## Abstract

This paper investigates the design of C-band double-cladding 2LP mode multicore fiber (2LP-MCF) with 125-µm cladding diameter by numerical simulation, in which a two-ring layout is used to improve crosstalk and critical bend radius. The numerical results show that the crosstalk can be less than ${-}{60}$ and ${-}{35}\;{\rm dB/km}$ for 2LP mode eight-core and 10-core fibers, respectively, and it is also shown that the crosstalk can be suppressed compared with that in heterogeneous 2LP-MCFs without a double-cladding layer.

© 2023 Optica Publishing Group

## 1. INTRODUCTION

Space division multiplexing (SDM) transmission technology has been gaining attention recently due to its potential to improve optical communication transmission capacity [1]. Multicore fiber (MCF) is one of the most active technologies in SDM fiber transmission. In terms of practical applications, MCFs with a cladding diameter (CD) of 125 µm have been developed [2,3].

The number of transmission lines can be increased without increasing the number of fibers when using MCFs that have multiple cores in one optical fiber. MCFs can be classified into two types. Coupled MCFs involve multiple cores transmitting signals that interfere with each other. Alternatively, each core can propagate the signal independently in an uncoupled MCF. Transmission systems based on the coupled MCF require multiple-input–multiple-output (MIMO) digital signal processing. Since the use of the MIMO receiver increases cost, the uncoupled MCF that does not require MIMO processing would be beneficial.

Uncoupled MCFs used for long-haul transmission require a reduction of intercore crosstalk (XT) because interference between the signals propagating through the cores reduces the optical signal to noise ratio (OSNR). Heterogeneous MCF (Hetero-MCF) is a fiber structure that consists of multiple fiber cores with different diameters and refractive indices. In conventional homogeneous MCF, the optical signals interfere with each other between different cores, resulting in signal distortion and reduced OSNR. In Hetero-MCF, the interference between them is greatly reduced because the core sizes and refractive indices in Hetero-MCF are designed to be different. In addition, the distance between cores in the Hetero-MCF can be adjusted to further reduce intercore interference. Hetero-MCFs have optimal distances between the core and cladding depending on the parameters of each core. Conventional Hetero-MCF does not emphasize this optimal distance, so a two-ring core layout Hetero-MCF was proposed, achieving XT of ${-}{47}$ and ${-}{23}\;{\rm dB/km}$ for single-mode cores with six and eight cores, respectively [4]. We obtained XT below ${-}{51}$ and ${-}{35}\;{\rm dB/km}$ for 2LP mode six-core and eight-core fibers, respectively, with the same two-ring core layout [5].

To further reduce the intercore XT, we used a double-cladding structure [6] and are ready to present a preliminary study at an international conference [7]. Considering our previously proposed two-ring core layout Hetero-MCF [5], the major distinction between our fiber structure and the one proposed in [6] is that our two cores are located at different distances from the outer layer of the fiber. So, the biggest problem in using outer cladding to further improve XT is the determination of the refractive index and width of the outer cladding to meet requirements such as the cutoff limit for both cores. In this paper, we will explain the detailed design procedure of the proposed fiber. Furthermore, compared with the six-core two-ring Hetero-MCF with double cladding proposed at the international conference [7], we increase the number of cores to 10 and simulate its intercore XT characteristics. The XT obtained for six- and eight-core fibers is superior compared to the previous two-ring Hetero-MCF with single cladding. For the guided-mode analysis of a fiber, the finite element method (FEM) by the commercial software program (COMSOL) is used.

## 2. CHARACTERISTICS OF STEP-INDEX DOUBLE-CLADDING 2LP-MODE HETERO-MCF

#### A. Core Layouts for Double-Cladding Hetero-MCF

Figures 1(a) and 1(b) illustrate the cross sections of six- and eight-core fibers that were utilized in our research on 2LP-mode Hetero-MCFs, along with the corresponding refractive index profile of the core. The cores were classified into two types: core H, with a higher core refractive index, and core L, with a lower core refractive index. Our previously proposed two-ring core layout with single cladding [5] is shown in Fig. 1(a), where core H and core L are situated on the larger and smaller circles shown by the dashed lines, respectively. Generally, larger core radii tend to have lower confinement loss to the outer coating material, which makes it possible to position core H outside of core L to enhance both the XT and confinement loss. The distance between the center of the core and the outer layer of the fiber, ${T_{\rm C}}$, plays a crucial role in achieving this objective. Specifically, ${T_{\rm C \text{-} H}}$ for core H, which has a higher effective index (${n_{\rm{eff}}}$), should be smaller than ${T_{\rm C \text{-} L}}$ for core L, which has a lower ${n_{\rm{eff}}}$, to obtain a two-ring core layout. When ${T_{\rm C \text{-} H}} = {T_{\rm C \text{-} L}}$, this arrangement is equivalent to a conventional one-ring core layout Hetero-MCF. Hereafter, ${T_{\rm C}}$ is used for determining the parameter of the specific core, and ${T_{\rm C \text{-} H}}$ and ${T_{\rm C \text{-} L}}$ are used for determining the core parameters of heterogeneous cores. To define the two-ring layout, we introduce the core pitch for core L as ${\Lambda _2}$, while core pitch $\Lambda$ indicates the distance between neighboring cores H and L. Given that the center of both circles coincides with that of the 125-µm cladding, the distance between the center of the core and outer layer of cladding ${T_{\rm C \text{-} H}}$ and ${T_{\rm C \text{-} L}}$ for all cores can be expressed as

where ${r_{{\rm coreH,}\,{\rm coreL}}}$ are 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 $N\; \ge \;{4}$), and the center of 125-µm diameter cladding is set as the origin of the polar coordinate system, the coordinate of the $i$th ($i = {1},\; \ldots ,\;N$) core position is expressed as (${r_{\rm coreH,\, coreL}}$, ${\theta _i}$), where ${\theta _i} = ({2}\pi /N)i$, and the ${r_{\rm coreH,\, coreL}}$ is given asFigure 1(b) shows the double-cladding structure we use in this paper. For this structure, the outer cladding (shown by the orange region) with thickness of ${t_{\rm{oc}}}$ is added to the outside of the inner cladding. When the outer cladding thickness ${t_{\rm{oc}}} = {0}$, the structure is the same as Fig. 1(a). Also, the diameter of the whole fiber is kept to 125 µm even when the outer cladding is used.

In this study, we assume a step-index profile of single cladding, as depicted in Fig. 1(a), where ${n_{\rm{core}}}$ (core H and core L) and ${n_{\rm{clad}}}$ represent the refractive indices of the core and inner cladding, respectively. The refractive index profile of the double-cladding structure is assumed to be as shown in Fig. 1(b), where ${n_{\rm{core}}}$ and ${n_{\rm{clad}}}$ are defined in the same way as in Fig. 1(a), and ${n_{\rm{oc}}}$ denotes the refractive index in the outer cladding. The value of ${n_{\rm{clad}}}$ is set to 1.45 to represent the refractive index of pure silica, while the refractive index outside of the cladding is set to 1.486, which corresponds to the coating material [8].

#### B. Definition of XT for Two Non-identical Cores

In this paper, our focus is on step-index Hetero-MCFs, depicted in Figs. 1(a) and 1(b), which feature two types of step-index cores aimed at reducing XT [9]. As illustrated in Fig. 2, it is well established that the XT of Hetero-MCF decreases significantly when the bending radius (${R_{\rm b}}$) exceeds a critical value (${R_{\rm{pk}}}$), and the XT converges to a certain value for sufficiently large ${R_{\rm b}}$ [9,10].

The critical bending radius for the XT between $p$th and $q$th modes can be expressed as [11]

where $\Lambda$ is core pitch, $\beta$ is the mean of propagation constants of $p$th and $q$th modes, and $\Delta {\beta _{{pq}}}$ is the difference of $\beta$ between $p$th and $q$th propagation modes. 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 by [13] where $L$ is the transmission length, and the calculation procedure for averaged power-coupling coefficient ${h_{\!{pq}}}$ is also described in detail in [13]. As shown in Fig. 2, which illustrates XT behavior, the correlation length $d$ becomes a dominant parameter for ${R_{\rm b}} \gt {R_{\rm{pk}}}$ compared to ${R_{\rm b}}$. However, it is challenging to determine the value of $d$, as it is dependent on the manufacturing precision, and different settings of $d$ may be necessary to match the calculated and experimental results. For instance, the results in [14] suggest that the estimated correlation length exceeds 1 m. Although we assumed a correlation length of $d = {1}\;{\rm m}$, consistent with a previous study on single-cladding fibers [5], $d$ varies depending on the actual manufacturing process, particularly in the $d$-dominant region. Here, the increase (decrease) of $d$ corresponds to a decrease (increase) of XT, as XT is proportional to $1/d$. To minimize the XT, using a $d$-dominant region (${R_{\rm b}} \lt {R_{\rm{pk}}}$) is preferable to an ${R_{\rm b}}$-dominant region (${R_{\rm b}} \lt {R_{\rm{pk}}}$). Increasing the core pitch $\Lambda$ shifts ${R_{\rm{pk}}}$ to a larger ${R_{\rm b}}$, although the XT defined by Eq. (5) will be small due to a decrease in ${h_{\!{pq}}}$, which entails a trade-off relationship. Moreover, MCFs with a limited CD need to consider the trade-off between $\Lambda$ and outer cladding thickness, which will be discussed in the following section. Note that since XT linearly increases, as shown in Eq. (5), we express the ${{\rm XT}_{{pq}}}$ [dB] for 1 km as ${{\rm XT}_{{pq}}}$ [dB/km] in this paper, i.e., ${{\rm XT}_{{pq}}}\;[{\rm dB}/{\rm km}] = {{\rm XT}_{{pq}}}\;[{\rm dB}]{|_{L\, = \,1 \;{\rm km}}}$.#### C. Parameter for Two-Ring Core Layout Hetero-MCF

According to our previously proposed single-cladding 2LP-mode Hetero-MCFs with two-ring core layouts in [5], the bending loss (BL) distribution at different core parameters is shown as colored lines in Fig. 3, where $a$ refers to the radius of the core, and $\Delta \;(= (n_{\rm{core}}^2 - n_{\rm{clad}}^2)/{2}n_{\rm{core}}^2)$ is the relative refractive index difference between the core and the cladding next to the core. To ensure 2LP mode operation, the ${{\rm LP}_{21}}$ mode must be cut off with a BL greater than 1 dB/m at $\lambda = {1530}\;{\rm nm}$ and ${R_{\rm b}} = {140}\;{\rm mm}$, represented by the colored dashed lines in Fig. 3. The excess loss (EL) of ${{\rm LP}_{11}}$ mode should be less than 0.01 dB/km at $\lambda = {1565}\;{\rm nm}$ and ${R_{\rm b}} = {140}\;{\rm mm}$, represented by the colored solid lines in Fig. 3. The black solid lines denote the effective areas (${A_{\rm{eff}}}$), where ${A_{\rm{eff}}} = {80}\;\unicode{x00B5}{\rm m}^2$ of ${{\rm LP}_{01}}$ mode at $\lambda = {1550}\;{\rm nm}$. To ensure homogeneous transmission properties, non-identical cores with equalized ${A_{\rm{eff}}}$ are preferred. Core parameters on the black solid line (${A_{\rm{eff}}} = {80}\;\unicode{x00B5}{\rm m}^2$) and surrounded by solid and dashed lines of the same color should be selected for the fiber design, referred to as the effective core region (ECR). For example, the orange area between points $M$ and $N$ in Fig. 3 represents the ECR at ${T_{\rm C}} = {33}\;\unicode{x00B5}{\rm m}$ for C-band use. In Fig. 3, ${A_{\rm{eff}}}$, EL, and cutoff limit coincide at the red point ($A$) when the distance between the center of the core and the outer layer of fiber ${T_{\rm C}}$ is 29 µm. The objective for designing Hetero-MCFs is to find non-identical core parameters with a large enough $\Delta {\beta _{{pq}}}$. Larger $\Delta {\beta _{{pq}}}$ leads to smaller ${R_{\rm{pk}}}$, indicating that XT will decrease more rapidly for larger ${R_{\rm b}}$. ECR disappears when ${T_{\rm C}}$ is less than 29 µm, and thus ${T_{\rm C}}$ has to be larger than 29 µm. As illustrated in [5], the core H of the two-ring layout Hetero-MCFs is fixed at ${T_{\rm C \text{-} H}} = {29}\;\unicode{x00B5}{\rm m}$ at point $A$ in Fig. 3. ${T_{\rm C \text{-} L}}$ is then increased from 29 to 35 µm to select non-identical core L for two-ring layout Hetero-MCF design, such as point $M$.

#### D. Parameter for Outer Cladding

Compared with traditional single-cladding MCFs, a double-cladding fiber has an additional outer cladding layer, defined by the width ${t_{\rm{oc}}}$ and relative refractive index difference ${\Delta _{\rm{oc}}}$ ($= (n_{\rm{oc}}^2 - n_{\rm{clad}}^2)/{2}n_{\rm{oc}}^2$) between the inner cladding and outer cladding. It was shown that ${T_{\rm C \text{-} H}} = {29}\;\unicode{x00B5}{\rm m}$ is the minimum value of MCF at ${t_{\rm{oc}}} = {0}\;\unicode{x00B5}{\rm m}$ [5], and we found in our previous study that the minimum value of ${T_{\rm C \text{-} H}}$ will be further reduced by appropriately increasing the outer cladding layer [7]. A smaller ${T_{\rm C \text{-} H}}$ will increase the size of the ECR causing a larger difference in the effective refractive indices of adjacent cores, leading to smaller XT. To evaluate the size of ECR, we define the effective refractive index (${n_{\rm{eff}}}$) difference of ${{\rm LP}_{11}}$ mode between the upper cutoff limit (${n_{\rm{eff}}}$ at the cutoff) and lower EL limit (${n_{\rm{eff}}}$ at the EL) of ECR as $\Delta {n_{\rm{ECR}}}$. If $\Delta {n_{\rm{ECR}}} \lt {0}$, the design region determined by the cutoff and EL in Fig. 3, such as the region between points $M$ and $N$, disappears. Therefore $\Delta {n_{\rm{ECR}}} \gt {0}$ is necessary for fiber design. Figure 4(a) shows the ${n_{\rm{eff}}}$ value for the upper and lower ECR limits of ${{\rm LP}_{11}}$ mode for single-cladding and double-cladding (${\Delta _{\rm{oc}}} = {0.34}\%$, ${t_{\rm{oc}}} = {14}\;\unicode{x00B5}{\rm m}$) MCFs, and Fig. 4(b) shows the relationship between $\Delta {n_{\rm{ECR}}}$ and ${T_{\rm C}}$. It can be found that compared with the single-cladding MCF of the black line, fluctuations of $\Delta {n_{\rm{ECR}}}$ can be seen in the double-cladding MCF in Fig. 4(b).

To explain why the fluctuation occurred, we have selected some sets of core parameters ($a$, $\Delta$) of double-cladding MCF by maintaining ${A_{\rm{eff}}} = {80}\;\unicode{x00B5}{\rm m}^2$. Then, we calculated the BL of ${{\rm LP}_{21}}$ mode separately at ${T_{\rm C}} = {29}$ to 31 µm using the core parameters obtained from these points, and the BL as a function of core radius $a$ is shown in Fig. 5. The star symbols represent the cutoff limit points of ECR at ${T_{\rm C}} = {29}$ to 31 µm with different colors, and the horizontal black dashed lines represent the cutoff limit of the ECR conditions. From Fig. 5, the following features can be extracted. First, for smaller ${T_{\rm C}}$, the BL becomes large due to the short distance to the outer cladding. Second, for smaller values of $a$ (at the same time, $\Delta$ is reduced to maintain ${A_{\rm{eff}}}$), the BL also becomes large due to weaker confinement. Third, there seems to be a plateau in the middle range of $a$. The last point can be explained by the coupling between core and cladding modes. Coupling between the two modes can be clearly observed by the electric field distributions, shown in Fig. 5 such as points A and B. Coupling causes BL modification and makes a plateau as the region between points C and D. In Fig. 5, for ${T_{\rm C}} = {30}\;\unicode{x00B5}{\rm m}$, the cutoff limit is at the red star symbol; however, for ${T_{\rm C}} = {31}\;\unicode{x00B5}{\rm m}$, due to the reduction of BL, the cutoff limit moves to the blue star symbol. Therefore, the cutoff condition of the ${{\rm LP}_{21}}$ mode (${\rm BL} \gt {1}\;{\rm dB/m}$) is changed significantly for the core parameters at ${T_{\rm C}} = {31}\;\unicode{x00B5}{\rm m}$, like the distance between the red and blue star symbols in Fig. 5. It leads to a significant change in ${n_{\rm{eff}}}$ of the cutoff limit core like the blue line in Fig. 4(a) and a decrease in $\Delta {n_{\rm{ECR}}}$, shown in Fig. 4(b) as well.

For our previously proposed two-ring core layout Hetero-MCF, we need to find a set of core parameters ($a$, $\Delta$) that can satisfy both the ${{\rm LP}_{21}}$ mode cutoff condition and the ${{\rm LP}_{11}}$ mode transmission condition. That means $\Delta {n_{\rm{ECR}}} = {0}$. To determine the proper fiber parameters, Fig. 6 shows the distribution of ${t_{\rm{oc}}}$ versus ${\Delta _{\rm{oc}}}$ for $\Delta {n_{\rm{ECR}}} = {0}$, where ${T_{\rm C}}$ decreases gradually from 28 to 27 µm. We used ${t_{\rm{oc}}} = {T_{\rm C}} - {3}a$ as in [6], and the range of ${T_{\rm C}}$ in a single-cladding Hetero-MCF is 29–36 µm, and the fiber radius $a$ is 5.2–5.7 µm according to [5]. The core radius $a$ increases and ${T_{\rm C}}$ decreases with the use of outer cladding. The minimum value of ${T_{\rm C}}$ is close to 27 µm, and the maximum value of $a$ is close to 6 µm. So, according to the previously mentioned relationship of ${t_{\rm{oc}}}$, ${T_{\rm C}}$ and $a$ (${t_{\rm{oc}}} = {T_{\rm C}} - {3}a$), we set the range of ${t_{\rm{oc}}}$ calculation to 8–18 µm. However, during the calculation, we observed that when ${T_{\rm C}} \lt {27.5}\;\unicode{x00B5}{\rm m}$ and ${t_{\rm{oc}}}$ values are 8 and 9 µm, there is no region for $\Delta {n_{\rm{ECR}}} \gt {0}$. So, in Fig. 6, we present the results of the calculations within the range of ${t_{\rm{oc}}}$ values from 10 to 18 µm. The upper region of the colored solid line represents $\Delta {n_{\rm{ECR}}} \gt {0}$, and the lower is the opposite. As can be seen from Fig. 6, ${T_{\rm C}}$ can also continue to decrease with a further increase in ${\Delta _{\rm{oc}}}$; also the core parameters ($a$, $\Delta$) are increasing. Considering the loss and fabrication issues of MCFs, the core parameters should be limited, and we limit the maximum value of core $\Delta$ to 1% in this paper. Although we assumed a core relative refractive index difference of $\Delta \lt {1}\%$, there will be other possibilities for fiber structures for further study. The colored dashed lines in Fig. 6 represent the $\Delta$ limit for each ${T_{\rm C}}$ value with the same color. The red points indicate $\Delta = {1}\%$ and $\Delta {n_{\rm{ECR}}} = {0}$, while the region below the dashed line represents the core $\Delta \lt {1}\%$. When ${T_{\rm C}} \gt {27.5}\;\unicode{x00B5}{\rm m}$, all ${\Delta _{\rm{oc}}}$ and ${t_{\rm{oc}}}$ values (range shown in Fig. 6) satisfy the condition $\Delta \lt {1}\%$. To minimize XT, we need to determine the smallest possible ${T_{\rm C}}$ value. Therefore, we extracted points $A$, $B$, $C$, and $D$ in Fig. 6, and present their parameters in Table 1. Hereafter, we consider these four structures. The distributions of $\Delta {n_{\rm{ECR}}}$ as a function of ${T_{\rm C}}$ for these structures are shown in Fig. 7, and we can find similar fluctuations as in Fig. 4.

As mentioned before, $\Delta {n_{\rm{ECR}}} \lt {0}$ means that there is not a possible candidate of the core parameter ($a$ and $\Delta$) satisfying two BL conditions, whereas for structure D (${\Delta _{\rm{oc}}} = {0.34}\%$ and ${t_{\rm{oc}}} = {14}\;\unicode{x00B5}{\rm m}$), $\Delta {n_{\rm{ECR}}}$ becomes positive for all ${T_{\rm C}}\; \ge \;{27.3}\;\unicode{x00B5}{\rm m}$ as shown in Fig. 7 as the blue line, which means that ${T_{\rm C}}$ can be reduced. As the previously mentioned XT equation shows, we should choose fiber parameters with larger differences to obtain lower XT. Therefore, we choose structure D and calculate XT between adjacent cores in the next section for all ${T_{\rm C \text{-} L}}$ values.

## 3. XT OF HETERO-MCF WITH DOUBLE CLADDING

#### A. XT of Two-Ring Hetero-MCF with Double Cladding

Starting with the two-ring layout 2LP-mode MCF with double cladding shown in Fig. 1(b), we need to determine both ${T_{\rm C \text{-} H}}$ and ${T_{\rm C \text{-} L}}$. As in Fig. 3, we generate a condensed BL distribution that highlights only the upper and lower limits of the ECR with ${\Delta _{\rm{oc}}} = {0.34}\%$ and ${t_{\rm{oc}}} = {14}\;\unicode{x00B5}{\rm m}$ in Fig. 8 and suggest that the possible upper limit of ECR (parameter of core H) can be obtained around the cross point for ${T_{\rm C}} = {27.3}\;\unicode{x00B5}{\rm m}$. To achieve a larger $\Delta {\beta _{{pq}}}$ and lower averaged power-coupling coefficient ${h_{\!{pq}}}$, we need to fix ${T_{\rm C \text{-} H}}$ and vary the parameter of core L by increasing ${T_{\rm C \text{-} L}}$ from the value of ${T_{\rm C \text{-} H}}$ to 35 µm. Fabrication errors can occur during the fiber production and manufacturing process in core radius $a$ and core $\Delta$, so we also evaluate a parameter of core radius $a$ about 0.01 µm inside the design area due to manufacturing errors.

Figure 9(a) shows the comparison of XT for ${{\rm LP}_{11}}$ mode (${{\rm XT}_{11 \text{-} 11}}$) between Hetero-MCFs with single-cladding [5] and double-cladding layouts, where $N = {6}$, $\lambda = {1550}\;{\rm nm}$, $d = {1}\;{\rm m}$, and ${T_{\rm C \text{-} H}} = {27.3}\;\unicode{x00B5}{\rm m}$. “H-L” denotes XT between adjacent cores of core H and core L, and “L-L” denotes XT between nearest cores L. Furthermore, we also investigate the case of $N = {8}\;{\rm in}$ Fig. 9(b). Error bars indicate XT errors due to the core radius variations about 0.01 µm inside the design area as mentioned before. As we can see, for all ${T_{\rm C \text{-} L}}$, XT of the double-cladding six- and eight-core fiber is improved compared with that of the single-cladding layout. Also, we can observe that the XT distribution among six-, eight- and 10-core Hetero-MCFs does not distribute monotonically. When ${T_{\rm C \text{-} L}}$ equals 27.3 µm for each fiber, because the ${R_{\rm{pk}}}$ of Hetero-MCFs is larger than 140 mm, XT is a relatively large value of the ${R_{\rm b}}$-dominant region, which is shown in Fig. 2. When ${T_{\rm C \text{-} L}}$ reaches 28 µm, the ${R_{\rm{pk}}}$ of Hetero-MCFs is smaller than 140 mm, so XT deceases into a low value of the $d$-dominant region as in Fig. 2. Due to the effect of intercore distance, XT increases with ${T_{\rm C \text{-} L}}$ for six- and eight-core fiber. However, the effect of intercore distance on XT is smaller than the difference between adjacent core parameters ($a$, $\Delta$) of 10-core fiber, so the XT of 10-core fiber will continue to decrease. For the six-core double-cladding layout MCF, the minimum ${{\rm XT}_{11 \text{-} 11}}$ of ${-}{97.9}\;{\rm dB/km}$ for adjacent cores can be obtained with structure D (${T_{\rm C \text{-} H}} = {27.3}\;\unicode{x00B5}{\rm m}$, ${\Delta _{\rm{oc}}} = {0.34}\%$, and ${t_{\rm{oc}}} = {14}\;\unicode{x00B5}{\rm m}$) with ${T_{\rm C \text{-} L}} = {29}\;\unicode{x00B5}{\rm m}$, indicating an improvement of 46.7 dB/km compared with the single-cladding layout. The minimum ${{\rm XT}_{11 \text{-} 11}}$ for the eight-core double-cladding layout MCF is ${-}{60.4}\;{\rm dB/km}$ at ${T_{\rm C \text{-} L}} = {30}\;\unicode{x00B5}{\rm m}$ for structure D (${T_{\rm C \text{-} H}} = {27.3}\;\unicode{x00B5}{\rm m}$, ${\Delta _{\rm{oc}}} = {0.34}\%$, and ${t_{\rm{oc}}} = {14}\;\unicode{x00B5}{\rm m}$), indicating an improvement of 24.5 dB/km compared with the single-cladding layout. Compared with our previous study of single-cladding six- and eight-core Hetero-MCFs in [5], the XT of the eight-core MCF was optimized by adding an outer cladding, so we increased the number of cores and wanted to confirm whether the 10-core fiber could also achieve an accepted XT value with the same fiber parameters. The results of the XT calculations for each ${T_{\rm C \text{-} L}}$ are shown in Fig. 9(c) for structure D (${T_{\rm C \text{-} H}} = {27.3}\;\unicode{x00B5}{\rm m}$, ${\Delta _{\rm{oc}}} = {0.34}\%$, and ${t_{\rm{oc}}} = {14}\;\unicode{x00B5}{\rm m}$). The lowest XT of 10-core MCF reaches ${-}{35.5}\;{\rm dB/km}$. This also means that with the current proposed double-cladding two-ring core layout fiber structure, the eight-core fiber is the one with the highest number of cores while maintaining XT values under ${-}{50}\;{\rm dB/km}$.

If we consider the quadrature phase shift keying (QPSK), 16 quadrature amplitude modulation (QAM), and 32QAM format signals, the allowable XT (${{\rm XT}_{\rm{al}}}$) is less than ${-}{16}$, ${-}\;{24}$, and ${-}{32}\;{\rm dB}$, respectively [15], and hence, the maximum transmission length ${L_{{\max}}}$ is estimated by

If the XT value is sufficiently low, the transmission length ${L_{{\max}}}$ may be limited by the differential mode delay (DMD), but considering only the effects of XT on ${L_{{\max}}}$, Fig. 10 shows the comparison of ${L_{{\max}}}$ between Hetero-MCFs with the different format signals, corresponding to ${{\rm XT}_{11 \text{-} 11}}$ for adjacent cores of structure D (H-L) in Figs. 9(b) and 9(c), where $N = {8}$ and 10. The eight-core fiber’s ${L_{{\max}}}$ for 64QAM reaches about 700 km, and the ${L_{{\max}}}$ of 10-core fiber for QPSK reaches about 90 km. The core parameters of six-, eight-, and 10-core fiber used for structure D of the lowest XT value are summarized in Tables 2 and 3.

In this study, we considered the correlation length $d$ of 1 m for each MCF’s calculations. Figure 11 presents the results obtained from our calculations where we observed variations in XT as a function of ${R_{\rm b}}$ for different correlation length values of eight-core fiber, where $\lambda = {1550}\;{\rm nm}$, and $d$ is assumed as 0.1, 1, and 10 m, respectively. We can observe that as $d$ increases or decreases from the reference value of 1 m, the XT levels experience noticeable changes and remain insensitive to ${R_{\rm b}}$ when ${R_{\rm b}} \gt {R_{\rm{pk}}}$. We also evaluate the average XT distribution for C-band use (1530–1565 nm) of the ${{\rm LP}_{11}}$ mode of six-, eight- and 10-core two-ring core layout Hetero-MCFs with double cladding in Fig. 12. The worst-case XT values of six- and eight-core fiber are also very low, indicating good performance in C-band use. The core parameters are the same as in Table 3.

Research on uncoupled 2LP mode MCFs with double cladding at the 125-µm standard diameter is still in its infancy. However, this study provides valuable insights into the performance of such fibers, and a comparison is made between the numerical simulation results presented in this paper and the simulation results of previous research, which are summarized in Table 4. The spatial channel count (SCC) is a key parameter that determines the number of channels that can be transmitted simultaneously. It is calculated as the product of the numbers of cores and modes. Figure 13 illustrates the relationship between SCC and XT for both single-cladding and double-cladding MCFs, where XT is for the highest-order modes in adjacent cores. Compared with the previous studies, the results of this research show significant improvement in XT with the same SCC, which is a promising development for the future of high-capacity optical communication systems.

## 4. SUMMARY

We investigated the 125-µm CD step-index 2LP-mode Hetero-MCF by utilizing the two-ring layout and double-cladding approach in the fiber’s design. The results showed that XT could be decreased by two-fold while maintaining the same level of SCC and can be effectively applied to C-band use. In our designed 2LP-mode, eight- and 10-core Hetero-MCFs with double cladding, even with the limited CD of 125 µm and simple step-index profile, were able to demonstrate XT suppression of less than ${-}{60}$ and ${-}{35}\;{\rm dB/km}$, respectively, at 1550 nm.

## Funding

China Scholarships Council (202108050017).

## Acknowledgment

Zheyu Zhao was supported by the China Scholarships Council.

## 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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