We propose a kind of heterogeneous multi-core fiber (Hetero-MCF) with trench-assisted multi-step index few-mode core (TA-MSI-FMC) deployed inside. After analyzing the impact of each parameter on differential mode delay (DMD), we design a couple of TA-MSI-FMCs with Aeff of 110 μm2 for LP01 mode. DMD of each TA-MSI-FMC is smaller than |170| ps/km over C + L band and the total DMD can approach almost 0 ps/km over C + L band if we adopt DMD managed transmission line technique by using only one kind of Hetero-TA-FM-MCF. For such Hetero-TA-FM-MCF, crosstalk is about –30 dB/100km at wavelength of 1550 nm as bending radius becomes larger than 15 cm, core number can reach 12, a relative core multiplicity factor (RCMF) is 15.7, and the RCMF can even reach 26.1 if we treat LP11 mode as two special modes thanks to the multiple-input-multiple-output technology.
© 2014 Optical Society of America
Several multiplexing technologies such as space-division multiplexing (SDM) using multi-core fiber (MCF)  and mode-division multiplexing (MDM) using few-mode fiber (FMF)  are being intensively investigated to overcome the capacity limit of the network traffic in the current conventional optical communication systems. In order to further increase the transmission capacity, the combination design of multi-core and few-mode has been discussed recently [3, 4].
For FMF, multiple-input-multiple-output (MIMO) digital signal processing (DSP) is applied to recover the transmitted signals. In order to decrease the MIMO-DSP complexity, we should guarantee as low differential mode delay (DMD) over C + L band as possible. FMF with low DMD is of benefit to MDM transmission utilizing MIMO. Furthermore, low DMD in the wide wavelength region is required for the wavelength division multiplexing (WDM) transmission applications . To realize low DMD, the 1st approach is using complex refractive index profile, such as multi-step index profile , graded index , and graded index profile with trench (T-GIP) , to add more degree of freedom to control the value of DMD. FMFs with T-GIP have realized ultra-low DMD less than 36 ps/km over the C band . The 2nd approach that realizes low DMD over the wide band is called DMD managed transmission line technique [5, 9]. The line consists of two kinds of FMFs with positive and negative DMDs to compensate for the total DMD. A transmission line which realizes low DMD within |3| ps/km over C band and L band has been proposed .
In this paper, we investigate and analyze appropriate index profile for few-mode core that support two LP modes respectively and then propose a relative optimum design scheme for heterogeneous trench-assisted FM-MCF (Hetero-TA-FM-MCF) with low DMD and large effective area. Additionally, in this work, trench layer is deployed around each core to realize low inter-core crosstalk even with small core pitch and heterogeneous cores are chosen to make the fiber insensitive to the curvature of fiber, which are explained in more detail in our last works [11, 12].
2. Design of trench-assisted multi-step index few-mode core (TA-MSI-FMC)
2.1 Profile of TA-MSI-FMC
In this work, we design trench-assisted few-mode core (TA-FMC) with two kinds of modes — LP01 mode and LP11 mode. Besides the number of mode transmitting in the core, we should also take into account the inter-mode crosstalk and the differential mode delay (DMD) . For a step index profile, it is impossible to design a fiber with low DMD over a transmission band like C and L bands due to its simple profile. Nevertheless, the multi-step index (MSI) profile that is shown as Fig. 1 has more degree of freedom to control the difference of group delays between LP01 mode and LP11 mode, since the DMD characteristics are sensitive to the change of the refractive index profile. In Fig. 1, a1, r1, r2, W, Δ1, Δ2 and Δt stand for inner core radius, outer core radius, the distance between the center of inner core and the inner edge of trench, the thickness of the trench layer, the relative refractive-index difference between inner core and cladding, the relative refractive-index difference between outer core and cladding, and the relative refractive-index difference between trench and cladding, respectively. In the following subsections, we analyze and discuss the relationship between these parameters and DMD and find out the appropriate set of a1, Δ1, r1/a1, Δd, r2/r1, W/r1, and Δt to obtain a couple of TA-MSI-FMCs with low DMD, low DMD slope, small inter-core crosstalk, and large effective area (Aeff).
2.2 TA-MSI-FMC with low DMD and low DMD slope
We define the DMD as a value obtained by subtracting the mode group delay of the fundamental mode (τLP01) from that of the higher-order mode (τLP11) and the expression of DMD is written as follows:
Figure 2 shows DMD and DMD slope as function of r2/r1 at λ = 1550 nm. Here, we fixed the value for a1, Δ1, r1/a1, Δd, W/r1, and Δt, which are assumed as 3.6 µm, 0.5%, 2.0, −0.13%, 1.0, and −0.7%, respectively. From Fig. 2, we can know that the location of trench layer has a big impact on the DMD and DMD slope. Furthermore, we can also observe that as r2/r1 increases, DMD and DMD slope are getting smaller and when r2/r1 = 1.6, the absolute value of DMD slope for λ = 1550 nm is the smallest and approximates to 0 ns/km/nm.
Figure 3 illustrates DMD and DMD slope as function of r1/a1 and Δd at λ = 1550 nm. Here, we assumed a1, Δ1, r2/r1, W/r1, and Δt to be 3.6 µm, 0.5%, 1.6, 1.0, and −0.7%, respectively. In Fig. 3, we can see that when we fix Δd and shift r1/a1, DMD does not change flexibly but DMD slope alters slowly. On the contrary, if we fix r1/a1 and shift Δd, DMD changes flexibly and DMD slope also alters slowly. The above-mentioned two approaches can both make DMD slope change, but only the second approach can help us control the DMD over the wide band. When r1/a1 is fixed, we can compensate the increment and decrement of DMD and DMD slope caused by altering r2/r1 via changing the value of Δd. This phenomenon implies that we can keep a suitable value for r1/a1 at first and then take advantage of both r2/r1 and Δd to find a reference point with relative low DMD and DMD slope. If we set r1/a1 to be 2.0, the Δd can be shifted from −0.17% to −0.11% so that the approximate range of DMD slope is −2 × 10−4 ~ + 2.7 × 10−4 ns/km/nm and that of DMD is −1 ~ + 1 ns/km. It indicts that r1/a1 of 2.0 is an appropriate design value, which make it possible for us to find suitable a1, Δ1 for inner core nearby the reference point — a1 of 3.6 µm and Δ1 of 0.5% to obtain both low absolute DMD and DMD slope under the condition that r2/r1 = 1.6 and Δd = −0.13%. Here, we do not analyze the impact of W/r1 and Δt on the DMD, which will be discussed in the following subsection.
Figure 4 shows DMD at λ = 1550 nm as function of a1 and Δ1 when r2/r1 = 1.6 and Δd = −0.13%. Here, we fixed the value for r1/a1, W/r1, and Δt, which are assumed as 2.0, 1.0, and −0.7%, respectively. Because different effective area (Aeff) in both cores will cause splice loss between different groups and different optical signal-to-noise ratio (OSNR) depending on the groups . In this work, we design two kinds of TA-MSI-FMCs, and it is very hard to ensure the Aeff of LP01 mode and LP11 mode to be the same in these two cores. Therefore in order to decrease such splice loss and OSNR as far as possible, we require the same Aeff of LP01 mode in both cores. Hence, we define the target value of Aeff of LP01 mode (Aeff_LP01) in both TA-MSI-FMCs to be 110 μm2. Through investigation we find that when Δd is around −0.13% under the condition that r2/r1 is 1.6, it is possible to find a1 and Δ1 which can make us obtain Aeff_LP01 of 110 μm2 and achieve low absolute value of DMD as well. In the Fig. 4, the black solid line and black dashed line represent Aeff_LP01 and effective index of LP01 mode (neff_LP01), which are both simulated based on full-vector FEM . The upper and lower white solid lines, white dashed lines and white dashed and dotted lines represent the cutoff of LP21 mode and the limit of LP11 mode at W/r1 of 0.2, 0.8 and 1.0, respectively. It should be noticed that W/r1 of 1.0 here is just an example value and the change of W/r1 will not influence the value of neff and Aeff to a large extent, but the two-mode operation region will shift as W/r1 alters. We set W/r1 to be 0.8 and 0.2 in order to make it probable to choose two sorts of TA-MSI-FMCs with same Aeff_LP01 of 110 μm2, low DMD and relative large difference between neff_LP01 in two TA-MSI-FMCs (Δneff_LP01). Here, to define the two-mode operation, the bending loss (BL) of LP21-like HOM should be > 1 dB/m at R = 140 mm, which is similar to the definition of BL of LP11-like HOM in  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-MSI-FMCs with low DMD and DMD slope in the two-mode operation regions at W/r1 of 0.2 and 0.8, which are shown as the filled circles in red and green in Fig. 4. For the filled circles in red which is designated as core 1, a1 = 3.81 μm, Δ1 = 0.406%, DMD at λ of 1550 nm is –160.90 ps/km and DMD slope at λ of 1550 nm is 0.27 ps/km/nm; For the filled circles in green which is designated as core 2, a1 = 3.92 μm and Δ1 = 0.458%, DMD at λ of 1550 nm is 168.30 ps/km and DMD slope at λ of 1550 nm is –0.27 ps/km/nm.
2.3 Impact of r2/r1 and Δd on DMD and DMD slope
As shown in Fig. 2, as r2/r1 shifts, DMD will decrease or increase. So in order to compensate for the decreased or increased DMD, we can find the solution in Fig. 3 that to alter the absolute Δd, which means that we can change r2/r1 and Δd to control the DMD. We analyzed and obtained the appropriate Δd for different r2/r1 that (a) r2/r1 = 1.3, Δd = −0.16%, (b) r2/r1 = 1.4, Δd = −0.15%, (c) r2/r1 = 1.5, Δd = −0.14%, (d) r2/r1 = 1.6, Δd = −0.13%, (e) r2/r1 = 1.7, Δd = −0.12%, (f) r2/r1 = 1.8, Δd = −0.11%.
Figure 5 shows DMD at λ = 1550 nm as function of a1 and Δ1 under these six situations. Here, we still assume r1/a1 to be 2.0 since in this case Δd that shifts within −0.17% ~−0.11% can guarantee the low absolute DMD and DMD slope. Moreover, we fixed W/r1 to be 1.0 and Δt to be −0.7% to simulate the DMD for the above-mentioned six situations. We also define the target Aeff_LP01 and Δneff_LP01 to be 110 μm2 and about 0.0008, and then we pick up six pairs of TA-MSI-FMCs with low DMD which are corresponding to the filled circles in Fig. 5.
Figure 6 illustrates the wavelength dependence of DMD for these six pairs of TA-MSI-FMCs. In Fig. 6, we can observe that as r2/r1 increases, the DMD slope is getting smaller and when it equals 1.6, the DMD slope is almost 0 ns/km/nm. Furthermore, when r2/r1 increases, the difference between the DMD in both cores becomes larger.
2.4 Impact of W/r1 and Δt on DMD and DMD slope
Here, we fixed the value for a1, Δ1, r1/a1 that are 3.6 µm, 0.5%, 2.0, respectively. Figures 7(a) and 7(b) show DMD and DMD slope dependence on W/r1. In Figs. 7(a) and 7(b), we can see that adjusting W/r1 is another way to control DMD and DMD slope, but when W/r1 becomes lager than ~0.5, it will not affect the DMD and DMD slope any more. Moreover, as r2/r1 increase, W/r1 has less impact on the DMD and DMD slope, which means that when the trench layer is deployed far away from the outer core, the thickness of trench will not influence the value of DMD and DMD slope. Figures 8(a) and 8(b) show DMD and DMD slope dependence on Δt. In Fig. 8(a), we can also observe a similar phenomenon that when r2/r1 is getting larger, the impact of Δt on the DMD will become smaller. In Fig. 8(b), we can see that when Δt increases, the DMD slope decreases no matter how we arrange the location of trench layer and design the refractive index of outer core.
In subsection 2.2, we have known that adjusting a1, Δ1, r1/a1, Δd, and r2/r1 can help us find the required TA-MSI-FMCs with low DMD, low DMD slope, and large Aeff. When these parameters are all set, we do not hope the both variables of trench structure — W/r1 and Δt affect the results too much. Instead, W/r1 and Δt can be used to control the bending loss of modes and the inter-core crosstalk. Therefore, we should deploy the trench layer far from the outer core, in other words, we need to increase r2/r1 as much as possible so that the change of W/r1 and Δt will not influence the DMD and DMD slope a lot. However, we can conclude from Fig. 6 that if r2/r1 becomes larger than 1.6, the maximum absolute DMD over C + L band will exceed |200| ps/km/nm, which is relative large value for DMD. Since MIMO system requires low DMD, we set |200| ps/km/nm to be the upper limit in this work. Hence, r2/r1 of 1.6 can be regarded as a relative optimum value, which can make TA-MSI-FMCs achieve relative low DMD over wide band and meanwhile have large tolerance of W/r1 and Δt. More interestingly, when we set r2/r1 to 1.6, we can obtain two TA-MSI-FMCs with positive and negative DMD (−160 and + 168 ps/km) whose absolute values are close to each other. This phenomenon implies that we can adopt DMD managed transmission line technique by using only one kind of Hetero-TA-FM-MCF and rotating one to splice different cores together to make the total DMD approach 0 ps/km over C + L band.
3. Layout of TA-MSI-FMCs in the fiber
According to the analyzation in section 2, we can treat that r2/r1 of 1.6, Δd of −0.13%, and r1/a1 of 2.0 as a relative optimum design scheme. Based on the design of these parameters, two kinds of TA-MSI-FMCs can be found which are shown in Fig. 4. The values of a1, Δ1, W/r1, Δt and the characteristics of effective index (neff), mode field diameter (MFD), effective area (Aeff), dispersion parameter, DMD and BL of core 1 and core 2 are summarized in Table 1. The difference between effective index of inter-modes (Δneff’) in core 1 and core 2 are 2.38 × 10−3 and 2.39 × 10−3, which proves that mode-coupling phenomena in core 1 and core 2 can be limited because both Δneff’ are larger than the critical value of 0.5 × 10−3 . On the other hand, the difference between effective index of LP01 mode in core 1 and effective index of LP01 mode in core 2 (Δneff_LP01) is about 0.0008. The difference between effective index of LP11 mode in core 1 and effective index of LP11 mode in core 2 (Δneff_LP11) is also about 0.0008.
3.1 The appropriate core pitch
As the proposals in our last works [11, 12], we arrange the TA-MSI-FMCs in a ring layout. If several layers of cores are set inside the fiber, the cut-off wavelengths of the interior cores tend to be longer than that of the exterior cores . This is due to the tight confinements in the interior cores which are caused by the trench structure deployed around each core. Moreover, excessive crosstalk degradation will also happen in the inner cores .
In order to accommodate as many cores as possible in the fiber, we should shorten the core pitch (Λ) to the largest extent and meanwhile make sure the low crosstalk and small Rpk. The crosstalk of Hetero-MCF decreases immediately after bending radius (R) reaching a critical value Rpk and then it converges to a certain value no matter how R increases . Therefore, we hope Rpk can be an extremely small value so that we can obtain a large non-phase-matching region with R > Rpk. In this non-phase-matching region, the bending extent doesn’t impact the crosstalk any more, which can make us design a kind of bend-insensitive Hetero-TA-FM-MCF.
Figure 9 shows inter-core LP01-LP01 crosstalk (XT01-01), LP01-LP11 crosstalk (XT01-11), LP11-LP01 crosstalk (XT11-01), and LP11-LP11 crosstalk (XT11-11) at λ = 1550 nm, R = 500 mm, and propagation length (L) = 100 km as function of core pitch. The reason why we set the R to be 500 mm is that this bending radius is much larger than the Rpk, which means that the crosstalk at R of 500-mm is the one in the bend-insensitive situation. In Fig. 9, we can find that XT11-11 is the largest crosstalk of the three kinds of inter-core crosstalk. The XT11-11 of less than −30 dB is realized when the core pitch becomes larger than about 37 µm. Figure 10 illustrates bending radius dependence of XT11-11 at λ = 1550 nm after 100-km propagation when Λ = 37 µm. As shown in Fig. 10, in the case that Λ = 37 µm, when R becomes larger than about 15 cm, the bend-insensitive characteristics of the Hetero-TA-FM-MCF in the practical applications can be guaranteed.
3.2 The appropriate core number
The cladding diameter (CD) can be determined by using the formula expressed as follows:3, 20]. Additionally, if we want to decrease the failure probability of a fiber in order to guarantee the mechanical reliability, CD should not be larger than 225 µm . The core pitch dependence of cladding diameter is shown as Fig. 11. From Fig. 11, we can find that when Λ equals 37 µm, Ncore of 12 is the upper limit and in this case CD is about 223 µm.
According to the redefinition of core multiplicity factor (CMF) for FM-MCF , CMF for Hetero-FM-MCF can be proposed as follows:
RCMF is a ratio between CMF of a FM-MCF and a standard single core single mode fiber with Aeff = 80 µm2 at 1550 nm and CD = 125 µm, which is shown as22], the RCMF can be further enhanced to be 26.1 which exceeds the record value of 14.8 for a two-LP mode seven-core fiber . Hence, we can design a kind of two-LP mode Hetero-TA-MCF with Ncore of 12, Λ of 37 µm, OCT of 40 µm, CD of 223 µm, RCMF of 15.7 for 24 special paths, and RCMF of 26.1 for 36 special paths.
Figure 12 shows the relationship between inter-core crosstalk (XT) and RCMF for the reported single-mode MCFs (SM-MCFs) [11, 12, 20, 21] and few-mode MCFs (FM-MCFs) . In Fig. 12, the black circles represent the SM-MCFs with only LP01 mode, the blue triangles mean the FM-MCFs with LP01 mode and LP11 mode, and the blue squares stand for the FM-MCFs with LP01 mode, LP11a mode and LP11b mode. It is obvious that the Hetero-TA-FM-12-core fiber presented in this work has the largest RCMF.
To design a TA-MSI-FMC, there are seven parameters that determine the profile — a1, Δ1, r1/a1, Δd, r2/r1, W/r1, and Δt. After analyzing how the r2/r1 and Δd affect the DMD and DMD slope for wavelength, we found that as r2/r1 changes, DMD and DMD slope will alter correspondingly and we can shift the absolute value of Δd to compensate the increment or decrement of the DMD and DMD slope. Furthermore, as r2/r1 increases, the maximum absolute DMD in two TA-MSI-FMCs over C + L band is getting larger but the impact of W/r1 and Δt on the DMD will become smaller. As a result, r2/r1 of 1.6 is regarded as a relative optimum design value since we should make sure not only the low DMD over wide band but also large tolerance of W/r1 and Δt.
For the application of fiber, we propose two kinds of strategies. In the first strategy, we use a single heterogeneous few-mode multi-core fiber (Hetero-FM-MCF) to transmit the signal for the long-haul transmission because of the easy deployment. The DMD of each core is about |170| ps/km over C + L band which is not so small for MIMO processing, but the absolute DMD can be further decreased by increasing Rpk and decreasing Aeff. In the second strategy, we only use one kind of Hetero-FM-MCF and rotate one to splice different cores together to form DMD-managed transmission line so that the total DMD of almost 0 ps/km can be achieved over C-L band.
After investigating the characteristics of crosstalk, Rpk, cladding diameter, and RCMF, we can design a kind of two-LP mode Hetero-TA-12-core fiber with XT of about –30 dB/100km at λ of 1550 nm as R becomes larger than 15 cm, RCMF of 15.7 for 24 special paths, and RCMF of 26.1 for 36 special paths. Actually, when the wavelength becomes larger, we can also obtain the low XT by scarifying Rpk and RCMF respectively.
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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