reported the use of a differential mode delay (DMD) management technique that did not require specialized fibers to achieve more than 6000 km with a 3-mode FMF and required memory lengths around 20 ns. In the case of few-mode MCFs, the reported transmission distances have so far been very short for a fair comparison. Nevertheless, [18] has recently reported more than 2500 km on a 12-core FMF with 3 modes per core, using the aforementioned DMD management technique, also with SDM-MIMO memory lengths around 20 ns. All of these works required some form of periodic compensation of the propagation delay between spatial channels. A noteworthy exception is the very promising work reported in [19] where CC-MCFs have allowed a reach of 5500 km using a 10 ns length SDM-MIMO. In this case, periodic compensation of the propagation delay between spatial channels was limited to matching the inline amplifier lengths and the impact of fiber nonlinearity was substantially reduced with respect to an equivalent single-mode fiber system. Furthermore, the reported value of required memory could potentially be reduced by handling limitations of the experimental setup that would not be part of a real system. As such, the use of CC-MCFs seems to be the most promising candidate for long-distance SDM transmission.

Perhaps the most significant limitation for the use of SDM-MIMO with ICS compensation in the field is the need to demultiplex and multiplex again the spatial channels in order to introduce appropriately tuned ODLs. This may be acceptable when using single-core fiber amplifiers but may become unpractical when moving towards SDM amplifiers. A solution that would not require spatial demultiplexing would certainly be desirable. Alternatively, the development of highly homogeneous MCFs with weaker crosstalk requirements but very low or null ICS would reduce or eliminate the need for ICS compensation, enabling the use of short memory length SDM-MIMO systems. This would also allow exploiting the fact that the complexity of the proposed MIMO systems grows linearly with the number of spatial channels rather than quadratically, as the SDM-MIMO systems required when using FMFs or CC-MCFs.

4. Conclusion

This work provided a detailed description of a method to use SDM-MIMO with homogeneous single-mode multi-core fibers. The proposed method required the compensation of the propagation delays between cores using optical delay lines after each span to impose a system memory length constant with the transmission distance. The validity of this approach was demonstrated using numerical simulation, showing that constant memory lengths may be achieved up to transmission distances corresponding to an accumulated crosstalk of approximately −11 dB. Our approach was also demonstrated experimentally, using a 7-core recirculating fiber loop with 3 cores under an SDM-MIMO. The memory length of the system was shown to be around 20 ns up to a transmission distance above 10000 km. This allowed the use of SDM-MIMO to reduce the crosstalk penalty after long distance transmission using 20 Gbaud PDM-QPSK, PDM-16QAM or PDM-64QAM signals, increasing the transmission reach by factors of 9% and 14%, respectively.