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Abstract
The numerical results of the step-index 2LP-mode 6-core and 8-core multi-core fibers (MCFs) with 125-µm cladding diameter and two-ring layout for C-band, show that the crosstalk less than −51 and −35 dB/km can be obtained respectively. It is revealed that the two-ring layout is adapted for improving the crosstalk and critical bending radius, also effective for few-mode MCFs as well as single-mode MCFs.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Space division multiplexing (SDM) is one of the essential technologies to handle data traffic growth. For scaling up the spatial channel count (SCC), the few-mode multi-core fiber (FM-MCF) is a promising approach because the lightwave can transmit in multiple cores with multiple modes within a single-fiber cladding. Although FM-MCFs with SCC exceeding 100 have been reported [1,2], their cladding diameters (CDs) are larger than 300 µm. It has been reported that the use of the standard 125-µm CD for single-mode MCFs has many advantages in the utilization and fabrication [3]. FM-MCFs are also preferable to use the 125-µm CD. There are, however, technical limitations in designing FM-MCF with 125-µm CD since a larger core pitch and cladding thickness are required to reduce the crosstalk (XT) and confinement losses of higher-order modes. Although a coupled-type MCF with a small core pitch, which allows the XTs and requires multiple-input multiple-output digital signal processing (MIMO-DSP), is also one of the promising ways, for the MIMO-free transmission system, uncoupled-type MCF still has been demanded.
So far, an uncoupled-type 2LP-mode four-core fiber with 125µm CD has been reported [4], in which air holes and trench structures are introduced and the XT of less than ‒30 dB/100 km was experimentally reported. However, it is difficult to expand the core number with enough low XT. To further reduce XT, the arrangement of heterogeneous cores is effective. In addition, using our previously proposed two-ring layout for step-index heterogeneous MCFs (hetero-MCFs) [5], the core number of single-mode MCF with 125-µm CD can increase to six and eight cores keeping low simulation XT results of −47.5 and −23.5 dB/km with correlation length d set to 1 m, respectively. If such a two-ring layout will be adapted to the 2LP-MCFs, the crosstalk should be successfully reduced. Furthermore, the hetero-MCF based on a two-ring layout does not require such an air hole or trench structure, which leads to ease of fabrication. In this study, the characteristics of step-index 2LP-MCF based on the two-ring layout with 125µm CD is numerically investigated, where the difference of the cases between single-mode and 2LP-mode hetero-MCF are investigated. Compared with single-mode hetero-MCFs, the SCC can be increased by 3-fold with small XT changes by utilizing 2LP modes.
This paper is organized as follows. In Section 2, the core layout of hetero-MCF is stated. In Section 3, the mathematical definition of XT is briefly described, and the step-index 2LP-MCF based on the one-ring and two-ring layouts with 125-µm CD are investigated. Finally, the obtained results are summarized in Section 4.
2. Core layouts for hetero-MCF
Figures 1(a) and (b) show schematics of the 6-, 8-core fiber cross sections of 2LP-mode hetero-MCFs, where two types of cores are called core H (with higher core refractive index) and core L (with lower core refractive index). As seen in Fig. 1(a), the conventional hetero-MCF can be categorized as a one-ring layout, where both cores H and L are placed on the same circle with the same core pitch Λ, but radius and refractive index are different for two types cores. Since the center of this circle coincides with that of the 125-µm cladding, the distance between the center of the core and the outer layer of cladding TC for all cores are the same as
where rcore is the distance between the center of the fiber and that of each core. When the number of cores is N (N is an even number and N ≥ 2), and the center of 125-µm diameter cladding is set as the origin of the polar coordinate system, the coordinate of the i-th (i = 1, …, N) core position is expressed as (rcore, θi), where θi = (2π/N)i and the rcore is given as
Fig. 1. Schematics of the fiber cross sections of 2LP-mode hetero-MCFs. (a) One-ring layout (b) Two-ring layout and (c) step-index refractive index distribution.
Figure 1(b) shows our previously proposed two-ring layout [5]. This two-ring layout hetero-MCF could be fabricated by conventional MCF fabrication methods such as drilling process [6]. In general, the larger core radius tends to have low confinement loss to outer coating material, which means that core H can be placed outside rather than core L. In other words, to improve both the XT and confinement loss, the distance between the center of the core and the outer layer of cladding for core H (TC‐H) with a higher effective index (neff) should be smaller than that for core L (TC‐L) with a lower neff. To define the two-ring layout, we introduce the core pitch for core L as Λ2, and the core pitch Λ indicates the distance between neighboring cores H and L. For the core number N (N is an even number and N ≥ 4), TC‐H and TC‐L can be expressed as
whereIn this study, the step-index profile is assumed as shown in Fig. 1(c), where ncore (core H and core L) and nclad are the refractive indices in core and cladding, respectively. The nclad is set to 1.45 for the refractive index of pure silica and the refractive index outside of the CD is set to 1.486 as a coating material [7].
3. Characteristics of step-index 2LP-mode hetero-MCF
3.1 Definition of XT for two non-identical cores
In this paper, we focus on the step-index hetero-MCFs as shown in Figs. 1(a) and (b), which have two types of step-index cores to reduce the XT [8]. It is known that, as illustrated by Fig. 2, the XT of hetero-MCF considerably decreases when the bending radius (Rb) becomes larger than a critical value (Rpk), and the XT converges on a certain value for enough larger Rb [8,9].
To reflect this effect in the calculation of XT, different equations have to be used depending on Rb and Rpk. The critical bending radius for the XT between p-th and q-th modes can be expressed as [10]
where Λ is core pitch, β is the mean of propagation constants of p-th and q-th modes, Δβpq is the difference of β between p-th and q-th propagation modes. If Rb > Rpk and the XT is very small, no phase-matching point arises [10,11]. XT in MCFs can be evaluated by the coupled mode theory with random phase function [12]. The random process can be characterized by an autocorrelation function and correlation length d. The autocorrelation function is usually modeled by the exponential autocorrelation function, and the averaged XT can be given by3.2 XT of hetero-MCF with the one-ring layout
We start with TC of 29 µm and increase it by 1 µm to select non-identical cores for hetero-MCFs design. For the guided-mode analysis of a fiber, the finite element method (FEM) by the commercial software program (COMSOL) is used. To guarantee 2LP mode operation, the LP21 mode is required to be cutoff. Thus, the bending loss (BL) of LP21 mode should be larger than 1 dB/m at λ = 1530 nm and Rb = 140 mm, which is represented by the colored dashed lines in Figs. 3(a) and (b), whereas the excess loss (EL) of LP11 mode is required to be less than 0.01 dB/km at λ = 1565 nm and Rb = 140 mm, which is represented by the colored solid lines in Figs. 3(a) and (b). The black dot and solid lines denote the contour lines for various neff and effective areas (Aeff), respectively, where Aeff = 80 µm2 of LP01 mode at λ = 1550 nm. Meanwhile, a refers to the radius of the core and Δ = (ncore2‒ nclad2)/2ncore2 is the relative refractive index difference between the core and the cladding. For the sake of homogeneity of transmission properties, the Aeff of non-identical cores are preferable to be equalized [9]. Therefore, core parameters that are on the black solid line (Aeff = 80 µm2) and also surrounded by colored dashed and solid lines of the same color should be selected for the fiber design, referred to as the effective core region (ECR) like the red area between points M and N in Fig. 3(a) for C-band use. From Fig. 3(a), Aeff, EL, and cutoff line intersect at the black point when TC = 29 µm, observing the Figs. 3(a) and (b), the ECR disappear when TC is less than 29 µm, and thus TC has to be larger than 29 µm. Point M should be below point N, but as will be explained in detail later, there is a suitable value for the distance between these two points. Due to the fact that errors can occur during the fabrication of the core radius and core Δ, we use one core whose core radius is 0.05 µm smaller than that of the upper limit core and core Δ is 0.025% lower than the upper limit core. Another one core whose core radius is 0.05 µm larger than that of the lower limit core and core Δ is 0.025% higher than that of the lower limit core to show the XT error. The Aeff of these cores are still close to 80 µm2 in Fig. 3(b) which represented by green circles with the same color as cores at the limits.
Fig. 3. The core parameters (core radius a and core Δ) and its relationship to cutoff region, EL limit, Aeff, and neff at λ = 1550 nm.
The main objective for designing hetero-MCFs is to find non-identical core parameters with a large enough Δβpq. According to Eq. (6), larger Δβpq leads to smaller Rpk, indicating that the XT will decrease more rapidly for larger Rb. Hence, we should select core parameters of core H and L from the upper and lower limit of ECR to minimize Rpk, respectively.
For example, the cases of TC = 32 and 34 µm are explained as follows. In Fig. 3(b), core parameters at the edges of ECR for TC = 32 µm correspond to the two blue dots (cores AH and AL), and for TC = 34 µm correspond to the two red dots (cores BH and BL). The subscriptions ‘H’ and ‘L’ correspond to the upper and lower limits of the ECR, respectively. To examine these two core parameters sets, we plot XTs for adjacent cores as a function of Rb in Figs. 4(a) and (b) by calculating Eqs. (7) and (8), where N = 6, λ = 1550 nm, and d = 1 m. XT11‐11 and XT01‐01 denote the XT between the LP11 and LP01 modes in two cores, respectively. XT11‐01 (XT01‐11) denotes the XT between the LP11 (LP01) mode in the lower limit core and the LP01 (LP11) mode in the higher limit core.
We can see that, as expected, XT11‐11 is the largest XT among all combinations of modes. When Rb > 100 mm, the XT for TC = 32 µm is better than that for TC = 34 µm. However, a relatively large Rpk can be seen for TC = 32 µm, this is because the Δneff between cores AH and AL is smaller than that between cores BH and BL. The dependency of XT11‐11 at Rb = 140 mm on TC is shown in Fig. 5(a), where N = 6, λ = 1550 nm, and d = 1 m, and error bars indicate the XT errors due to the core parameter variations. We can see that XT11‐11 is suppressed well when TC ≥ 31 µm because Rpk < 140 mm satisfies. The minimum XT11‐11 is −42.7 dB/km when TC = 31 µm. As TC further increases, the core pitch Λ decreases, and then the mode coupling strength Kpq in Eq. (8) increases, resulting in the degradation of XT.
Fig. 5. XT11‐11 and maximum transmission distance, where N = 6, λ = 1550 nm, d = 1 m, and Rb = 140 mm.
If we consider the QPSK format signals, the XT less than −16 dB is allowed [14], and hence, the maximum transmission length LQPSK is estimated by
The maximum transmission length for the 16-QAM (64-QAM), L16QAM (L64QAM) is also estimated by replacing −16 dB in Eq. (9) to −24 dB (−32 dB). Figure 5(b) shows LQPSK, L16QAM, and L64QAM corresponding to Fig. 5(a). When TC = 31 µm, LQPSK, L16QAM, and L64QAM are 680, 108, and 17 km, respectively. Noting that, XTs are calculated at λ = 1550 nm, and therefore, it is expected that the maximum distances would be practically much lower for longer wavelength in C-band.
3.3 XT of hetero-MCF with the two-ring layout
From now, we investigate the 2LP-mode MCF with the two-ring layout as shown in Fig. 1(b). Unlike the case of one-ring layout, both TC‐H and TC‐L must be determined. As seen from Figs. 3(a) and (b), the possible upper limit of ECR (the parameter of core H) can be obtained around the cross point of colored solid and dashed line for TC = 29 µm. To obtain larger Δβpq and lower Kpq, we should fix TC‐H = 29 µm and change the parameter of core L by increasing TC‐L from 29 µm. Figure 6(a) shows the comparison of XT11‐11 between the hetero-MCFs with the one-ring and two-ring layout, where N = 6, λ = 1550 nm, d = 1 m, and TC‐H = 29 µm. ‘H‐L’ denotes the XT between adjacent cores of core H and core L, and ‘L‐L’ denotes the XT between nearest cores L. For all TC‐L, XTs of the two-ring layout are improved compared with those of one-ring layout. Such a tendency is similar to single-mode hetero-MCFs [5]. For the two-ring layout, the minimum XT11‐11 of −51.2 dB/km for adjacent cores can be obtained, indicating the improvement of 8.5 dB/km compared with one-ring layout. The XT of identical cores are also sufficiently suppressed. Figure 6(b) shows the comparison of LQPSK between the hetero-MCFs with the one-ring and two-ring layout, corresponding to XT11‐11 for adjacent cores (H‐L) in Fig. 6(a). When TC = 31 µm, LQPSK for the one-ring and two-ring layout are 680 and 3330 km, respectively. By introducing the two-ring layout, about a 5-fold improvement is obtained. Interestingly, LQPSK = 3330 km for 2LP-mode hetero-MCF with two-ring layout is larger value compared with LQPSK for single-mode hetero-MCFs [5]. The reason is considered as the effective index difference for 2LP-mode hetero-MCF is much larger.
Fig. 6. Comparison of characteristics between the hetero-MCFs with the one-ring and two-ring layout, where N = 6, λ = 1550 nm, d = 1 m, and TC‐H = 29 µm. ‘H‐L’ denotes the XT between adjacent cores of core H and core L, and ‘L‐L’ denotes the XT between nearest cores L.
We also investigate the case of N = 8. Figure 7(a) shows XT11-11 as a function of TC-L, where the parameters are the same as Figs. 6(a) and (b) except for N = 8. When N = 6, the XT peak is seen at TC‐L = 30 µm, whereas this peak is shift to smaller TC‐L ≤ 29 µm for N = 8. This is because, by fixing TC‐H and increasing N, core pitches Λ and Λ2 decrease, and then Rpk also decreases. The minimum XT11-11 for one-ring and two-ring layouts are −18.2 dB/km at TC‐L = 31 µm and −35.5 dB/km at TC‐L = 33 µm, respectively, indicating the improvement of 17.3 dB/km. Figure 7(b) shows the comparison of LQPSK corresponding to Fig. 7(a). At TC‐L = 33 µm, LQPSK reaches about 10 km that is larger value rather than the single-mode hetero-MCF similar to N = 6. A list of core parameters for the hetero-MCF used in this paper are summarized in Tables 1 and 2.
Fig. 7. Comparison of characteristics between the hetero-MCFs with the one-ring and two-ring layout, where N = 8 and other parameters are the same as Fig. 6.
Table 1. Core Parameters for Designed Hetero-MCF
Table 2. Fiber Parameters of the Designed Hetero-SI-MCFS
Finally, although there have been many studies on uncoupled MCFs in recent years, the study of uncoupled 2LP mode MCFs at 125-µm standard diameter is really limited. A comparison has been made between the numerical simulation results of this paper and those experimental and simulation results of previous research. The following Table 3 contains various fiber data. And we briefly discuss the SSC, which is defined by the multiplication of the numbers of cores and modes. Figure 8 shows the relationship between SSC and XT in single-mode and 2LP-mode MCF, where XTs are for the highest-order modes in adjacent cores. Compared with single-mode hetero-MCFs, the SCC can be increased by 3-fold with small XT changes by utilizing 2LP modes.
Table 3. Comparison of Parameters between Single-Mode and 2LP-Mode Hetero-MCFs
4. Summary
The step-index 2LP-mode hetero-MCF with 125-µm CD is investigated by introducing the two-ring layout approach to the 2LP-mode hetero-MCF design. It is revealed that the SCC can be increased by 3-fold keeping the almost same level of XT. In the designed 2LP-mode 6-core Hetero-MCF, even in the limited CD of 125µm and the simple step-index profile, the XT suppression of less than −51 dB/km at 1550 nm can be demonstrated.
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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