We propose and design a kind of heterogeneous rod-assisted and trench-assisted multi-core fiber (Hetero-RA-TA-MCF) with 32 cores arranged in square-lattice structure (SLS), and then we introduce the design method for Hetero-RA-TA-MCF. Simulation results show that the Hetero-RA-TA-32-Core-Fiber achieves average effective area (Aeff) of about 74 μm2, low crosstalk (XT) of about −31 dB/100km, threshold value of bending radius (Rpk) of 7.0 cm, relative core multiplicity factor (RCMF) of 8.74, and cable cutoff wavelength (λcc) of less than 1.53 μm.
© 2017 Optical Society of America
Nowadays, by employing time-, wavelength-, polarization-division multiplexing, as well as multi-level modulations, the current optical fiber transmission systems have achieved ultra-high transmission capacity of 100 Tb/s per fiber [1–3]. However, the transmission capacity of single mode single core fiber (SM-SCF) is rapidly approaching the fundamental limit owing to the limitation of amplifier bandwidth, nonlinear noise and fiber fuse phenomenon [4, 5], it is predicted that capacity crunch will happen in the near future. In order to overcome the capacity limit, space division multiplexing (SDM) techniques based on weakly-coupled multi-core fiber (MCF) is one of promising candidates to enlarge the capacity . In MCFs, the suppression of crosstalk (XT) between neighboring cores becomes an issue if the number of cores increases in a limited cladding space. In order to decrease XT in MCFs, the coupling coefficient between cores has to be reduced. Therefore, the core design which has strong confinement of modes is important for the suppression of the mode coupling coefficient. Core structure with high-index and small-diameter is one of the options, but it degrades the effective area (Aeff) and increases the nonlinear noise [6, 7]. Recently, several research groups have investigated specially designed refractive index profiles to meet the requirement of low XT, such as trench-assisted (TA) profile [8–12], hole-assisted (HA) profile [13, 14], which are shown as Figs. 1(a) and 1(b). Besides, low index rod-assisted (RA) profile shown as Fig. 1(c) can also be used to suppress the inter-core XT. In this work, we adopt RA profile instead of TA profile and HA profile, since rod has smaller size than trench so that it can suppress XT as well as shorten the cable cutoff wavelength (λcc) of higher order mode in the core, especially in the central core. Furthermore, a type of optical fiber called heterogeneous MCF (Hetero-MCF) has been proposed to realize much lower XT, in which there are not only identical cores but also non-identical cores [8, 9, 15, 16]. Besides the core structure, we also need to analyze and carefully design the core arrangement for MCFs. Figures 2(a)-2(c) show three kinds of core arrangement for high core density MCFs, which are hexagonal closed-packed structure (HCPS) , ring-lattice structure (RLS)  and square-lattice structure (SLS) . Here, the red, blue and green circles stand for non-identical cores. However, both HCPS and RLS needs three kinds of cores with RA profile to produce a heterogeneous relation for all the adjacent cores with similar Aeff, and each central core needs six low index rods to suppress XT, resulting in lengthening λcc of the inner cores. Furthermore, it is very difficult to pick out three heterogeneous RA cores with relative large effective index difference (Δneff) and ensure similar Aeff simultaneously. On the other hand, SLS shown as Fig. 2(c) can arrange 32 cores with only two kinds of cores with RA profile.
Therefore, in this paper, we adopt SLS instead of HCPS and RLS for high core density MCFs with 32 cores arranged inside it, and investigate the characteristics such as Aeff, XT, threshold value of bending radius (Rpk), relative core multiplicity factor (RCMF) and λcc to prove the feasibility and advantages of square-lattice structured 32-core fiber. The discussion of Rpk, RCMF will be introduced later. In Section 2, we firstly propose Hetero-RA-32-Core fiber based on SLS. Then, we introduce the design method and analyze the rationality of the designed MCFs. In Section 3, we present an optimization scheme−Hetero-RA-TA-32-Core-Fiber, then introduce the design method and give the core parameter design. In Section 4, we investigate the characteristics including Aeff, XT, Rpk and RCMF of Hetero-RA-TA-32-Core-Fiber, and compare the proposed Hetero-RA-TA-32-Core-Fiber with other reported MCFs to point out the superiority of our design. In Section 5, we conclude our work.
2. Design of Hetero-RA-32-Core-Fiber
2.1 Profile of Hetero-RA-32-Core-Fiber
Figure 3 shows the schematic structure and core index profile of Hetero-RA-32-Core-Fiber with SLS. In Fig. 3(a), outer cladding thickness (OCT) is the distance from the center of outer core to the circumference of cladding, Λ is the core-to-core distance, Dcl is the cladding diameter. In Fig. 3(b), r1, rh, dcr, Δ1, Δ2, and Λ correspond to core radius, rod radius, the distance between the center of core and the center of rod, the relative refractive index difference between core and cladding, the relative refractive index difference between cladding and rod, and the core-to-core distance, respectively.
2.2 Design of low-index rod
In the simulation of this work, the operation wavelength is assumed to be 1550 nm. And in order to arrange more cores in a limited cladding space, we set Λ to 30 μm for making sure it is weakly-coupled MCF . To simplify the fiber design scheme, dcr is set to 15 μm which means that the rod is located at the center of two non-identical cores. Considering the design of Hetero-RA-32-Core-Fiber, we only need to focus on rod parameters and core parameters, since the core arrangement, Λ and dcr are fixed.
Figure 4 shows the numerically calculated mean value of XT between two non-identical RA cores after 100-km propagation as function of rod radius when Δ2 = −0.7%, where the calculated XT values are obtained by coupled-power theory (CPT) . In this work, the fiber is assumed to be twisted at twisting rate of 5 turns per 100 m  and all the XT values are calculated by using CPT with the correlation length d being set as 0.05 m, based on which the calculated average XT values agree well with the experimental results [17, 19, 21]. And we set core with (r11, Δ11) of (5.2 μm, 0.37%) as Core 1, core with (r21, Δ21) of (4.6 μm, 0.31%) as Core 2 for calculation. In Fig. 4, we can find that as rh increases, XT level decreases. Therefore, we set rh to 6.0 μm as a compromise value, since larger rh can suppress XT but lengthen λcc.
Figure 5 illustrates the XT calculated by CPT  between Core 1 and Core 2 as function of bending radius (R) at various Δ2 when rh = 6.0 μm. In Fig. 5, we can see that with the increase of R, the XT of Hetero-MCF firstly increases, however after R reaching a threshold value, which is called Rpk, the XT decreases immediately and then it converges to a certain value no matter how R increases . There is a relationship between Rpk, neff, Δneff and Λ, which is given by Eq. (1), we can know that Rpk increases as Δneff decreases when neff and Λ are fixed. Therefore, a large Δneff will be required for pushing the value of Rpk to sufficiently small range, so that we can obtain a large non-phase-matching region, R>Rpk, in which the bending extent will not impact the crosstalk any more.
From Fig. 5, we can also obtain that when we fix rh and shift Δ2, the XT level decreases with the decrease of Δ2. Thus, Δ2 is set to −0.7% for simulation, which is the structural parameter used in the fabricated index trench assisted MCFs .
2.3 Selection of RA core
We begin to design core parameters, such as core r1 and core Δ1, since all the low index rod structural parameters are designed before. Figure 6 illustrates the neff and Aeff of LP01 mode at 1550-nm wavelength as function of core r1 and core Δ1 when rh = 6.0 μm and Δ2 = −0.7%. And color map means the neff values. The black solid lines and the black dashed lines represent certain special values of neff and Aeff of LP01 mode at 1550-nm wavelength, which are simulated by full-vector finite element method (FEM) . The upper and lower white bold solid lines in Fig. 6 correspond to the cutoff limit of LP11 mode and the bending loss (BL) limit of LP01 mode. On the one hand, according to the deployment configuration in IEC 60793-1-44 document, we think that LP11 mode has been mostly cut off if BL of LP11 mode is larger than 1 dB/m at R = 140 mm and λ = 1530 nm, which can make the total BL>20 dB/22 m; on the other hand, the BL of LP01 mode should be smaller than 0.5 dB/100 turns at R = 30 mm and λ = 1625 nm according to ITU-T recommendations G.655 and G.656, which can be thought that LP01 mode has been totally confined inside the core. The field which is surrounded by the upper and lower white bold solid lines is called single mode operation region (SMOR) [9, 23]. Thus, for single-mode operation from C-band to L-band, we should choose two non-identical RA cores in such field, noticing that the two non-identical RA cores should better have the same Aeff, because the different Aeff will cause splice loss . In Fig. 6, we can also observe that if we enlarge the expected Aeff, the maximum Δneff we can obtain between non-identical cores will decrease, which means there is a trade-off relationship between Δneff and Aeff, and this kind of trade-off relationship also exists similarly between Rpk and Aeff. Therefore, if the required Δneff is set to 0.001 and the target Aeff is 80 μm2, we designate core with (r1, Δ1) of (4.65 μm, 0.375%) as Core 1, core with (r1, Δ1) of (4.19 μm, 0.328%) as Core 2, which are shown as the filled circles in Fig. 6.
2.4 Analysis of requirement for OCT
Subsequently, we analyze the BL dependence of LP01 mode at R = 140 mm and λ = 1625 nm on OCT for Core 1 and Core 2 when rh = 6.0 μm and Δ2 = −0.7% and the results are shown as Fig. 7. In Fig. 7, we can observe that for Core 1 and Core 2, OCT should be larger than 37.1 μm and 41.8 μm to ensure BL at λ = 1625 nm and R = 140 mm of smaller than 0.001 dB/km . In addition, the relationship between Dcl, Λ, and OCT of SLS in Fig. 3 (a) is given by
It is known that Dcl of MCFs should be smaller than around 230 μm to satisfy the limit of failure probability . However, according to the Eq. (2), the structural upper limit of OCT is 28 μm when Dcl is smaller than 230 μm and Λ is set to be 30 μm. Thus, we need to optimize core index profile and core parameters to meet the requirement of OCT of 28 μm.
3. Optimization scheme—-Hetero-RA-TA-32-Core-Fiber
3.1 Profile of Hetero-RA-TA-32-Core-Fiber
As core with RA profile at outer layer cannot satisfy the lower limit of OCT, in order to make stronger confinement on mode for outer layer cores, we deployed TA profile instead of RA profile for the eight corner outer cores in SLS. Figure 8 shows a schematic structure and heterogeneous core types of the proposed Hetero-RA-TA-32-Core-Fiber. Core 1 and Core 2 are RA cores and Core 3 is TA core. This structure shares the same parameters—r1, dcr, rh Δ1, Δ1, Λ, OCT, Dcl with Fig. 3(b), and for Core 3, it also shares the same parameters—r2, W with Fig. 1(a), where r2, W in TA profile stand for the distance between the center of core and inner edge of trench, the thickness of the trench layer, respectively. Figure 8(b) shows eight types of heterogeneous core combinations, in which four types are RA core combinations and four types are combinations of RA core and TA core. The eight types will be discussed later in the XT characteristics.
3.2 Design of low-index trench and rod
According to the previous discussion, we should first determine the trench parameters and then decide the rod parameters, so that we can calculate the cutoff limit of LP11 mode and the BL limit of LP01 mode for TA core and RA core, and obtain the core selection region.
Firstly, we analyze r2/r1 and W/r1 of the trench layer for TA cores when we set core r1 and core Δ1 to 4.90 μm and 0.40%. Figures 9(a)-9(c) show BL, Aeff and neff of LP01 mode as function of W/r1 at various r2/r1, respectively. We can find that with the increase of r2/r1, BL decreases, but Aeff and neff stay almost unchanged. With the increase of r2/r1, BL decreases, but both Aeff and neff increase. Figure 9(a) can also help us know that to ensure BL of smaller than 0.001 dB/km, r2/r1 and W/r1 should be larger than or equal to 1.5 and 1.2, respectively.
Figure 10 show cutoff limit and OCT lower limit of 28 μm as function of r1 and Δ1 when (a) r2/r1 = 1.5, (b) r2/r1 = 1.6 and (c) r2/r1 = 1.7 at various W/r1 respectively. In Figs. 10(a)-10(c), the black dashed lines and bold black solid lines represent values of neff and Aeff of LP01 mode at 1550 nm wavelength, which are also simulated by using full-vector FEM , where (a) r2/r1 = 1.5, (b) r2/r1 = 1.6 and (c) r2/r1 = 1.7. The group of blue, red and green solid lines and the group of blue, red and green dashed lines correspond to the cutoff limits of LP11 mode at W/r1 = 1.1, 1.2 and 1.3 when R = 140 mm and λ = 1530 nm and the OCT lower limits of LP01 mode at W/r1 = 1.1, 1.2 and 1.3 when R = 140 mm and λ = 1625 nm, respectively, where (a) r2/r1 = 1.5, (b) r2/r1 = 1.6 and (c) r2/r1 = 1.7. According to Fig. 10, we need to know that we should select TA core above the OCT lower limit lines and under the cutoff limit lines simultaneously to meet the requirements of OCT of smaller than 28 μm and cable cutoff wavelength of smaller than 1530 nm. In order to show the intersections of the OCT lower limit lines, cutoff limit lines and Aeff lines more clearly, we enlarge the line intersections region in Fig. 10, which are shown in Fig. 11. Based on Fig. 11, when the target value of Aeff is 74 μm2, and at the same time to keep the obtained neff value as large as possible, here maximum neff = 1.4533, the choices of (r2/r1, W/r1) can be (1.5, 1.2) and (1.6, 1.2), which correspond to TA Core 1 in Fig. 11(a) and TA Core 2 in Fig. 11(b). Comparing Fig. 11(a) with Fig. 11(b), we can know that with r2/r1 increased from 1.5 to 1.6, we can obtain TA core with the same Aeff and neff but higher refractive index Δ1 and smaller core radius r1 at the same time, which is effective in suppressing inter-core XT. Thus, we will choose (r2/r1, W/r1) of (1.6, 1.2) for Aeff = 74 μm2. When the target value of Aeff is 78 μm2, the choice of (r2/r1, W/r1) will be (1.5, 1.3), which correspond to TA Core 3 in Fig. 11(a). Thus, we present two options for TA core: (1) r2/r1 = 1.6, W/r1 = 1.2, (2) r2/r1 = 1.5, W/r1 = 1.3.
Secondly, we need to figure out the suitable value of rod radius. Figure 12 illustrates BL of LP11 mode at λ = 1530 nm and R = 140 mm and BL of LP01 mode at λ = 1625 nm and R = 30 mm as function of r1 and Δ1 for RA core when (a) rh = 7.0 μm, (b) rh = 7.5 μm, and (c) rh = 8.0 μm, and also describes the SMOR dependence on rod radius rh. The black solid lines and the black dashed lines represent certain values of neff and Aeff of the LP01 mode at 1550 nm wavelength, which are simulated by full-vector FEM . Here, all the Aeff values of RA cores are calculated by using core type 1 with four surrounding rods in Fig. 8(b). In Fig. 12, we can observe that as rh increases, the SMOR becomes narrow, then the Max Δneff we can obtain becomes small. When determining the suitable rh, there are two things to be considered: the first one is rh should be as large as possible to suppress XT between two adjacent Hetero-RA-cores which can be found in Fig. 5, the second one is that Δneff between each pair of Hetero-RA-cores should be as large as possible to decrease Rpk. Thus, we choose rh = 7.5 μm to compromise this tradeoff relationship.
3.3 Selection of TA core and RA core
In the following discussion, we will introduce the core parameters selection in two parts, one is r2/r1 = 1.6, W/r1 = 1.2, rh = 7.5 μm, Aeff = 74 μm2, and the other one is r2/r1 = 1.5, W/r1 = 1.3, rh = 7.5 μm, Aeff = 78 μm2.
Figure 13 shows BL of LP11 mode at λ = 1530 nm and R = 140 mm and BL of LP01 mode at λ = 1625 nm and R = 30 mm as function of r1 and Δ1 with target Aeff is 74 μm2 when (a) TA core, r2/r1 = 1.6, W/r1 = 1.2, rh = 7.5 μm and (b) RA core, r2/r1 = 1.6, W/r1 = 1.2, rh = 7.5 μm. According to the principle of TA core-selecting described before, we designate core with (r1, Δ1) of (4.79 μm, 0.408%) as Core 3 when r2/r1 = 1.6, W/r1 = 1.2, rh = 7.5 μm, which is shown as the red filled circles in Fig. 13(a). For RA core-selecting, we should make sure that Aeff of RA cores is 74 μm2, and at the same time, selecting RA cores which can ensure sufficiently large Δneff between each set of non-identical cores so that fiber becomes bend-insensitive with small Rpk. Thus, we designate core with (r1, Δ1) of (4.39 μm, 0.371%) as Core 1 and core with (r1, Δ1) of (4.05 μm, 0.330%) as Core 2 when r2/r1 = 1.6, W/r1 = 1.2, rh = 7.5 μm, which is shown as the green and blue filled circles in Fig. 13(b). In this case, the designed fiber is called Fiber A.
Figure 14 shows BL of LP11 mode at λ = 1530 nm and R = 140 mm and BL of LP01 mode at λ = 1625 nm and R = 30 mm as function of r1 and Δ1 with target Aeff is 78 μm2 when (a) TA core, r2/r1 = 1.5, W/r1 = 1.3, rh = 7.5 μm and (b) RA core, r2/r1 = 1.5, W/r1 = 1.3, rh = 7.5 μm. Following the same principles of core-selecting as Aeff = 74 μm2, when Aeff = 78 μm2, we designate core with (r1, Δ1) of (4.98 μm, 0.378%) as Core 3 when r2/r1 = 1.5, W/r1 = 1.3, rh = 7.5 μm, which is shown as the red filled circles in Fig. 14(a). We designate core with (r1, Δ1) of (4.47 μm, 0.341%) as Core 1 and core with (r1, Δ1) of (3.94 μm, 0.301%) as Core 2 when r2/r1 = 1.5, W/r1 = 1.3, rh = 7.5 μm, which is shown as the green and blue filled circles in Fig. 14(b). At this time, the designed fiber is called Fiber B.
Table 1 and Table 2 summarize the structural parameters of Fiber A and Fiber B. Those structural parameters ensure that 32-core MCFs realize Aeff of around 74 μm2 and 78 μm2 at λ = 1.55 μm, respectively, λcc of smaller than 1.53 μm, and BL of smaller than 0.5 dB/100 turns at λ = 1.625 μm with R = 30 mm.
4. Characteristics of Hetero-RA-TA-32-Core-Fiber
Figure 15 shows the numerically calculated mean value of inter-core XT results of Fiber A and Fiber B at 1.55 μm with eight types of heterogeneous core combinations after 100-km propagation as a function of bending radius R, where the calculated results are obtained by coupled-power theory (CPT) .
From Figs. 15(a) and 15(b), we can see that the 100-km worst XT of Fiber A is the XTtype5 which is still lower than −30 dB, while the 100-km worst XT of Fiber B is the XTtype2 and XTtype3 which are −28 dB. It is interesting to notice that the XTtype1 is very close to XTtype4 in both Fiber A and Fiber B, which separate from the XTtype2 and XTtype3. Thus, we can conclude that the XT between RA core and RA core is sensitive to the surrounding low index structures. On the other hand, the XTtype5 is very close to XTtype7, and XTtype6 is very close to XTtype8 in both Fiber A and Fiber B. Then we can know that the XT between RA core and TA core is irrelevant to the surrounding low index structures. According to Eq. (1), the calculated Rpk of Fiber A and Fiber B are 7.0 cm and 6.5 cm, respectively. Thus, in order to ensure low crosstalk transmissions with high-order modulation format , Fiber A is our final design choice for Hetero-RA-TA-32-Core-Fiber. And RCMF of Fiber A is estimated to be 8.74. According to the definition of core multiplicity factor (CMF) for MCF , the CMF is given by24]
Figure 16 shows comparison of spatial multiplicity and inter-core XT at λ of 1550 nm between the reported high-density MCFs [17, 26–29] and the proposed Hetero-RA-TA-32-Core-Fiber in this work. Figure 16(a) and Fig. 16(b) show RCMF and core count, respectively. In Figs. 16(a) and 16(b), circle, diamond and square mean RLS, HCPS and SLS, respectively. It can be found that both 32-core fiber with SLS and 30-core fiber with HCPS  can make RCMF reach above 8 and 100-km XT of smaller than −31 dB/100 km which implies that 64-QAM can be used in 100 km transmission . However, the λcc of 30-core fiber with HCPS is less than 1570 nm , which is larger than 1530 nm, while the λcc of the proposed Hetero-RA-TA-32-Core-Fiber with SLS is smaller than 1530 nm, which can ensure single-mode transmission over the C + L band from 1530 nm to 1625 nm.
In this paper, we first introduced a heterogeneous rod-assisted MCF with 32 cores arranged in SLS. After investigation, we found that the rod-assisted structure was unable to satisfy the requirement of OCT in the limited cladding space. Thus, we presented an optimized scheme for this SLS-MCF, which is a heterogeneous rod-assisted as well as trench-assisted 32-core multi-core fiber (Hetero-RA-TA-32-Core-MCF). For the optimized Hetero-RA-TA-32-Core-MCF, we introduced the design method for core parameters in detail. Simulation results showed that Hetero-RA-TA-32-Core-Fiber achieved average Aeff of about 74 μm2, low XT of about −31 dB/100 km, Rpk of 7.0 cm, RCMF of 8.74, and λcc of less than 1.53 μm. In summary, Hetero-RA-TA-32-Core-Fiber with SLS can realize high-density transmission with improved performance.
National Natural Science Foundation of China (NSFC) (Grant No. 61501027, 61370191, 61671053); Fundamental Research Funds for the Central Universities (Grant No. FRF-TP-15-031A1); Hong Kong Scholars Program (XJ2016029); Foundation of Beijing Engineering and Technology Center for Convergence Networks and Ubiquitous Services.
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