We investigated the inter-core differential mode delay (DMD) characteristic of a weakly-coupled homogeneous multi-core fiber with a view to utilizing inter-core crosstalk compensation with MIMO processing. We confirmed experimentally that the bend induced inter-core DMD is lower than the simulated results, which we expected owing to the twist of the fiber. We also revealed numerically that the refractive index profile variation of each core greatly increases inter-core DMD. Finally, we conducted a 4 × 4 MIMO transmission experiment using a weakly-coupled 4-core fiber and successfully compensated for the inter-core crosstalk.
© 2014 Optical Society of America
Space division multiplexing (SDM) technologies have been intensively investigated with the aim of greatly improving the transmission capacity. Multi-core fiber (MCF) is one of the transmission media designed for use in SDM systems. Various crosstalk suppressed MCF designs [1–4] have been reported that make it possible to transmit multiple signals simultaneously through each core without inter-core crosstalk, and a transmission exceeding 1 P bit/s was achieved by using a low crosstalk 12-core fiber . An important parameter for MCFs is space utilization efficiency (SUE). One example is core multiplicity, namely the core number per unit area. An effective way of improving SUE and increasing the capacity per fiber is to reduce the core pitch of the MCF. However, the crosstalk and core pitch have a tradeoff relationship. Various core profiles such as trench- [2–4] or hole-assisted structure  have been studied with a view to reducing the core pitch. There is another way of improving SUE, which is to tolerate inter-core crosstalk and to design MCFs coupled with high spatial density [6, 7]. This means that we employ MIMO processing at the receivers to compensate for the crosstalk occurring in the fiber. This approach is equivalent to a MIMO transmission system using few-mode fibers (FMFs) when we treat the coupled multi-cores as one core in which multiple super-modes propagate. Therefore, the inter-core DMD becomes an important parameter as well as the intra-core DMD in FMF [8, 9] to reduce the MIMO processing complexity. It has been reported in  that the inter-core DMD increases as the inter-core mode coupling increases even though in homogeneous-core MCF, and we have proposed designing weakly-coupled MCF where the inter-core DMD variation owing to the mode coupling is negligibly small. It has also been reported that the inter-core DMD can be affected by fiber bending .
In this study, we investigated the inter-core DMD characteristic of a weakly-coupled homogeneous multi-core taking bending/twisting or manufacturing error of the fiber into account, and evaluated its applicability to a transmission system employing inter-core crosstalk compensation with MIMO processing. We first evaluated the inter-core DMD characteristic of fabricated 4-core fiber and obtained a maximum 125-ps/km delay at 1550 nm between modes propagated through different cores even though our fiber was fabricated as MCF with homogeneous cores. We next confirmed experimentally that the bend induced inter-core DMD was lower than the simulated results, which we had expected owing to the twist of the fiber. We also revealed numerically that variations in the refractive index profile, and in particular that of the core, greatly increased the inter-core DMD. Finally, we conducted a 4 × 4 MIMO transmission experiment using weakly-coupled 4-core fiber and successfully compensated for the inter-core crosstalk.
2. MIMO transmission with coupled multi-core fiber for dense space-division multiplexing
Figure 1 shows the normalized channel multiplicity as a function of the cladding diameter of recently reported MCFs. The normalized channel multiplicity Mc is the spatial mode number per unit area compared with that of single-mode fiber. The early studies of MCF were based on non-coupled single-mode cores [1–5], and MCFs were realized with an Mc value of more than 6. To increase the per fiber capacity further, non-coupled few-mode-core based MCFs [11–13] were then investigated and an Mc value exceeding 10 was obtained. On the other hand, it becomes difficult to increase the core number simply by extending the cladding region because the cladding diameter is limited to less than about 220 μm in terms of the mechanical reliability of the fiber. Thus, moderately-coupled few-mode-core-based MCFs have been recently proposed  to greatly improve the space utilization efficiency.
Figure 2(a) shows a conceptual diagram of a transmission system using coupled MCF. For simplicity, this figure illustrates a case where a coupled 2-core fiber is used. As mentioned above, there is considerable crosstalk between the signals transmitted through different cores in coupled MCFs. Therefore, a MIMO equalizer must be deployed on the receiver side to compensate for the inter-core crosstalk as shown in the figure. The MIMO equalizer consists of finite impulse response (FIR) filters and the recovered signals are obtained by utilizing the received signals from Rx1 and Rx2 (the MIMO equalizer shown in the figure has a 2 × 2 configuration). Figure 2(b) shows a schematic diagram of conventional FIR filters. Each tap in the FIR filter consists of a multiply unit, an adder unit and a delay unit with a delay of a symbol or half-symbol interval τ. The required tap number depends on the group delay difference between the signals, and each tap coefficient is appropriately determined for the equalization by the adaptive algorithm. However, the computation becomes more complex as the total tap number increases. Thus, the inter-core DMD of coupled MCF is an important parameter for a system offering inter-core crosstalk compensation with MIMO processing. As mentioned in the introduction, we focus on the DMD characteristic in weakly-coupled MCFs because strong inter-core mode coupling results in an increase of inter-core DMD.
3. DMD characteristic of homogeneous 4-core fiber
We utilized a homogeneous single-mode core based 4-core fiber to characterize the inter-core DMD properties for simplicity. Figure 3 shows a cross-sectional image and the refractive index profile of our fabricated fiber. It has 4 cores with a trench-assisted profile and the cladding diameter is 125 μm. This fiber was designed and fabricated as a homogeneous 4-core fiber as shown in Table 1. The fiber is 10 km long and the core pitch is 30 μm, which is smaller than that of reported non-coupled MCFs (typically 40 μm ~50 μm) to realize weakly-coupled MCF.
Figure 4 shows the measured crosstalk as a function of wavelength from core 1 to the other cores when the fiber was wound onto the bobbin with an 80 mm radius. Here, we defined the crosstalk as the output power ratio between the cores. Each crosstalk value was obtained by spectrally averaging data with 10 pm resolution over a 1 nm window. The crosstalk value at 1550 nm between adjacent cores was about −5 dB/10 km and that between opposite cores was about 3 dB lower than those between adjacent cores, which were agreed with our calculation results within a few decibels. The crosstalk value increases as the wavelength increases and converges on 0 dB in the longer wavelength region. These characteristics are the same for the crosstalk between other core combinations. Therefore, our MCF causes serious signal degradation when there is no crosstalk compensation, thus requiring MIMO processing at the receivers. On the other hand, we have also numerically confirmed that the increase of inter-core DMD owing to mode coupling in our MCF is negligible when we assume that each core structure is completely same and take account of super-mode aspect.
We next measured the impulse responses in the 4-core fiber at 1550 nm as shown in Fig. 5. We created an optical impulse signal with a 100 ps duration using a laser source and an intensity modulator, and measured the impulse response with an oscilloscope. Figure 5(a) shows the impulse responses when we input the impulse signals from each core and received them individually at the corresponding through cores. We found that there was a maximum 1.25 ns delay (125 ps/km) even though our MCF was designed and fabricated as homogeneous 4-core fiber. Although we numerically calculated the impact of the marker (Δ was −0.3%) on the DMD characteristic, we found no remarkable DMD change owing the marker. We also measured the impulse response when we input the impulse signal from only core 4 and received the signals at all the cores as shown in Fig. 5(b). We observed that the crosstalk pulses broadened over the 1.25 ns width at cores 1 ~3, which indicates that MIMO equalizer requires FIR filters with tap number to cover these broadened crosstalk pulses if we are to compensate for the crosstalk correctly. However, the tap number is closely related to the MIMO processing complexity and should be reduced as with the intra-core DMD in FMF.
We then discussed the factors affecting the inter-core DMD in multi-core fibers and investigated their impact on the DMD change. We consider that fiber bending and core profile variation owing to manufacturing error as possible reasons for the DMD change. First, we evaluated both numerically and experimentally the effect of bending on the inter-core DMD characteristic. The bend-induced effective index change in weakly-coupled homogeneous-core multi-core fiber is described as Figure 6(a) shows the maximum inter-core DMD property as a function of bending radius. Here, we analyzed the same 4-core structure shown in Table 1 with 2-dimensional full vector finite element method for our calculation and showed maximum DMD value between the modes. The solid line is a calculated result and the dots show measured results for a bending radius of 80, 110 or 140 mm. We found that the measured inter-core DMD was greatly inferior to the simulated result. Figure 6(b) shows only the measured results for the inter-core DMD between cores 1~3 and core 4. Although a maximum 43-ps/km DMD variation was observed between three measurements with different bending radii, the group delay in each core showed almost no bending radius dependence. We consider that the discrepancy between the calculated and measured results arose from the twist of the fiber. The group delay of each core varied periodically as a function of bending direction θ as shown in Fig. 7. This figure shows calculated group delays normalized by that without applying a bend when the bending radius and direction were 80 mm and 0 ~180 degrees, respectively. The structural parameters are the same as in Table 1. Although the twisting was not purposely applied to the fiber during the fabrication and spooling of the fiber, the fibers are spontaneously twisted longitudinally. Thus, the bend-induced group delay variation is expected to be mitigated even if the fiber is wound onto a bobbin and bent throughout its length. A slight change in the inter-core DMD with a different bending radius is considered to be due to a change of non-uniform twisting angle distribution along the fiber. Thus, we expected a bend-induced inter-core DMD change to have less impact for a relatively long-distance transmission system with randomly twisted fiber.
Since a delay exceeding a hundred ps/km was still observed in our measurements, we investigated the influence of core profile variation on the inter-core DMD. Figure 8 shows the calculated inter-core DMD and DMD slope changes at 1550 nm as a function of the relative variation in a1 or Δ1. In these calculations, the base profile is assumed to be the same, as shown in Table 1, and only a1 or the refractive index of the core was changed. The inter-core DMD obtained with our fiber could be generated by a relative variation of 2% in a1 or 0.5% in Δ1, which corresponds to an a1 variation of about 10−1 μm or a Δ1 variation of about 10-3% in this case. These tolerances seem to be more severe than those of the intra-core DMD in FMF . In particular, the variation in the refractive index of the core causes a great increase in inter-core DMD, which has little impact on intra-core DMD in FMFs. We also found that the inter-core DMD slope was more affected by a1 variation than Δ1 variation.
We then confirmed the wavelength dependence of the inter-core DMD. Figure 9 shows the measured group delay properties normalized by the group delay at 1460 nm where the bending radius is 110 mm. The calculated value is also shown in the figure when we assumed that the observed inter-core DMD (150 ps/km) resulted solely from variation in a1 or Δ1, which was 7.5 × 10−2 μm or 2.6 × 10-3%. The measured results fall within the two calculated results, which means that fabricated MCF has fabrication errors related to both the core radius and refractive index. Although, the fabrication accuracy of the core radius and refractive index are both important if we are to obtain a low inter-core DMD value over a broad wavelength range, a delay of a few hundred ps/km could be generated if the typical fabrication error of the core profile is taken into account. In the next section, we confirmed experimentally the applicability of MIMO processing to a weakly-coupled MCF with such a few hundred ps/km delay.
4. 4 × 4 MIMO transmission experiment
Finally, we conducted a MIMO transmission experiment with a 10 km-long weakly-coupled 4-core fiber to confirm the feasibility of inter-core crosstalk compensation. Figure 10 shows our experimental setup for a 40 Gbps QPSK 4 × 4 MIMO transmission. We used a tunable laser source operating at 1530 ~1565 nm and generated single polarization 40 Gbps QPSK signals with 220−1 PRBS bit patterns. The signals were then split into four through a power splitter and each signal was appropriately delayed (0, 2, 5, 10 m) for signal decorrelation. Each signal was launched into each core using a fan-in device and was transmitted through a 10 km-long 4-core fiber. After transmission through the fiber, each signal was again split into four by a fan-out device and was detected at the coherent receiver. Although we adjusted polarization states of the transmitted four signals with the polarization controllers to maximize the detected power, a few decibel loss could be observed in the receivers because signals with only one polarization was detected in our experiment. The 4 × 105 symbols stored by the oscilloscope were then recovered using a 4 × 4 MIMO equalizer. Our equalizer consists of 4 half-symbol spaced FIR filters with a tap number of 140 and we determined each tap coefficient with a recursive least squares algorithm . We used 4000 training symbols to converge the tap coefficient and switched the equalization mode from the training sequence to the decision directed mode  to compensate for the carrier phase noise.
Figure 11 shows the Q factors of back-to-back and transmitted signals. The signal wavelength was 1530, 1540, 1550 or 1565 nm and the total received power was adjusted to −9 dBm with variable optical attenuators. Although there was an average power penalty of 2.3 dB, we successfully obtained Q factors exceeding 11.3 dB after a 10-km transmission. The power penalty increased as the wavelength increased. We assumed that the degradation was because there were insufficient taps for the equalization. In theory, we need a tap number corresponding to the maximum DMD / symbol period (or half symbol period) × N (N is an integer). The N value depends on the amount of crosstalk and a larger N may be needed to compensate for the large crosstalk. In our experiment, we set N at 1 taking into account the calculation time so the Q factors at longer wavelengths tend to be lower than at shorter wavelengths because the amount of crosstalk increases as the wavelength increases (As is also described in , the power penalty increases due to the increase of the crosstalk even if the tap number is sufficient to cover the impulse response spread). Figure 12 shows the bit error rate (BER) properties of the recovered signals when the MIMO configuration was 1 × 1, 3 × 3 or 4 × 4. Constellation maps obtained with each configuration are also shown in this figure. We observed a high BER value with the 1 × 1 configuration because a 1 × 1 MIMO equalizer cannot compensate for the inter-core crosstalk. On the other hand, the inter-core crosstalk for all possible core combinations could be compensated for with a 4 × 4 configuration and lower BER characteristics were successfully obtained with a 4 × 4 MIMO equalizer. In addition, we found that a 3 × 3 configuration was even effective when we only compensated for the crosstalk between adjacent cores. This configuration has superior BER characteristics but less MIMO processing complexity than with the 4 × 4 configuration. Thus, such a configuration may be applicable if the power penalty induced by the crosstalk level between opposite cores is acceptable. Although further study will be needed to clarify the relationship between the amount of crosstalk and the maximum DMD needed to reduce the power penalty with MIMO processing, we successfully confirmed the feasibility of using a MIMO transmission system over weakly-coupled MCF with inter-core crosstalk compensation. It should be noted that an 8 × 8 configuration will be required if we are to utilize all the spatial and polarization modes in 4-core fiber.
We investigated the inter-core DMD characteristic of weakly-coupled homogeneous multi-core fiber and evaluated its applicability to a transmission system with MIMO processing. We experimentally evaluated the inter-core DMD characteristic of the fabricated 4-core fiber and confirmed that there was a delay exceeding a hundred picoseconds per kilometer between the modes that propagated through the different cores even though the fiber was designed with homogeneous cores. We next confirmed experimentally that bend-induced inter-core DMD variation was mitigated by the twist of the fiber. We also revealed numerically that refractive index profile variation, and in particular that of the core, caused a great increase in the inter-core DMD. Finally, we conducted a 4 × 4 MIMO transmission experiment using weakly-coupled 4-core fiber and successfully compensated for the inter-core crosstalk.
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