## Abstract

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 *A*_{eff} of 110 μm^{2} for LP_{01} 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 LP_{11} mode as two special modes thanks to the multiple-input-multiple-output technology.

© 2014 Optical Society of America

## 1. Introduction

Several multiplexing technologies such as space-division multiplexing (SDM) using multi-core fiber (MCF) [1] and mode-division multiplexing (MDM) using few-mode fiber (FMF) [2] 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 [5]. To realize low DMD, the 1st approach is using complex refractive index profile, such as multi-step index profile [6], graded index [7], and graded index profile with trench (T-GIP) [8], 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 [8]. 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 [10].

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 — LP_{01} mode and LP_{11} 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) [13]. 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 LP_{01} mode and LP_{11} mode, since the DMD characteristics are sensitive to the change of the refractive index profile. In Fig. 1, *a*_{1}, *r*_{1}, *r*_{2}, *W*, Δ_{1}, Δ_{2} and Δ_{t} 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 *a*_{1}, Δ_{1}, *r*_{1}/*a*_{1}, Δ_{d}, *r*_{2}/*r*_{1}, *W/r*_{1}, and Δ_{t} to obtain a couple of TA-MSI-FMCs with low DMD, low DMD slope, small inter-core crosstalk, and large effective area (*A*_{eff}).

#### 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 (τ_{LP01}) from that of the higher-order mode (τ_{LP11}) and the expression of DMD is written as follows:

*c*is the light velocity in a vacuum,

*n*

_{eff}is the effective index, and λ means free space wavelength. Since we should also ensure the low DMD over C + L bands transmission, we need to design two kinds of TA-MSI-FMCs with not only low DMD at a certain operating wavelength but also low DMD slope for the wavelength (λ).

Figure 2 shows DMD and DMD slope as function of *r*_{2}/*r*_{1} at λ = 1550 nm. Here, we fixed the value for *a*_{1}, Δ_{1}, *r*_{1}/*a*_{1}, Δ_{d}, *W/r*_{1}, and Δ_{t}, 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 *r*_{2}/*r*_{1} increases, DMD and DMD slope are getting smaller and when *r*_{2}/*r*_{1} = 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 *r*_{1}/*a*_{1} and Δ_{d} at λ = 1550 nm. Here, we assumed *a*_{1}, Δ_{1}, *r*_{2}*/r*_{1}, *W/r*_{1}, and Δ_{t} 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 Δ_{d} and shift *r*_{1}/*a*_{1}, DMD does not change flexibly but DMD slope alters slowly. On the contrary, if we fix *r*_{1}/*a*_{1} and shift Δ_{d}, 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 *r*_{1}/*a*_{1} is fixed, we can compensate the increment and decrement of DMD and DMD slope caused by altering *r*_{2}*/r*_{1} via changing the value of Δ_{d}. This phenomenon implies that we can keep a suitable value for *r*_{1}/*a*_{1} at first and then take advantage of both *r*_{2}/*r*_{1} and Δ_{d} to find a reference point with relative low DMD and DMD slope. If we set *r*_{1}/*a*_{1} to be 2.0, the Δ_{d} 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 *r*_{1}/*a*_{1} of 2.0 is an appropriate design value, which make it possible for us to find suitable *a*_{1}, Δ_{1} for inner core nearby the reference point — *a*_{1} of 3.6 µm and Δ_{1} of 0.5% to obtain both low absolute DMD and DMD slope under the condition that *r*_{2}/*r*_{1} = 1.6 and Δ_{d} = −0.13%. Here, we do not analyze the impact of *W/r*_{1} and Δ_{t} on the DMD, which will be discussed in the following subsection.

Figure 4 shows DMD at λ = 1550 nm as function of *a*_{1} and Δ_{1} when *r*_{2}/*r*_{1} = 1.6 and Δ_{d} = −0.13%. Here, we fixed the value for *r*_{1}/*a*_{1}, *W/r*_{1}, and Δ_{t}, which are assumed as 2.0, 1.0, and −0.7%, respectively. Because different effective area (*A*_{eff}) in both cores will cause splice loss between different groups and different optical signal-to-noise ratio (OSNR) depending on the groups [14]. In this work, we design two kinds of TA-MSI-FMCs, and it is very hard to ensure the *A*_{eff} of LP_{01} mode and LP_{11} 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 *A*_{eff} of LP_{01} mode in both cores. Hence, we define the target value of *A*_{eff} of LP_{01} mode (*A*_{eff_LP01}) in both TA-MSI-FMCs to be 110 μm^{2}. Through investigation we find that when Δ_{d} is around −0.13% under the condition that *r*_{2}/*r*_{1} is 1.6, it is possible to find *a*_{1} and Δ_{1} which can make us obtain *A*_{eff_LP01} of 110 μm^{2} and achieve low absolute value of DMD as well. In the Fig. 4, the black solid line and black dashed line represent *A*_{eff_LP01} and effective index of LP_{01} mode (*n*_{eff_LP01}), which are both simulated based on full-vector FEM [15]. The upper and lower white solid lines, white dashed lines and white dashed and dotted lines represent the cutoff of LP_{21} mode and the limit of LP_{11} mode at *W*/*r*_{1} of 0.2, 0.8 and 1.0, respectively. It should be noticed that *W*/*r*_{1} of 1.0 here is just an example value and the change of *W*/*r*_{1} will not influence the value of *n*_{eff} and *A*_{eff} to a large extent, but the two-mode operation region will shift as *W*/*r*_{1} alters. We set *W*/*r*_{1} to be 0.8 and 0.2 in order to make it probable to choose two sorts of TA-MSI-FMCs with same *A*_{eff_LP01} of 110 μm^{2}, low DMD and relative large difference between *n*_{eff_LP01} in two TA-MSI-FMCs (Δ*n*_{eff_LP01}). Here, to define the two-mode operation, the bending loss (*BL*) of LP_{21}-like HOM should be > 1 dB/m at *R* = 140 mm, which is similar to the definition of *BL* of LP_{11}-like HOM in [16] and we assume the limit value of the *BL* of LP_{11}-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 *R*_{pk} which is a critical value of bending radius [17], we define the required Δ*n*_{eff_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*/*r*_{1} 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, *a*_{1} = 3.81 μm, Δ_{1} = 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, *a*_{1} = 3.92 μm and Δ_{1} = 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 r_{2}/r_{1} and Δ_{d} on DMD and DMD slope

As shown in Fig. 2, as *r*_{2}/*r*_{1} 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 Δ_{d}, which means that we can change *r*_{2}/*r*_{1} and Δ_{d} to control the DMD. We analyzed and obtained the appropriate Δ_{d} for different *r*_{2}/*r*_{1} that (a) *r*_{2}/*r*_{1} = 1.3, Δ_{d} = −0.16%, (b) *r*_{2}/*r*_{1} = 1.4, Δ_{d} = −0.15%, (c) *r*_{2}/*r*_{1} = 1.5, Δ_{d} = −0.14%, (d) *r*_{2}/*r*_{1} = 1.6, Δ_{d} = −0.13%, (e) *r*_{2}/*r*_{1} = 1.7, Δ_{d} = −0.12%, (f) *r*_{2}/*r*_{1} = 1.8, Δ_{d} = −0.11%.

Figure 5 shows DMD at λ = 1550 nm as function of *a*_{1} and Δ_{1} under these six situations. Here, we still assume *r*_{1}/*a*_{1} to be 2.0 since in this case Δ_{d} that shifts within −0.17% ~−0.11% can guarantee the low absolute DMD and DMD slope. Moreover, we fixed *W/r*_{1} to be 1.0 and Δ_{t} to be −0.7% to simulate the DMD for the above-mentioned six situations. We also define the target *A*_{eff_LP01} and Δ*n*_{eff_LP01} to be 110 μm^{2} 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 *r*_{2}/*r*_{1} increases, the DMD slope is getting smaller and when it equals 1.6, the DMD slope is almost 0 ns/km/nm. Furthermore, when *r*_{2}/*r*_{1} increases, the difference between the DMD in both cores becomes larger.

#### 2.4 Impact of W/r_{1} and Δ_{t} on DMD and DMD slope

Here, we fixed the value for *a*_{1}, Δ_{1}, *r*_{1}/*a*_{1} that are 3.6 µm, 0.5%, 2.0, respectively. Figures 7(a) and 7(b) show DMD and DMD slope dependence on *W*/*r*_{1}. In Figs. 7(a) and 7(b), we can see that adjusting *W*/*r*_{1} is another way to control DMD and DMD slope, but when *W*/*r*_{1} becomes lager than ~0.5, it will not affect the DMD and DMD slope any more. Moreover, as *r*_{2}/*r*_{1} increase, *W*/*r*_{1} 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 Δ_{t}. In Fig. 8(a), we can also observe a similar phenomenon that when *r*_{2}/*r*_{1} is getting larger, the impact of Δ_{t} on the DMD will become smaller. In Fig. 8(b), we can see that when Δ_{t} 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 *a*_{1}, Δ_{1}, *r*_{1}/*a*_{1}, Δ_{d}, and *r*_{2}/*r*_{1} can help us find the required TA-MSI-FMCs with low DMD, low DMD slope, and large *A*_{eff}. When these parameters are all set, we do not hope the both variables of trench structure — *W*/*r*_{1} and Δ_{t} affect the results too much. Instead, *W*/*r*_{1} and Δ_{t} 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 *r*_{2}/*r*_{1} as much as possible so that the change of *W*/*r*_{1} and Δ_{t} will not influence the DMD and DMD slope a lot. However, we can conclude from Fig. 6 that if *r*_{2}/*r*_{1} 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, *r*_{2}/*r*_{1} 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*/*r*_{1} and Δ_{t}. More interestingly, when we set *r*_{2}/*r*_{1} 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 *r*_{2}/*r*_{1} of 1.6, Δ_{d} of −0.13%, and *r*_{1}/*a*_{1} 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 *a*_{1}, Δ_{1}, *W*/*r*_{1}, Δ_{t} and the characteristics of effective index (*n*_{eff}), mode field diameter (MFD), effective area (*A*_{eff}), dispersion parameter, DMD and *BL* of core 1 and core 2 are summarized in Table 1. The difference between effective index of inter-modes (Δ*n*_{eff}’) 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 Δ*n*_{eff}’ are larger than the critical value of 0.5 × 10^{−3} [2]. On the other hand, the difference between effective index of LP_{01} mode in core 1 and effective index of LP_{01} mode in core 2 (Δ*n*_{eff_LP01}) is about 0.0008. The difference between effective index of LP_{11} mode in core 1 and effective index of LP_{11} mode in core 2 (Δ*n*_{eff_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 [18]. 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 [19].

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 *R*_{pk}. The crosstalk of Hetero-MCF decreases immediately after bending radius (*R*) reaching a critical value *R*_{pk} and then it converges to a certain value no matter how *R* increases [17]. Therefore, we hope *R*_{pk} can be an extremely small value so that we can obtain a large non-phase-matching region with *R* > *R*_{pk}. 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 LP_{01}-LP_{01} crosstalk (*XT*_{01-01}), LP_{01}-LP_{11} crosstalk (*XT*_{01-11}), LP_{11}-LP_{01} crosstalk (*XT*_{11-01}), and LP_{11}-LP_{11} crosstalk (*XT*_{11-11}) 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 *R*_{pk}, 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 *XT*_{11-11} is the largest crosstalk of the three kinds of inter-core crosstalk. The *XT*_{11-11} of less than −30 dB is realized when the core pitch becomes larger than about 37 µm. Figure 10 illustrates bending radius dependence of *XT*_{11-11} 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:

*N*

_{core}is the number of core and the outer cladding thickness (

*OCT*) is the radial distance between the center of the outer core and the cladding edge. In order to reduce the micro-bending loss, the

*OCT*has different required minimum value corresponding to the different

*A*

_{eff}. Here, we set the

*OCT*to be at least 40 µm [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 [21]. The core pitch dependence of cladding diameter is shown as Fig. 11. From Fig. 11, we can find that when Λ equals 37 µm,

*N*

_{core}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 [3], *CMF* for Hetero-FM-MCF can be proposed as follows:

*A*

_{eff-}

*is effective area of*

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

*p*,

*A*

_{eff-}

*is effective area of*

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

*q*,

*l*is the number of mode,

*CD*

_{FM-MCF}is the cladding diameter of FM-MCF.

*RCMF* is a ratio between *CMF* of a FM-MCF and a standard single core single mode fiber with *A*_{eff} = 80 µm^{2} at 1550 nm and *CD* = 125 µm, which is shown as

*RCMF*of the two-LP mode Hetero-TA-12-core fiber (whose

*A*

_{eff-1}of LP

_{01}mode and LP

_{11}mode are ~110 µm

^{2}and ~225 µm

^{2};

*A*

_{eff-2}of LP

_{01}mode and LP

_{11}mode are ~110 µm

^{2}and ~219 µm

^{2}) is 15.7. If we use the degenerated LP

_{11}mode as two different special modes thinks to MIMO technology [22], 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 [3]. Hence, we can design a kind of two-LP mode Hetero-TA-MCF with

*N*

_{core}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) [3]. In Fig. 12, the black circles represent the SM-MCFs with only LP_{01} mode, the blue triangles mean the FM-MCFs with LP_{01} mode and LP_{11} mode, and the blue squares stand for the FM-MCFs with LP_{01} mode, LP_{11a} mode and LP_{11b} mode. It is obvious that the Hetero-TA-FM-12-core fiber presented in this work has the largest *RCMF*.

## 4. Conclusion

To design a TA-MSI-FMC, there are seven parameters that determine the profile — *a*_{1}, Δ_{1}, *r*_{1}/*a*_{1}, Δ_{d}, *r*_{2}/*r*_{1}, *W/r*_{1}, and Δ_{t}. After analyzing how the *r*_{2}/*r*_{1} and Δ_{d} affect the DMD and DMD slope for wavelength, we found that as *r*_{2}/*r*_{1} changes, DMD and DMD slope will alter correspondingly and we can shift the absolute value of Δ_{d} to compensate the increment or decrement of the DMD and DMD slope. Furthermore, as *r*_{2}/*r*_{1} increases, the maximum absolute DMD in two TA-MSI-FMCs over C + L band is getting larger but the impact of *W/r*_{1} and Δ_{t} on the DMD will become smaller. As a result, *r*_{2}/*r*_{1} 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*/*r*_{1} and Δ_{t}.

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 *R*_{pk} and decreasing *A*_{eff}. 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, *R*_{pk}, 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 *R*_{pk} and *RCMF* respectively.

## Acknowledgments

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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