## Abstract

High spatial multiplicity fiber designs are presented for homogeneous and heterogeneous 4LP-mode multicore fibers (MCFs) that support six spatial modes per core. The high-spatial-density 4LP-mode MCF design methodology is explained in detail. The influence of the number of cores on the cladding diameter (*D*_{cl}) and relative core multiplicity factor (RCMF) is investigated. The optimal core designs and MCF layouts with square and triangular lattices maintain glass fiber reliability (maximum *D*_{cl} = 250 μm). For homogeneous 4LP-mode MCFs, a 19-core triangular-lattice fiber gives the highest RCMF of 61.7. For heterogeneous 4LP-mode MCFs, an RCMF of 65.4 is obtained for a 21-core square-lattice fiber.

© 2017 Optical Society of America

## 1. Introduction

Space-division multiplexing (SDM) has attracted considerable attention as a method for overcoming the capacity limits of conventional single-mode single-core fibers as well as for enhancing optical fiber transmission systems [1–6]. Moreover, SDM based on multicore fibers (MCFs), few-mode fibers (FMFs), and few-mode MCFs (FM-MCFs) has been proposed to increase spatial multiplicity, which is SDM channel density per fiber, and transmission capacity [7–28].

The main issue in weakly coupled FM-MCFs is the suppression of inter-core crosstalk (XT) while achieving high spatial multiplicity, as well as the reduction of the cable cutoff wavelength (*λ*_{cc}) to less than 1530 nm [29]. In addition, the differential mode delay (DMD) should be reduced in order to relax the complexity of multiple-input and multiple-output (MIMO) processing at the receiver [30]. So far, various weakly coupled homogeneous and heterogeneous FM-MCFs have been fabricated with the aim of achieving more than 100 spatial channels [17, 18, 25]. The number of spatial channels can be increased by enlarging the cladding diameter (*D*_{cl}). However, to satisfy mechanical reliability, *D*_{cl} should be less than the limiting value of around 250 μm [25, 31]. Thus, the number of cores for FM-MCFs is limited by *D*_{cl} which is in turn determined by the core pitch (Λ) and outer cladding thickness (*t*). Therefore, it is important to focus not only on the spatial channel count but also on the spatial multiplicity of the FM-MCFs. Recently, we have proposed a 4LP-mode 19-core fiber with a triangular lattice layout, which has the highest reported relative core multiplicity factor (RCMF) of more than 60 and 100 spatial channels with *D*_{cl} of less than 250 μm [25].

In this work, we describe the detailed design and characteristics of the highest spatial multiplicity homogeneous and heterogeneous 4LP-mode MCFs with low DMD, which were not reported in our earlier experimental work [31]. Here, 4LP-mode includes the following six spatial modes: LP_{01}, LP_{11a}, LP_{11b}, LP_{21a}, LP_{21b}, and LP_{02} modes. Since each mode is degenerate in terms of the polarization, we only consider one-polarization for each mode for calculating the fiber characteristics. The full-vector finite-element method [32] was used for the calculation. The design starts by determining isolated core parameters (delta, radius, etc.) in which the effective area (*A*_{eff}) and DMD of the fiber are the target characteristics. Next, the core pitch is decided by considering XT and cable cutoff wavelength trade-off for a square or triangular lattice layout, since the square or triangular lattice is the best option for dense core arrangement. Finally, the outer cladding thickness is calculated to satisfy low bending loss (BL) requirements. Based on this design procedure, a comprehensive analysis of the 4LP-mode MCF is presented. In particular, the influence of the number of cores on *D*_{cl} and RCMF is thoroughly investigated, and the optimal core designs and MCF layouts for both square and triangular lattices are presented that maintain glass fiber reliability (maximum *D*_{cl} = 250 μm). For homogeneous 4LP-mode MCF, a 19-core fiber with a triangular lattice layout yields the highest RCMF of 61.7, which was experimentally realized in our previous work [25]. For heterogeneous 4LP-mode MCF, we newly propose a 21-core fiber with a square lattice layout that yields an RCMF of 65.4.

## 2. Homogeneous 4LP-mode MCFs

The left panel of Fig. 1 shows the refractive index profile of the trench-assisted graded index core, which is used for each core in the 4LP-mode MCF. The center and right panels of Fig. 1 show cross-sections of 12-core fibers with square and triangular lattice layouts. Here, *r*_{1}, *r*_{2}, *W*, *Δ*_{1}, *Δ*_{t} ( = −0.7%), *t*, and Λ stand for the core radius, the distance between the core center and the inner edge of trench, the thickness of the trench layer, the peak relative refractive index difference between the core and cladding, the relative refractive index difference between the trench and cladding, outer cladding thickness, and core pitch, respectively. The *Δ*_{t} value of −0.7% might be the limit for the regular vapor axial deposition (VAD) and outside vapor deposition (OVD) processes [33]. Here, *Δ*_{t} was set to −0.7% for fabricated MCFs in our previous works [6, 15, 34]. *Δ* stands for the relative refractive index differences between the core and cladding and is defined as *Δ* = *Δ*_{1} [1 − (*r*/*r*_{1})* ^{α}*], where

*α*and

*r*represent shape factor and the radial coordinate, respectively. In this work,

*α*is set to 2.0 because low DMD can be achieved with a parabolic shape [35]. In Fig. 1, the cross-sections of a 12-core fiber are shown with square and triangular lattice layouts. We investigated 4LP-mode MCFs with square and triangular lattice layouts with the number of cores ranging from 12 to 27.

#### 2.1 Core for homogeneous 4LP-mode MCFs

Figure 2 shows various fiber characteristics as a function of *r*_{1} and *Δ*_{1} at the wavelength of 1550 nm, where *Δ*_{t} is −0.7% and *α* is 2.0. The target characteristics of the 4LP-mode MCF are given as follows: the maximum *D*_{cl} is 250 μm [25], the effective area of LP_{01} mode (*A*_{effLP01}) is 80 μm^{2}, the maximum DMD (max DMD) is less than 100 ps/km, and the total XT is lower than −30 dB/100 km at bending radius *R* = 140 mm and wavelength of 1565 nm. The max DMD indicates the group velocity difference between the fastest and slowest propagation modes. In addition, total XT indicates the XT between LP_{02} modes, which has the worst XT of all modes, from four neighboring cores in the square lattice or six neighboring cores in the triangular lattice layout. Note that the mode coupling coefficient between LP_{02} modes in neighboring cores is the largest among 4LP-modes, resulting in the largest XT of LP_{02} mode. In Fig. 2, the black solid line shows *r*_{1} and *Δ*_{1} at which *A*_{effLP01} = 80 μm^{2}, while the dashed lines show *r*_{1} and *Δ*_{1} at which max DMD is 100 ps/km for *r*_{2}/*r*_{1} = 1.14, 1.15, and 1.16 for *W*/*r*_{1} = 0.6. Note that max DMD characteristics are insensitive to *W*/*r*_{1} if *W*/*r*_{1} is larger than 0.5. For each dashed line, the region indicated by the arrow has a max DMD that is less than 100 ps/km. The red, blue, and green shaded areas show the region where the max DMD is less than 100 ps/km for *r*_{2}/*r*_{1} = 1.14, 1.15, and 1.16, respectively. The lowest max DMD is obtained in the middle of each shaded area. Finally, the *r*_{1} and *Δ*_{1} values should be chosen from the black solid line considering the max DMD. In addition, we must consider cutoff wavelength requirement (BL of LP_{31} mode must be greater than 1 dB/m at 1530 nm for *R* = 140 mm) and low BL requirement (BL of LP_{02} mode must be less than 0.5 dB/100 turns at 1565 nm for *R* = 30 mm) according to International Telecommunication Union Telecommunication (ITU-T) recommendations G.654 [29].

When we select *r*_{1} and *Δ*_{1} from the green area, both BL and XT will be large since *Δ*_{1} from the green area is relatively small. On the other hand, *Δ*_{1} from red area is so high that the cutoff wavelength of LP_{31} mode, *λ*_{cc}, will be longer. Therefore, it is best to select *r*_{1} and *Δ*_{1} from the blue area. By considering these requirements, (*r*_{1}, *Δ*_{1}) = (9.6 μm, 0.82%) with *r*_{2}/*r*_{1} = 1.15, shown by red pentagram in Fig. 2, is selected for this study, at which point the max DMD is less than 18 ps/km. For these *r*_{1} and *Δ*_{1} values, all required target characteristics are satisfied if the proper *W*/*r*_{1} is selected.

Table 1 shows the parameters and *A*_{eff} of each mode at 1550 nm for homogeneous 4LP-mode MCFs, except for *W*/*r*_{1}. For *W*/*r*_{1}, if *W*/*r*_{1} is small while the cutoff wavelength requirement is relaxed, XT will be increased. For large *W*/*r*_{1}, we find the opposite tradeoff. The value of *W*/*r*_{1} should be set carefully by considering the tradeoff, as shown in the next section. For heterogeneous 4LP-mode MCFs, as shown in section 3.1, the black and white pentagrams in Fig. 2 are selected as the two types of non-identical cores with similar *A*_{eff} with *r*_{2}/*r*_{1} = 1.15. Figure 3 illustrates the max DMD as a function of wavelength for the homogeneous 4LP-mode MCFs. Moreover, Fig. 3 indicates that the max DMD over the C band is less than 20 ps/km.

#### 2.2 XT and λ_{cc} characteristics

Here, we investigate the total XT and *λ*_{cc} characteristics of homogeneous 4LP-mode 12-core fibers with the square and triangular lattice layouts. Figure 4 illustrates XT and *λ*_{cc} as a function of core pitch, which are used to determine the optimal *W*/*r*_{1} value. The solid and dashed lines represent XT and *λ*_{cc}, respectively. The dashed-dot lines show the target XT and *λ*_{cc} values. To ensure 4LP-mode operation over the C band, target total XT is less than −30 dB/100 km, and *λ*_{cc} should not exceed 1530 nm. Red, blue, and green lines correspond to cases when *W*/*r*_{1} ≅ *W*/*r*_{1opt}, *W*/*r*_{1} > *W*/*r*_{1opt}, and *W*/*r*_{1} < *W*/*r*_{1opt}, respectively. Here, *W*/*r*_{1opt} stands for the optimal *W*/*r*_{1} that can minimize Λ for FM-MCFs, and Λ_{XT} and Λ_{λ}_{cc} represent required Λ to satisfy XT and *λ*_{cc} requirements, respectively. In Fig. 4, there is a large gap between Λ_{XT} and Λ_{λ}_{cc} when *W*/*r*_{1} is larger or smaller than *W*/*r*_{1opt}. Hence, we must determine *W*/*r*_{1opt}, where Λ_{XT} ≅ Λ_{λ}_{cc}, by varying *W*/*r*_{1} considering the XT and *λ*_{cc} requirements.

Figure 5 shows the core pitch as a function of *W*/*r*_{1} for 12-core fibers with square and triangular lattice layouts, where the core parameters are given in Table 1. Red solid and blue dashed lines represent Λ_{XT} and Λ_{λ}_{cc}, respectively. Figure 5 indicates that *W*/*r*_{1} of around 0.65 can minimize the core pitch Λ for 12-core fibers with square and triangular lattice layouts, where Λ = 40.7 and 41.5 μm, respectively. Figure 6 shows the relationship between the bending loss *α*_{b} of LP_{02} mode and the outer cladding thickness *t* in the outmost core at the wavelength of 1565 nm and *R* = 140 mm. Figure 6 indicates that *t* should be larger than 37.3 μm for the 12-core fibers in order to suppress *α*_{b} to less than 10^{−3} dB/km [36].

From the definition of core multiplicity factor (CMF) [36], the CMF for homogeneous and heterogeneous FM-MCFs is described as follows [12, 37]:

*N*

_{c}is number of cores,

*l*is number of spatial modes (

*l*= 6 for 4LP-mode core), and

*A*

_{eff-}

_{p}_{-}

*and*

_{m}*A*

_{eff-}

_{q}_{-}

*correspond to effective area of*

_{m}*m*-th spatial mode in core

*p*and core

*q*, respectively. Moreover, RCMF is a ratio between CMF of FM-MCF and that of a conventional single-mode fiber, where

*A*

_{eff}and

*D*

_{cl}are 80 μm

^{2}and 125 μm, respectively, which is given by

When Λ is set at 40.7 and 41.5 μm and *t* is set at 37.3 μm for 12-core fibers with square and triangular lattice layouts, *D*_{cl} is 203.3 and 201.4 μm, respectively. In this case, RCMF is 56 for the 12-core square-lattice fiber and 57.1 for the 12-core triangular-lattice fiber. The same design procedures were performed to find high-spatial-density 4LP-mode MCF structures with core numbers ranging from 12 to 27.

#### 2.3 RCMF of homogeneous 4LP-mode MCFs

In this section, RCMF of 4LP-mode MCFs with square and triangular lattice layouts with various numbers of cores are investigated. For each 4LP-mode MCF, the parameters of isolated cores are given in section 2.1. Core pitch and *W*/*r*_{1} are determined as shown in section 2.2. Figure 7 shows *W*/*r*_{1} and Λ as a function of the number of cores for homogeneous 4LP-mode MCFs. Solid and dashed lines represent *W*/*r*_{1} and Λ, while red and blue lines correspond to the results obtained for square and triangular lattice layouts, respectively. The minimum value of Λ that satisfies both the XT and *λ*_{cc} requirements is selected for each fiber. Here, *W*/*r*_{1} decreases as the number of cores increases to relax the confinement of an inner core. Furthermore, the minimum Λ tends to increase as the number of cores increases to compensate for XT degradation.

Figure 8 demonstrates RCMF as a function of *D*_{cl} for homogeneous 4LP-mode MCFs. The red and blue lines show the results obtained for the square and triangular lattice layouts, respectively. The number of spatial channels is noted in brackets. The relationship for *D*_{cl} of square lattice layout (*D*_{cl_s}) and triangular lattice layout (*D*_{cl_t}), Λ, *t*, and *N*_{c} in the MCFs is described as follows:

In the triangular lattice layout, the peak value of RCMF seems to be achievable at 27-core fibers; however, *D*_{cl} is 286 μm, which exceeds the upper limit of *D*_{cl} for mechanical reliability [30]. In the square lattice layout, the peak value of RCMF is at 21-core fiber. If the limit values of *D*_{cl} are set to 250 μm, the highest RCMF of 61.7 can be achieved in the 19-core triangular-lattice fiber. Therefore, the homogeneous 4LP-mode 19-core triangular-lattice fiber is the best possible candidate for achieving an RCMF of more than 60 and 100 spatial channels with a reliable *D*_{cl}. We should note that the fabrication error of the core parameters (*r*_{1}, *Δ*_{1}, etc.) would be in the order of several %. Even in this case, the designed MCF satisfies both the XT and *λ*_{cc} requirements while the max DMD varies within a few hundred picoseconds per kilometer [31].

## 3. Heterogeneous 4LP-mode MCFs

So far, we have numerically investigated the influence of an increased number of cores on the *D*_{cl} and RCMF of homogeneous 4LP-mode MCFs with square and triangular lattice layouts. To further increase the spatial multiplicity, employing a heterogeneous arrangement is a promising approach [38]. When designing heterogeneous MCFs, the effective index differences between each non-identical core (*Δn*_{eff}) should be larger than 0.5 × 10^{−3} to suppress the threshold bending radius (*R*_{pk}) [39] to less than 100 mm [22]. Due to the difficulty of selecting more than three types of non-identical cores with *Δn*_{eff} larger than 0.5 × 10^{−3} while achieving low DMD, we consider heterogeneous MCFs with two types of non-identical cores in this study. Moreover, to realize the heterogeneous arrangement, a minimum of two and three non-identical cores are required for the square and triangular lattice layouts, respectively. Therefore, we only demonstrate the detailed design method and characteristics for heterogeneous 4LP-mode MCFs with the square lattice layout.

#### 3.1 Core design for heterogeneous 4LP-mode MCFs with square lattice layout

Using the same technique as that used for designing the homogeneous core, we select two non-identical cores, where (*r*_{1}, *Δ*_{1}) = (9.50 μm, 0.80%) is Core 1 and (9.73 μm, 0.84%) is Core 2 from the blue area in Fig. 2 in section 2.1. In these non-identical cores, *Δn*_{eff} of our target modes is approximately 0.6 × 10^{−3} for the entire C band. Table 2 shows the parameters, except for *W*/*r*_{1}, and *A*_{eff} of each mode at the wavelength of 1550 nm for the heterogeneous 4LP-mode MCFs. Here, a heterogeneous 21-core fiber with square lattice layout is considered as an example of a heterogeneous MCF.

For *W*/*r*_{1}, we investigated the dependence of several pairs of *W*/*r*_{1} of Core 1 (*W*/*r*_{1 Core 1}) and *W*/*r*_{1} of Core 2 (*W*/*r*_{1 Core 2}) on RCMF, as shown in Table 3. Here, *W*/*r*_{1 Core 2} must be lower than that of the homogeneous 21-core fiber to relax the tight confinement of the inner core. Table 3 shows the highest RCMF of 65.4 for the heterogeneous 21-core fiber can be obtained when *W*/*r*_{1 Core 1} = 0.65 and *W*/*r*_{1 Core 2} = 0.40.

Figure 9 shows the total XT and *λ*_{cc} as a function of the core pitch for the heterogeneous 21-core fiber with *W*/*r*_{1 Core 1} = 0.65 and *W*/*r*_{1 Core 2} = 0.40. The solid and dashed lines represent XT and *λ*_{cc}, respectively. Red and blue dashed lines correspond to *λ*_{cc} of Core 1 and Core 2, respectively. Here, the XT and *λ*_{cc} requirements are the same as those for homogeneous 4LP-mode MCFs. The inset in Fig. 9 is a schematic of the heterogeneous 21-core fiber with square lattice layout. The *λ*_{cc} values for both Core 1 and Core 2 were calculated in the center core, where the confinement is the highest in the fiber, and we used Core 1 as the center core as an example. Figure 9 indicates that a minimum Λ of 38.3 μm is required for the 21-core fiber with square lattice, noting that minimum Λ is restricted by Core 1 when Core 1 is used as the center core. In this case, *λ*_{cc} for Core 2, which is adjacent to the center core, is shorter than 1500 nm and satisfies the *λ*_{cc} requirement.

Figure 10 shows the relationship between BL *α*_{b} of LP_{02} mode and the outer cladding thickness in the outmost core at the wavelength of 1565 nm and *R* = 140 mm, where *W*/*r*_{1} = 0.65 and 0.40 for Core 1 and Core 2, respectively. The red solid and blue dashed lines correspond to the BL of the fiber of Core 1 and Core 2, respectively. Figure 10 indicates that the minimum *t* of 39.2 μm is required in order to suppress *α*_{b} to less than 10^{−3} dB/km [36] for the 21-core square-lattice fiber. Note that the outmost core of the fiber is Core 2 in this case. When Λ is set at 38.3 μm and *t* is set at 39.2 μm for the 21-core square-lattice fiber, *D*_{cl} is 249.1 μm, resulting in RCMF of 65.4.

Figure 11 shows the max DMD as a function of wavelength for heterogeneous 4LP-mode MCFs, where *W*/*r*_{1} = 0.65 and 0.40 for Core 1 and Core 2, respectively. Red solid and blue dashed lines correspond to DMD of Core 1 and Core 2, respectively. Figure 11 indicates that the max DMD over the C band is less than 70 ps/km in both cores.

#### 3.2 RCMF of heterogeneous 4LP-mode MCFs with square lattice layout

Table 4 shows the optimized structural parameters for heterogeneous 4LP-mode MCFs. The parameters of Core 1 and Core 2 are given in Table 2 for each 4LP-mode MCF. The value for *W*/*r*_{1 Core 1}, *W*/*r*_{1 Core 2}, core pitch, and the outer cladding thickness are determined as shown in section 3.1. Figure 12 demonstrates the RCMF as a function of *D*_{cl} for heterogeneous 4LP-mode MCFs. The number of spatial channels is noted in brackets. The peak value of RCMF is achieved with the 21-core fiber. If the limit value of *D*_{cl} is set to 250 μm, the highest RCMF of 65.4 can be achieved in 21-core fibers. Therefore, the 4LP-mode 21-core fiber with a square lattice layout is a possible heterogeneous fiber candidate for achieving an RCMF of more than 65 with 100 spatial channels and a reliable *D*_{cl}.

## 4. Conclusion

The optimized fiber designs of high spatial multiplicity homogeneous and heterogeneous 4LP-mode MCFs were presented and the design methodology of the high-spatial-density 4LP-mode MCF has been explained in detail. Comprehensive analysis on the influence of the number of cores on *D*_{cl} and RCMF was performed to reveal the optimal layout for obtaining the highest spatial density while maintaining the mechanical reliability. For the homogeneous 4LP-mode MCF, a 19-core fiber with a triangular layout produces the highest RCMF of 61.7, which was experimentally realized in [25]. For the heterogeneous 4LP-mode MCF, we proposed a 21-core fiber with a square lattice layout, which achieved an RCMF of 65.4. The MCF design strategy presented here can be applied to MCFs with any number of modes. However, for MCFs with larger number of mode (for example, 6LP-mode), some requirements may need to be relaxed.

## Funding

Part of this work was supported by the National Institute of Information and Communication Technology (NICT) Japan, under “Research and Development of Innovative Optical Fiber and Communication Technology.”

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