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Abstract
We investigated the relationships between the capacity limit and various figures of merit (spatial channel density, aggregate effective area ratio to cladding area, and bandwidth density) of few-mode multi-core fibers (FM-MCFs) where the modes in each core are weakly coupled. The capacity limit was estimated based on the Gaussian noise model for nonlinear impairment of single-mode fibers by neglecting crosstalk and intermodal nonlinear effects for simplicity; therefore, the estimated capacity can be the upper bound of the weakly-coupled FM-MCF capacity. When we take account of the transmission wavelength band of the FM-MCFs and the wavelength band where efficient amplification is available, the bandwidth density had a good correlation to the estimated capacity, but the spatial channel density and the aggregate effective area ratio often overestimate the FM-MCF capacity. Thus, we propose the bandwidth density for the figure of merit of FM-MCFs rather than the spatial channel density or the aggregate effective area ratio. We also investigated the relationship between the bandwidth density and fiber design, and found that the supporting transmission bandwidth and the core count are the dominant factors for the bandwidth density, and the mode count per core has a small impact for the bandwidth density, when the core Δ is fixed.
© 2017 Optical Society of America
1. Introduction
Spatial division multiplexing (SDM) is a strong candidate technology to overcome the capacity limit of single-mode fiber transmission systems [1], and a factor of 100 of improvement of spatial channel count has been realized by few-mode multi-core fibers (FM-MCFs) [2–4]. However, these FM-MCFs have been evaluated by various figures of merit (FoMs), and it is difficult to understand which fiber can realize the maximum capacity per fiber cross-sectional area. In addition, the reported FM-MCFs with more than 100 modes [2–4] do not support more than 100-times larger bandwidth (BW) because of their BW-limited designs, as discussed later.
In this paper, we discuss the relationship between the maximum fiber capacity and various FoMs, and propose the bandwidth density as the FoM of FM-MCFs. After that, we investigate the relationship between the BW density and the various FM-MCF designs.
2. Figure of merit for FM-MCF
Various figures of merit have been proposed for the SDM fiber performance. In the case of uncoupled single-mode MCF, the FoM based on the OSNR without nonlinear/crosstalk (NL/XT) compensation can be obtained in very analytical forms [5,6], thanks to the Gaussian noise (GN) model for single-mode fiber (SMF) transmission systems [7]. On the other hand, FM-MCFs where the modes in each core are uncoupled or weakly-coupledhave no FoM that directly relate to the fiber capacity, because the NL effects in such fibers are still under study [8,9] and no simple closed-form expression with fiber parameters for the NL noise has been proposed. Therefore, some simple parameters have been proposed for the FoM for such fibers, e.g., the spatial channel density (DSC: total spatial channel/mode count divided by fiber cross-sectional area):
and aggregate effective area ratio (REA: the sum of the effective areas of all the spatial modes divided by fiber cross-sectional area):where NSDM is the number of the spatial channels of the fiber, ACS is the fiber cross-sectional area, and Aeff,agg is the sum of the effective area (Aeff) of the spatial modes of the fiber. In this paper, the parameters normalized by the standard SMF (SSMF) are expressed by bold typefaces. REA has been referred to as “relative core multiplicity factor (RCMF)” and used in many papers [3,4,10,11]. One might think REA can evaluate the spatial capacity of the fiber more quantitatively than DSC since the NL effect seems to be included in the Aeff terms, but it is incorrect. To understand this, we would like to discuss the relationship between the fiber capacity, NSDM and Aeff,agg.The Shannon limit of aggregate spectral efficiency (SEagg) of an SDM fiber can be expressed as
where SNRn is the signal-to-noise ratio (SNR) of the n-th spatial channel, and the logarithm term is multiplied by 2 for polarization multiplexing. For the simplicity of discussion, we may regard the SNR as the OSNR whose BW equals the signal BW by assuming a negligible transceiver noise and ideal filtering. Now, we assume negligible inter-modal NL effects as we have no analytical solutions for them, and assume the same optical characteristics for all the modes, also for simplicity. In actual weakly-coupled few-mode cores, inter-modal NL effects may be cannot be ignored [12], but the assumption may be used for estimating the upper limits of OSNR and SE. Then, by using the GN model [5,13], the achievable SEagg of a certain transmission system (system parameters such as span length and span count are fixed) is maximized at the optimum launch power aswhere β2 = −λ2D/(2πc) is the chromatic dispersion in [(unit time)2/(unit length)], Leff = [1−exp(−αLs)]/α is the effective length, α is the power loss coefficient, Lspan is the span length, γ = 2πn2/(λAeff) is the NL coefficient, n2 the NL refractive index, Nspan the number of spans, Csystem is the constant depending on the system configuration. Equation (4) can be further simplified asby assuming the parameters except NSDM and Aeff are constant and including them in a constant Cpar. From Eq. (5), we can see that 2-times Aeff improvement results in 22/3-times OSNR improvement, which corresponds to ~1.33-bit/Hz SE increase when SNR >> 1. In contrast, 2-times NSDM increase results in the 2-times SE increase. Both cases increase the Aeff,agg two times, but the SE increases are different. In addition, the Aeff,agg does not change but the SEtot changes when the NSDM doubles and Aeff halves. This is because SEtot is linearly proportional to NSDM but logarithmically proportional to Aeff. Thus, the NSDM straightforwardly correlates to the SEtot. On the other hand, the Aeff,agg can be a misleading FoM parameter because the SEtot is not proportional to Aeff,agg but (almost) proportional to the sum of the logarithms of Aeffs. Therefore, NSDM is better than Aeff,agg to estimate the FM-MCF capacity improvement.However, NSDM is not enough for the capacity estimation, because the transmission capacity is the product of SEagg and the signal BW. So, the aggregate BW (Bagg)—the product of the NSDM and the available transmission BW B in the efficient amplification band (C + L band: 1530–1625 nm, 11.46 THz)— is a more proper parameter to estimate the fiber capacity than NSDM. Then, we investigated the relationships between Bagg, NSDM, Aeff,agg, and the normalized fiber capacity Cfig of various FM-MCFs—which are newly designed ones shown in Section 3—, as shown in Fig. 1, where the capacity Cfib was calculated by using
by neglecting the effects of the XT and inter-modal NL—thus, Cfib can be regarded as the upper bound of the weakly-coupled FM-MCF capacity estimation. It should be noted that strong and random coupling may increase the NL tolerance of few-mode cores [9] if it is realized, but this paper focus on the weak coupling regime as the strong coupling seems very challenging— and by assuming α and β2 are the same between the FM-MCFs and the reference SMF. For Fig. 1, we assumed the Aeff as 80 µm2 for the reference SMF, and calculated Cfibs for when the SNR of the reference SMF () is 5 dB, 10 dB, 15 dB, or 20 dB. However, we cannot see a significant difference due to the SNRref difference. Though the Cfib values in Fig. 1 can be regarded as the upper bound of the capacity estimation, the parameters NSDM and Aeff,agg often greatly exceed Cfib. In contrast, Bagg agrees well with Cfib. We can also understand these from the correlation coefficients between Cfib and Bagg, NSDM, or Aeff,agg, which are shown in Fig. 2. So, Bagg is a good parameter for estimating the fiber capacity but NSDM or Aeff,agg may overestimate the fiber capacity.
Fig. 1 The relationships between the various FoMs (Bagg, NSDM, and Aeff,agg) vs. the normalized fiber capacity Cfib, where Cfib was calculated using Eq. (6) by assuming the Aeff of the reference SMF is 80 µm2 and the SNR (SNRref) in the reference SMF system is (a) 5 dB, (b) 10 dB, (c) 15 dB, or (d) 20 dB. No significant difference was observed due to the SNRref difference.
Fig. 2 The correlation coefficients between the normalized fiber capacity Cfib and the various FoMs (Bagg, NSDM, or Aeff,agg), whose dependence on the SNRref is little.
Since the previously-reported FM-MCFs with more than 100 spatial modes were designed for C-band (1530–1565 nm, ~4.38 THz) operation [2–4]—the XT and the coating-leakage loss were designed at 1565 nm, we should evaluate them by Bagg (not by NSDM or Aeff,agg). Figure 3 shows the comparison of the Baggs of various FM-MCFs. The 3-mode 36-core fiber [2] and the 6-mode 19-core fiber [3,4] for C-band have NSDM of more than 100 but Bagg of only ~40. The 3-mode 19-core fiber for C + L-band has NSDM of only 57 but its Bagg can be larger than Bagg of the C-band FM-MCFs supporting ~100 modes.
Fig. 3 The aggregate bandwidths of various FM-MCFs. Horizontal axis shows the wavelength band, mode count in a core, and core count in a fiber, from the bottom.
Therefore, we employed the BW density—i.e., the spatial density of aggregate BW— as the FoM of FM-MCFs, and investigated the relationship between the BW density and fiber design.
3. Bandwidth density of FM-MCFs
For the BW density investigation, we designed 3-mode (3M) MCFs and 6-mode (6M) MCFs with various core counts for C- or C + L-band operation. In the fiber designs, the trench-assisted graded-index cores with the refractive index contrast Δ of 1% and an index gradient parameter α of ~1.94 were employed. The trench Δ was −0.7%. The core–trench separation and trench width were optimized so as to suppress the differential mode delay. The refractive index profiles and optical properties of the designed cores are shown in Fig. 4 and Table 1, respectively. Since the core Δ is fixed, the Aeff depends on the mode count, and the designed 6M core has larger Aeffs than those of the designed 3M core.
Table 1. Designed optical properties of the few-mode cores.
We designed FM-MCFs with 7, 19, or 37 cores using the 3- or 6-mode core design. We employed the center core plus one/multiple ring layouts, as shown in Fig. 5. The first, second, and third rings, from inner to outer, have 6, 12, and 18 cores, respectively. The radius of each ring was designed so that the total crosstalk to a mode of a core from the other cores can be suppressed to be −45 dB/100 km at 1565 nm for C-band designs or at 1625 nm for C + L-band designs. The cladding diameter (OD) was designed so that the coating-leakage loss can be less than 0.001 dB/km at 1565 nm for C-band designs or at 1625 nm for C + L-band designs. The designed FM-MCF dimensions are shown in Table 2.
Table 2. The dimensions of the designed FM-MCFs.
Now, we can investigate the relationship between the normalized aggregate BW—the C + L-band SSMF is the reference— and the cladding diameter for the designed FM-MCFs, which are shown in Fig. 6 for the FM-MCFs with the OD of <300 µm. The core count and the supporting transmission band (C or C + L) have large impacts on the aggregate BW, but the mode count per core has less impact on the aggregate BW.
Fig. 6 The relationships between the normalized aggregate BW and the cladding diameter for the designed FM-MCFs.
The BW density is different from the aggregate BW. The core count difference has a quite small impact on the BW density without the coating, as shown in Fig. 7(a), which means the effect of the core–coating separation for the coating-leakage loss suppression is very small. If we take account of the cross-sectional area of the coating—we assumed the coating thickness of 60 µm which is equivalent to those of standard fibers—, the BW density becomes more dependent on the core count, as shown in Fig. 7(b), because the ratio of the coating area unutilized for core arrangement can be reduced by increasing the core count or the utilized cladding area. Thus, for improving the BW density in a real use case, we should set the cladding diameter as large as possible in terms of mechanical reliability, mass-productivity, and cost efficiency. In terms of the mechanical reliability, for example, Ref [3]. claimed that the cladding diameter of around 300 μm can still keep the mechanical reliability equivalent to that of the standard optical fiber by increasing the proof-test level from 1% strain to 2% strain and by relaxing the minimum bend radius from 30 mm to 40 mm, and Ref [4]. claimed that 250 µm is the upper limit of the cladding diameter for keeping the mechanical reliability comparable to the conventional standard optical fiber without changing the minimum bend radius when one assumes the upper limit of the proof level is 2% strain and the proof level for the standard fiber is 1% strain.
Fig. 7 The relationships between the normalized BW density and the cladding diameter for the designed FM-MCFs. The BW densities were calculated with (a) bare fibers, or (b) coated fibers with a 60-µm coating thickness.
4. Conclusions
We clarified that the spatial channel density and the aggregate Aeff ratio are not suitable for the FoM of weakly-coupled FM-MCFs for evaluating the achievable spatial capacity (total capacity per fiber cross-sectional area), and proposed to use the BW density for a simple FoM of weakly-coupled FM-MCFs. In addition, we investigated the relationship between the BW density and fiber design, and found that the supporting transmission BW and the core count are the dominant factors for the BW density, and the mode count per core has a small impact for the BW density, when the core Δ is fixed.
To estimate the FM-MCF performance or system transmission capacity more precisely, we have to take account of the NL effects—especially, the inter-modal NL interference within each core could affect the transmission capacity and is expected to be expressed in analytical forms. However, at present, the BW density may be a good FoM for the FM-MCFs.
Funding
National Institute of Information and Communications Technology (NICT), Japan (1700103).
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