## Abstract

We present a novel dense hole-assisted structure (DHAS) in homogeneous few-mode multi-core fibers (FM-MCFs) for significantly suppressing inter-core crosstalk (XT) and bending loss while realizing a high spatial density in limited cladding diameter. The fabrication methods of DHAS FM-MCFs are illustrated and the equivalent model of DHAS is proposed. To point out the superiority, the XT of a DHAS 7-core 4-LP-mode fiber is investigated by an average power-coupling coefficients analytical expression in DHAS model and the equivalent model, respectively. A simple derived analytical expression for XT estimation in the trench-assisted homogeneous MCFs is introduced to verify the change of XT with fiber parameters in the equivalent model. The results imply that the XT obtained by the equivalent model is in good agreement with the one through DHAS model. Furthermore, the bending loss and chromatic dispersion dependences on DHAS are calculated by the finite element method (FEM). Through numerical simulations, we show that the DHAS has great contribution to meet the XT value requirement of lower than -30 dB/100km and make the bending loss values satisfy the ITU-T recommendations of G. 654 in a 7-core 4-LP-mode fiber with a 125-μm cladding diameter. The designed structure targets applications in space division multiplexing (SDM) fibers with independent transmission channels.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The use of space as an additional multiplexing dimension, known as space division multiplexing (SDM), is of great interest as a method to increase the number of independent data channels and thus expanding the transmission capacity of a single mode fiber. Weakly-coupled multi-core fibers (MCFs) based on SDM technique have received enough attention since it can effectively use the space within a fiber cladding [1–4].

The main issue of high capacity transmission in MCFs is how can we increase the numbers of cores while keeping the inter-core crosstalk (XT) low to allow higher multi-level modulation formats and longer transmission distance [5–8]. As we know, more spatial paths inevitably lead to serious modes coupling. A couple of ways have been developed to address this problem. The approach of differential core-cladding index between neighbor cores in heterogeneous MCFs is proposed, which uses phase mismatch to achieve lower XT than homogeneous MCFs [9–11]. An optimized design scheme of heterogeneous trench-assisted 14-core fiber has been proposed, which reached XT of lower than -50 dB/100km [10]. Some special fiber structures are also demonstrated in these years. The trench-assisted structure, based on its strong mode confinement capability, was applied to each core in MCFs for XT reduction [12–15]. Soon after, hole-assisted structure is designed to further improve the core isolation because the largest refractive index contrast exists between the cores and the air-holes [16,17]. The fabricated hole-assisted multicore fiber shows XT of below -60 dB at a distance of approximately 1 km experimentally [17]. Rod-assisted structure, as an alternative way of trench-assisted structure, has been investigated to achieve XT of about -31 dB/100km in a 32-core fiber [18].

Unfortunately, as the cores and supporting modes increase in number, it is getting more difficult to suppress XT in MCFs by heterogeneous cores, trenches, rods or air holes assisted structures. Additionally, the transmission MCFs should be designed with other necessary properties. Random bending tends to occur in the actual use of the fiber, and the control of bending loss has become one of the major concerns in practical applications [19]. The cut-off wavelength and chromatic dispersion in MCFs should also be optimized to keep stable signal transmission in the corresponding band. Researches show that the MCFs with 125 $\mathrm{\mu}\textrm{m}$ cladding diameter is best suited for current optical fiber systems [20]. As a result, an ideal weakly-coupled MCF should be designed with powerful light confinement ability while keeping a large number of spatial channels in a 125-$\mathrm{\mu}\textrm{m}$ cladding range.

In this paper, we present a novel dense hole-assisted structure (DHAS) in homogeneous FM-MCFs to significantly enhance the ability of mode fields restriction among individual cores, thereby reducing XT and making the fiber bend insensitive. COMSOL Multiphysics are used to simulate the modes and then the data are integrated to MATLAB for analyzing. Firstly, the fabrication methods of DHAS are introduced and the equivalent calculation model of DHAS is proposed in section 2 to explain our designing ideas. Then, in order to understand the XT dependence on DHAS parameters, the power-coupling coefficients expression for mean XT estimation in DHAS model and the equivalent model are investigated, respectively. A derived analytical formula for XT estimation in the trench-assisted homogeneous MCFs is introduced to verify the change of XT with fiber parameters in the equivalent model. From section 3, the results come from these two models agree very well. They reveal that the DHAS can reach the XT standard of lower than -30 dB/100km for transmission fibers. Further, section 4 discusses the impact of DHAS on other fiber properties by finite element method (FEM). Through numerical simulations, we confirm that the DHAS makes it possible to design a 7-core 4-LP-mode MCF with satisfied XT and bending loss values, appropriate cut-off wavelength and dispersion in a 125-$\mathrm{\mu}\textrm{m}$ cladding diameter. Moreover, the spatial density in our proposed DHAS FM-MCF evaluated by the relative core multiplicity factor (RCMF) can reach 84.92, due to the massive spatial channels within the standard fiber cladding diameter. Finally, we briefly discuss the fabrication tolerance of the DHAS and it shows limited impact on fiber performance.

## 2. DHAS and its equivalent model

The proposed DHAS is created by means of differential diameter air-holes in which the large diameter air-holes is surrounded by multiple small ones, as shown in Fig. 1. Several sides of junction are reserved as cladding aiming at increasing the core unit strength. The particularity of our design is that the differential air-holes form a groove-like structure around the core, which can evidently prevent the leaking of core energy and isolate the light from other cores. More importantly, this groove-like DHAS has the maximum index difference to the cores, thus it exhibits better light confinement capability than the trench-assisted and rod-assisted structures. What’s more, compared with normal hole-assisted structure, the DHAS features a more comprehensive structural advantage.

While the above mentioned benefits of DHAS are unquestionable, some downsides of such a structure also need to be considered. First, the DHAS may modify propagation characteristics such as chromatic dispersion. Then, a second waveguide may also be created by the DHAS, named as cladding modes [17]. The discussion of dispersion and cladding modes is unneglectable, and we will extend the analysis of dispersion and the confinement loss of cladding modes in section 4.

Figure 2 shows the equivalent calculation model of the DHAS, where *r _{1}*,

*r*,

_{2}*w*, ${\Delta _1}$ and

*p*are the core radius, the distance between core and DHAS, the width of DHAS, the index difference between core and cladding and the width of junction, respectively. The reason why we made this equivalent is that the differential air-holes around the cores are dense enough, and the air-groove model can sufficiently replace the DHAS for following calculations. The step-index cores supported 4-LP-mode (LP

_{01}, LP

_{11}, LP

_{21}and LP

_{02}) are deployed.

Considered the mechanical reliability, the fabrication methods of such FM-MCF with DHAS is worth noting. Stack-and-draw technique is one of the suitable ways to fabricate DHAS MCFs, according to the practical technology and experience [17,21–23]. For example, a hole-assisted 7-core few-mode fiber has been successfully demonstrated by stack-and-draw technique and data transmission over 1 km of this fiber has been demonstrated with negligible penalty [16]. Figure 3 shows the partial schematic diagram of a stacked preform assembly of the DHAS FM-MCF, where the hollow circles with different sizes (glass capillary) correspond to the differential air-holes. The geometry specifications of the GeO_{2}-doped preforms, glass rods and glass capillaries can be determined by the ratio of various structural parameters to the cladding diameter, according to the actual requirements of different commercial drawing devices. After that, the fiber can be fabricated after melting and drawing process. Another approach to make this fiber, as mentioned in Ref. [24], is 3D printed silica die. The 3D printing is a unique technology that can be used to fabricate micro-structured optical fiber via soft glass preform. The combination of 3D printed dies and structured capillary stacking allows for precisely control of the DHAS FM-MCFs fabrication. In terms of the examples of the fibers made before, we believe that the above methods can be used to make DHAS MCFs with our target goals. The impact of fabrication tolerance on fiber performance will be given in section 4.

## 3. XT charateristics

Since the air-groove structure is utilized as the equivalent model of DHAS, it is essential to evaluate the change of XT with fiber parameters in the DHAS model and the equivalent model. The average power-coupling coefficients in the two models are simulated, which is one of the most common ways for mean XT estimation [25]. For homogeneous MCFs, the average power-coupling coefficients can be written as [26]:

where $k_{mn}^{\prime}$ is the mode coupling coefficient in electromagnetic form and ${R_b}$ is the bending radius. The $k_{mn}^{\prime}$ is written as [10]:*n*, * denotes the complex conjugate and ${u_z}$ is a unit vector.

Using the average power-coupling coefficients, the XT between two cores with length L is easily estimated as [25]

In order to prove the reliability of the air-groove in the equivalent model, we apply a derived analytical expression for the XT estimation in homogeneous trench-assisted MCFs [26]. Note that this method has a unique simple and convenient advantage in calculating the XT without the need of numerical simulations in normal step-index MCFs, we use it to illustrate the XT of fundamental mode in the equivalent model in comparison to the XT values obtained by the mean XT calculation method. According to the Ref. [26], the mode coupling coefficient between two neighboring cores (core *m* and core *n*) with trench-assisted structures can be expressed as:

^{nd}kind with 1

^{st}order, $\mathrm{\Lambda}$ is the core pitch, $k = 2\pi /\lambda $ is the wave number, λ is the operation wavelength, $\beta $ is the propagation constant. The refractive indices for the core, cladding and trench are ${n_{core}}$, ${n_{clad}}$ and ${n_{trench}}$, respectively.

Figure 4 shows the cross-sectional view of a hexagonally arranged DHAS 7-core 4-LP-mode fiber, where *D _{cl}* and OCT represents cladding diameter and outer cladding thickness, respectively. The cores are numbered clockwise and the initial fiber parameters are listed in Table 1. The core parameters are given to keep 4-LP-mode operation, and the parameters of the equivalent model will be adjusted.

The threshold of XT here is set to be -30 dB/100 km, which can satisfy the XT required for the long-haul transmission [14,19,27–30]. Limited by the fluorine-doped technology, the trench-cladding index contrast can reach as low as -0.7%. Therefore, $\varLambda $ of larger than 40 μm is of necessary to suppress XT in trench-assisted FM-MCFs [14]. We believe that the DHAS can alter this trend. The XT as the function of $\varLambda $ is investigated in Fig. 5. The ${R_b}$ is assumed to be 80 mm, which is the same as that of the fiber spool. It can be observed that the XT of LP_{02} mode is less than -30 dB/100 km when $\varLambda $ of larger than 32 μm. Figures 6(a) and (b) show the relationship between XT and DHAS parameters. The XT degrades linearly as *W* increases, and a lower XT can be obtained with the decrease of *p*. In other words, the smaller junction width, the less energy will be leaked out.

Bending has great influence on XT, we can utilize the phase mismatch induced by the bend for suppressing XT in homogeneous MCFs [31]. The dependence of XT on ${R_b}$ with the $\varLambda $ of 35 μm at 1550 nm has been considered. As shown in Fig. 7, the XT increases along with the increase of ${R_b}$, and it is estimated to be lower than -30 dB/100 km when ${R_b}$ ≤ 500 mm. Note that a small ${R_b}$ results in a larger difference of propagation constant between the adjacent cores for homogeneous MCFs. As ${R_b}$ increases, this difference will become smaller and smaller, making the XT increase slowly with larger ${R_b}$. When the ${R_b}$ is smaller than 100 mm, the XT values increases sharply with the enlarging ${R_b}$, due to the rapidly decreasing propagation constant difference. However, as ${R_b}$ continues to be larger, the trend of increasing XT becomes slower and the curves become relatively flatter. We believe that when the ${R_b}$ is greater than 100 mm, the influence of bending on XT will become limited.

However, due to the presence of random structural fluctuations in the longitudinal direction of a real fiber, the XT will eventually saturate to a constant XT value even for a homogeneous MCFs. This threshold bending radius is usually on the order of several centimeters, and all the modes have almost the same correlation length. Furthermore, the XT values of all the 4-LP mode between adjacent cores (such as core 1 to core 3) at 1550 nm when ${R_b}$ = 80 mm have been summarized in Table 2. Note that the XT estimation based on the DHAS model and on the equivalent model are in good agreement as they almost overlap with each other in the simulation, and the XT variation of LP_{01} mode in the equivalent model calculated by the derivation formula Eq. (4) is almost consistent with the above results.

## 4. Bending loss and other fiber properties

The bending loss in the outer cores of MCFs is a major and complex task which requires insight study. Considering the results that the XT of lower than -30 dB/100km can be reached in the DHAS FM-MCF with $\varLambda $ = 35 μm so that it is possible to design a MCF with 125-μm cladding diameter due to the good mode fields confinement performance of DHAS. Here, the ITU-T recommendations G. 654 is adopted as the standard of bending loss in MCFs and the finite element method (FEM) is applied [9]. To keep 4-LP-mode operation in C band, the bending loss should be lower than 0.5 dB/100 turns (${R_b}$ = 30 mm) for LP_{02} at 1565 nm while it should be larger than 1 dB/m (${R_b}$ = 140 mm) for unwanted mode (LP_{31} mode) at 1530 nm. The bending loss can be calculated by [32]:

The bending loss of the outer cores with DHAS are simulated with different *W*, *p* and OCT in Figs. 8(a) and (b), respectively. The black and purple lines correspond to the bending loss of LP_{02} after 100 turns at 1565 nm (${R_b}$ = 30 mm) and the bending loss of LP_{31} after 1-m transmission at 1530 nm (${R_b}$ = 140 mm), respectively. From Fig. 8(a), *W* > 3.2 μm is needed to satisfy the demand of BL < 0.5 dB/100 turns of LP_{02}, while it should be smaller than 5.4 μm to keep bending loss of unwanted mode larger than 1 dB/m. Here, the width of DHAS can be set as 4.0 μm in the consideration of the XT and bending loss values. In Fig. 8(b), the OCT of smaller than 26.5 μm no longer meets the bending loss target. As a result, the trade-off relationship between $\varLambda $ and OCT can be balanced, the $\varLambda $ = 35 μm and OCT = 27.5 μm is acceptable.

Due to multiple layers of air-holes, as mentioned in section 2, leaking of the light may excites cladding modes between the core and DHAS. In addition to enhancing the fiber strength, another role of the junction is to achieve high cladding modes loss. Three junctions can effectively leak the cladding modes, and more junctions may weaken the mode field confinement ability of DHAS. Figure 9 shows the propagation modes and possible cladding modes in the DHAS FM-MCF. It is found that the cladding modes between core and DHAS only exist when *r _{2}* is much larger than

*r*. When

_{1}*r*is as similar as

_{2}*r*, the cladding modes disappears regardless of the parameters of the DHAS. Thus, we calculate the relationship between the confinement loss of cladding modes and

_{1}*r*in Fig. 10. As shown, they are of over 10

_{2}^{4}dB/m ensure cladding modes high suppression when

*r*is smaller than twice of

_{2}*r*. Specifically, the ${\lambda _{cc}}$ here is defined as the wavelength at which the bending loss of LP

_{1}_{31}mode equals 1 dB/m at ${R_b}$ = 140 mm. The ${\lambda _{cc}}$ of central core are calculated because it has much higher ${\lambda _{cc}}$ values than that of outer cores. The influence of DHAS parameters on ${\lambda _{cc}}$ is shown in Fig. 11. It is estimated to be lower than 1510 nm when

*W*≤ 5 μm.

Chromatic dispersion is a major factor causing optical pulse broadening in the transmission fibers. It is necessary to evaluate the effect of DHAS on fiber dispersion. The dispersion is composed of two components: material dispersion *D _{m}* and waveguide dispersion

*D*, which can be written as Eq. (7), where

_{w}*c*is the velocity of light in a vacuum,

*n*is dependent on

_{M}*λ*in dispersive media,

*Re(n*is the real part of the ${n_{eff}}$, respectively [33].

_{eff})Material dispersion refers to the wavelength dependence of the refractive index of material caused by the interaction between the optical modes and the state of material. The refractive indices of SiO_{2} and GeO_{2}-SiO_{2} at different wavelengths can be found in Ref. [33]. Waveguide dispersion depends among others on the core and DHAS parameters. The degree of influence of DHAS on waveguide dispersion is an important part in determining the total dispersion in our design. As one can see in Fig. 12, the slope of the dispersion curve of LP_{01} mode changes insignificantly with the added DHAS. The dispersion values of other higher-order modes follow the similar trend with the fundamental mode as the increase of the wavelength, and they will be summarized in Table 4.

As demonstrated above, the DHAS helps to achieve low XT and bending loss in a 7-core 6-mode fiber within 125-um cladding diameter, and leaves no significant change on dispersion performance. However, this not all-solid structure puts a harsh requirement on the fabricating accuracy. The major challenge faced in the manufacturing process is the occlusion of the air-holes.

The occlusion of the air-holes seems to be inevitable in the fabrication of hole-assisted fibers and it may change fiber performance to some extent. In our proposed DHAS, each large air-hole surrounded by multiple small air-holes to form a groove-like structure. Several closed holes among the dense enough air-holes structure will not alter the groove-like structure so much that the occlusion of air-holes cause limited sacrifice on the light confinement ability of DHAS.

Note that the fabrication errors may cause the not perfectly smooth interface between silica and air-holes, which can result in scattering. We simulate the normalized amplitude variation of the proposed fiber along the normalized radius in Fig. 13. The most power of propagation modes is distributed within the normalized radius of 1.4 and it is weak enough in the boundary of the DHAS (≥ 1.54). Furthermore, there is no power abruption at the surface of DHAS and silica, which means the structure roughness will not generate scattering effectively. In this case, the light confinement ability of the cores is greatly related to the presence of DHAS, even if they may be slightly distorted by fabrication imperfections.

The core multiplicity factor (CMF) is one of the parameters to define the spatial density in MCFs. In this paper, we evaluate the spatial density of the DHAS 7-core 6-mode fiber using the relative core multiplicity factor (RCMF), which is the CMF normalized by that for standard single-mode fiber [20]. It is worth noting that an FM-MCF with both acceptable spatial channels of 42 and more than 80 RCMF values can be achieved by employing the DHAS in the standard fiber cladding diameter. Finally, we summarize the optimal fiber parameters with fabrication tolerance and performance of DHAS 7-core 4-LP-mode fiber in Table 3 and Table 4, respectively.

## 5. Conclusion

We propose a novel DHAS in MCFs to dramatically improve the light confinement ability of cores. According to former researches and experience, stack-and-draw technique and 3D printing are acceptable as the fabrication methods of our design. The groove-like equivalent model for DHAS is proposed for numerical simulation. To show the talents of DHAS on XT suppression, the mean XT estimating method by average power-coupling coefficients is applied in DHAS model and equivalent model separately. A derived analytical expression for XT estimating in the trench-assisted homogeneous MCFs is introduced in the equivalent model to verify the change of XT with fiber parameters. The results of two models agree very well and they show that the DHAS can realize a XT of lower than -30 dB/100 km in a 7-core 4-LP-mode homogenous fiber when ${\boldsymbol{\varLambda} }$ = 35 μm. Furthermore, the major fiber properties, including bending loss, cut-off wavelength and chromatic dispersion dependences on DHAS parameters are calculated by the FEM. Through numerical simulations, we show that the DHAS makes it possible to design a 125-μm 7-core 4-LP-mode fiber with satisfied XT (lower than -30 dB/100 km for transmission MCFs) and bending loss (meets the ITU-T recommendations G. 654), appropriate dispersion and high density of cores. With the viewpoints of these properties, we believe that the DHAS can be used in SDM fiber designs that require independent signal channels.

## Funding

National Natural Science Foundation of China (61827817).

## Disclosures

The authors declare no conflicts of interest.

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