We propose a kind of heterogeneous trench-assisted graded-index few-mode multi-core fiber with square-lattice layout. For each core in the fiber, effective area (Aeff) of LP01 mode and LP11 mode can achieve about 110 μm2 and 220 μm2. Absolute value of differential mode delay (|DMD|) is smaller than 100 ps/km over C + L bands, which can decrease the complexity of digital signal processing at the receiver end. Considering the upper limit of cladding diameter (Dcl) and cable cutoff wavelength of LP21 mode in the cores located at the inner layer, we set core pitch (Λ) as 43 μm. In this case, Dcl is about 220.4 μm, inter-core crosstalk (XT) is lower than −40 dB/500km and the relative core multiplicity factor (RCMF) reaches 15.93.
© 2015 Optical Society of America
Internet traffic in optical fiber communication systems has been increasing rapidly, so the conventional single-core single-mode fibers utilized in current system may approach the fundamental limit over the next decade or so. Therefore, in order to deal with this capacity crunch, multiplexing technologies such as space-division multiplexing (SDM) using multi-core fiber (MCF) and multi-mode fiber (MMF) are being intensively investigated [1–3]. For dense space-division multiplexing (DSDM) utilizing both multi-core and multi-mode, few-mode multi-core fibers (FM-MCFs) were recently proposed to further improve the multiplicity and increase the transmission capacity [4–6].
In order to make the multiple signal streams transmit simultaneously over each core in FM-MCFs, we can use multiple-input multiple-output (MIMO) system to enhance the spectral efficiency of an optical communication system. For MIMO system, if the differential mode delay (DMD) between propagation modes becomes very large, it needs complex computation to recover the signals at the receiver end [7–10]. Hence, to design a kind of fiber with low DMD is one of the requirements in MIMO processing. So far, it has been reported that total DMD can approximate to 1 ps/km by using DMD compensation technology [11,12]. However, DMD can still be smaller than 100 ps/km by not using DMD compensation technology [13,14].
In our previous work , we proposed an optimal design scheme for two-mode supporting heterogeneous trench-assisted few-mode multi-core fiber (Hetero-TA-FM-MCF) with multi-step index profile and absolute DMD in each core can be smaller than 170 ps/km over C + L bands. However, in this paper, we adopt graded-index profile and numerically investigate the characteristics of the trench-assisted graded-index few-mode core (TA-GI-FMC) in which LP01 mode and LP11 mode are supported. Furthermore, we also discuss optimal index profile and propose the appropriate arrangement for this Hetero-TA-FM-MCF in order to further lower the DMD. Here, non-identical cores deployed in Hetero-TA-FM-MCF can make the inter-core crosstalk be insensitive to the bending after bending radius reaches the threshold value and trench structure surround each core can suppress the crosstalk, which are also the distinguishing features of this work [16,17].
2. Design method of trench-assisted graded-index few-mode core (TA-GI-FMC)
2.1 Profile of TA-GI-FMC
Figure 1 shows the refractive index profiles of TA-GI-FMC which have been investigated by many research groups [1,2,18]. r1, r2, W, and Δt stand for core radius, the distance between the center of core and the inner edge of trench, the thickness of the trench layer, and the relative refractive-index difference between trench and cladding, respectively. Δ can be expressed as Δ = Δ1 [1 − (r/r1) α], where α represents shape factor, Δ1 stands for the peak relative refractive index difference between core and cladding, and r is the radial coordinate. We define the DMD as a value obtained by subtracting the mode group delay of the fundamental mode from that of the higher-order mode.
2.2 TA-GI-FMC with low DMD and low DMD slope
Figure 2 is a concept figure which illustrates DMD as function of r1 and Δ1. The blue dashed-and-dotted lines represent DMD and red solid line denotes effective area for LP01 mode (Aeff_LP01) of 110 μm2. Here, we selected three coordinate points which are shown as the black circle dots in Fig. 2 to investigate how r2/r1 and α affect DMD. Three selected coordinate values for the black circle dots are (a) (r1, Δ1) = (9.0 µm, 0.4%), (b) (r1, Δ1) = (9.5 µm, 0.446%), and (c) (r1, Δ1) = (10.0 µm, 0.5%), which do not only achieve Aeff_LP01 of 110 μm2 but also make sure two modes (LP01 mode and LP11 mode) are supported in the TA-GI-FMC. Subsequently, we analyze the dependence of DMD at these three dots on r2/r1 and α. After knowing how DMD changes at these three dots and obtaining the appropriate set of r2/r1 and α, we can find the suitable values of r1 and Δ1 for TA-GI-FMC in this red segment.
Figure 3 illustrates DMD as function of r2/r1 and α at λ = 1550 nm when (a) (r1, Δ1) = (9.0 µm, 0.4%), (b) (r1, Δ1) = (9.5 µm, 0.446%), and (c) (r1, Δ1) = (10.0 µm, 0.5%). Here, we set that Δt = −0.7%, W/r1 = 1.0 and refractive index of cladding ncl = 1.45 for the simulation of DMD. The numerical results are all simulated based on full-vector FEM . In Fig. 3, the black, blue and red solid lines represent DMDs of 0 ns/km and the red pentagrams which stand for (r2/r1, α) of (1.3, 2.2) can obtain DMDs of −0.0081 ns/km, −0.0209 ns/km, and + 0.0348 ns/km respectively when wavelength (λ) equals 1550 nm. For (r2/r1, α) of (1.3, 2.2), the DMD slopes are all smaller than |4| × 10−4 ns/km/nm under condition (a), (b) and (c).
Putting three curves of DMD = 0 ns/km in Fig. 3 together, we can obtain a new figure which is shown as Fig. 4. Figure 4 shows curves of DMD = 0 ns/km as function of r2/r1 and α at λ = 1550 nm. The area filled with pattern can be regarded as an appropriate design region for r2/r1 and α. In this area, we can choose suitable set of r2/r1 and α to make low DMD for the above-mentioned three nodes in Fig. 2 at same time. We have known that (r2/r1, α) of (1.3, 2.2) which locates at the black solid line in Fig. 4 can obtain DMDs of −0.0081 ns/km, −0.0209 ns/km, and + 0.0348 ns/km for the three dots respectively when λ = 1550 nm. Therefore, we set (r2/r1, α) as (1.3, 2.2) and then find the suitable sets of (r1, Δ1) in the red segment shown in Fig. 2 for TA-GI-FMC.
Figure 5 shows DMD at λ of 1550 nm, r2/r1 of 1.3, α of 2.2 and Δt of −0.7% as function of r1 and Δ1 where (a) W/r1 = 0.7 and (b) W/r1 = 0.2. For the non-identical TA-GI-FMCs, we also set the same Aeff of LP01 mode (Aeff_LP01) in both cores and define the target value of Aeff_LP01 in both TA-GI-FMCs as 110 μm2. In Fig. 5, the black solid line and black dash line represent Aeff_LP01 and effective index of LP01 mode (neff_LP01). In Fig. 5(a), the upper and lower white solid lines represent the cutoff of LP21 mode and the lower limit of LP11 mode at W/r1 of 0.7 and in Fig. 5(b), the upper and lower white solid lines stand for the cutoff of LP21 mode and the lower limit of LP11 mode at W/r1 of 0.2. We set W/r1 as 0.7 and 0.2 in order to make it probable to choose two sorts of TA-GI-FMCs with same Aeff_LP01 of 110 μm2, low DMD and relative large difference between neff_LP01 in two TA-GI-FMCs (Δneff_LP01). Here, to define the two-mode operation region, the bending loss (BL) of LP21-like HOM should be > 1 dB/m at R = 140 mm, which can make the total BL be larger than 20 dB/22m according to the deployment configuration in IEC 60793-1-44 document and we assume the limit value of the BL of LP11-like HOM to be 0.5 dB/100 turns at R = 30 mm according to the description of BL of fundamental mode in ITU-T recommendations G.655 and G.656. To ensure a relative small Rpk which is a critical value of bending radius , we define the required Δneff_LP01 to be about 0.0008. In this case, we can select two kinds of TA-GI-FMCs with low DMD and DMD slope in the two-mode operation regions, which are shown as the filled circles in red and green in Fig. 5(a) and 5(b) correspondingly. For the filled circle in red which is designated as core 1, r1 = 9.22 μm, Δ1 = 0.420% ; For the filled circle in green which is designated as core 2, r1 = 9.78 μm and Δ1 = 0.473%. The values of a1, Δ1, W/r1, Δt, α, r2/r1 and the characteristics of effective index (neff), effective area (Aeff), DMD, DMD slope and BL of core 1 and core 2 are summarized in Table 1. The dependence of DMD on λ is shown as Fig. 6 and we can observe that |DMD| values of core 1 and core 2 are both smaller than 100 ps/km over C + L bands.
3. Arrangement of TA-GI-FMCs in the fiber
3.1 Appropriate outer cladding thickness
Figure 7 shows a cross section of four-core fiber model. The coating has higher refractive index than that of the cladding in fiber, which would cause bending loss in the core if the outer cladding thickness (OCT) is not large enough. It indicates that we have to find out the appropriate OCT to suppress the bending loss. Figure 8(a) and 8(b) show the dependence of bending loss of LP11 mode at bending radius (R) of 140 mm on OCT when λ is 1625 nm, 1565 nm, and 1550 nm for core 1 and core 2, respectively. In Fig. 8(a) and 8(b), we can observe that for core 1, the OCT should be larger than 48.0 µm, 42.2 µm, and 40.2 µm at λ of 1625 nm, 1565 nm, and 1550 nm respectively to suppress bending loss at R of 140-mm less than 0.001dB/km, which is regarded as a upper limit here ; For core 2, the OCT should be larger than 45.3 µm, 42.2 µm, and 42.1 µm at λ of 1625 nm, 1565 nm, and 1550 nm respectively to suppress bending loss at R of 140-mm less than 0.001dB/km. Here, we take λ of 1565 nm as the operation wavelength in the simulation, so we can know that the OCT should be at least 42.2 µm.
3.2 Optimal layout
We investigated two kinds of layouts for 12-core Hetero-TA-FM-MCF with graded-index profile — one-ring layout and square-lattice layout, which are shown as Fig. 9(a) and 9(b). Cladding diameter (Dcl) should not be larger than 225 µm in order to satisfy the limit of failure probability . The relationship between Dcl and core pitch (Λ) for these two layouts are shown as the expressions below. According to that, we can obtain the upper limit of Λ. Hence, according to the Eq. (1) and Eq. (2) which are corresponding to one-ring layout and square-lattice layout respectively, the upper limits of Λ for each layout are 35 µm and 44 µm.
Figure 10 illustrates the dependence of inter-core crosstalk (XT)  on core pitch at length (L) of 500 km and bending radius (R) of 500 mm. The reason why we set R as 500 mm is that this bending radius is much larger than Rpk, which means that the crosstalk at R of 500-mm is the one in the bend-insensitive situation. Moreover, we define L as 500 km to see whether crosstalk can be small after long transmission distance. From Fig. 10, we can find that if we expect the largest XT11-11 to be lower than −30 dB/500 km, Λ should be larger than 37 µm, which is greater than the upper limit Λ of one-ring layout. Moreover, we also investigated the square-lattice structure for our model and we found that after meeting the requirement of the upper limit of Dcl and lower limit of OCT, Λ still has space to increase, since this arrangement makes full use of the center region of fiber.
Therefore, we choose square-lattice layout for this Hetero-TA-FM-MCF. For the square-lattice layout, the trench structure around each core causes tight confinement of the higher-order mode in the cores located at the inner layer. Hence, the cable cutoff wavelength (λcc) of LP21 mode in the cores located at the inner layer must be considered and we define λcc as the one that can make BL of LP21 mode equal to 1 dB/m. In this case, the total BL of LP21 mode based on the deployment configuration in IEC 60793-1-44 document must be larger than 20 dB/22m. Figure 11 illustrates λcc of core 1 and core 2 in the area surrounded by the dash line as function of Λ. In Fig. 11, we can find that the Λ should be larger than 43 µm to make λcc of the inner cores be smaller than 1.53 µm which is the shortest operating wavelength. According to Eq. (2), if we set Λ as 43 µm which is smaller than the above-mentioned upper limit, Dcl is 220.4 µm. In this case, the relative core multiplicity factor (RCMF)  for this Hetero-TA-FM-MCF reaches 15.93 which is a bit higher than that of our proposed Hetero-TA-FM-MCF with multi-step index profile . Moreover, Fig. 12 shows the dependence of XT11-11 on R when λ = 1565 nm and Λ = 43 µm and we can observe that when R is larger than about 15 cm, crosstalk becomes insensitive to the bending radius. Additionally, a kind of 12-core × 3-mode FM-MCF with square-lattice layout which was based on our design has been fabricated and proposed .
We designed a kind of 12-core heterogeneous trench-assisted few-mode multi-core fiber (Hetero-TA-FM-MCF) with graded-index profile. For each core in this Hetero-TA-FM-MCF, |DMD| is smaller than 100 ps/km over C + L bands which can decrease the complexity of digital signal processing to a large extent at the receiver end. After the investigation about the arrangement, we found that the square-lattice layout is better than one-ring layout, because it can ensure cladding diameter (Dcl) not exceed 225 µm if core pitch (Λ) is not larger than 44 µm and meanwhile make sure the inter-core crosstalk (XT) be lower than −40 dB/500km. Additionally, we also found that if Λ is larger than 43 µm, the cable cutoff wavelength (λcc) is not larger than the shortest operating wavelength of 1530 nm. When we set Λ as 43 µm, the relative core multiplicity factor (RCMF) for this Hetero-TA-FM-MCF reaches 15.93.
This work was partially supported by the National Institute of Information and Communication Technology (NICT), Japan under “R&D of Innovative Optical Fiber and Communication Technology”.
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