Design and Characterization of Nanopore- Assisted Weakly-Coupled Few-Mode Fiber for Simpler MIMO Space Division Multiplexing

We report on a novel weakly-coupled 6-LP-mode fiber that features a nanopore-assisted step-index core with double-cladding profile. The purpose of this fiber design is to alleviate modes coupling and bend sensitivity, thus simplifies the multiple-input multiple-output (MIMO) process and the environmental requirements in which the fiber is used. Through numerical simulations, we show that the minimum effective index difference ( $\Delta n_{eff}$ ) of larger than 1.8*10−3 among all LP-mode over C and L band can be achieved by a simple nanopore with 170 nm radius in the center of the core. Furthermore, the dependences of both bending loss and chromatic dispersion on fiber structures are investigated by the finite element method (FEM) to show the good broadband performance. The proposed fiber targets applications in simpler MIMO space division multiplexing (SDM) systems such as data center and short-reach large-capacity transmission.


I. INTRODUCTION
As a promising way to overcome capacity limitations of optical systems based on single-mode fibers (SMFs), space division multiplexing (SDM) transmission through multi-core fibers (MCFs) and few-mode fibers (FMFs) has attracted worldwide concern [1]- [5]. Compared with MCFs, FMF has the advantage of boosting the transmission capacity via a single enlarged core in the cladding. A variety of FMFs that support several modes have been investigated, which have different core parameters and structures [6]- [10].
Mode coupling is a major obstacle to the use of FMFs in SDM systems. Most FMF transmission experiments have been demonstrated using complex multipleinput multiple-output (MIMO) techniques to mitigate the random mode coupling, especially in long-haul transmissions [10]- [12]. Although the applying of graded-index fibers with minimal differential modal delay (DMD) is one of effective approaches, the 2M * 2M equalizers are also necessary The associate editor coordinating the review of this manuscript and approving it for publication was San-Liang Lee. to separate all M spatial channels (each channel has two polarizations) for the higher-order LP modes. In contrast, in data centers, where direct detection is common, it is urgent to reduce the complexity of MIMO and decrease the number of operations and its power consumption. Weakly coupled fibers have the unique superiority. They reduce the dependence on MIMO process via minimizing mode coupling [13]- [15]. For a better understanding, each LP mode in weakly coupled fibers can be detected separately using simple 2 * 2 (non-degenerate LP modes) or 4 * 4 (two-time degenerate LP modes) MIMO technology, regardless of the number of modes. In this approach, signals can be transmitted and received independently, which makes the number of MIMO facilities decline dramatically.
Generally, simpler MIMO transmissions require that the degeneracy between spatial modes should be lifted. The effective index difference ( n eff ) between adjacent LP modes should be larger than 1 * 10 −3 [13]. Some efforts have been made to solve this problem. The high core-cladding index difference design was found to expand mode spacing. Yet, core-cladding index difference of more than 1.6% is needed VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ to keep n eff = 1 * 10 −3 for a 6-LP-mode step-index fiber [1]. The special fiber structures, including ring-assisted cores, are introduced to increase the n eff [6], [13], [16]- [18]. A latest record that the n eff of higher than 1.49 * 10 −3 is achieved by a ring-core FMF with 6-LP-mode [13]. The fact is that the excessive changes in the core structure will alter fiber properties, such as transmission loss and increase the fabricating difficulty. Some elliptical cores are also designed to separate degenerate modes [9], [19], [20]. However, breaking the degeneracy usually brings an even smaller n eff , which will restrict the number of independent modes. Besides, weakly coupled FMFs should be designed with other necessary properties. Random bending tends to occur in the actual fiber use, so the bending loss of FMFs has become one of the major concerns in its practical application [21]. The DMD and chromatic dispersion in FMFs should also be noted to keep stable signal transmission in the corresponding band. Therefore, a weakly coupled FMF should be designed with simple core structures and multiple spatial channels while suppressing mode coupling and bending loss.
In this paper, a novel nanopore-assisted 6-LP-mode fiber with double-cladding profile has been demonstrated to address the above issues. By the using of finite element method (FEM), COMSOL Multiphysics is adopted to simulate the mode field distribution and then the data are integrated to MATLAB for analyzing. Firstly, we introduce the design principle of weakly-coupled FMF and the nanoporeassisted core structure in section 2. It is revealed that, the minimum n eff of larger than 1.8 * 10 −3 over C and L band can be achieved by a 170 nm-radius nanopore in the center of step-index core. Then, in order to understand the broadband performance of our designed fiber, the bending loss and chromatic dispersion dependences on fiber parameters and wavelengths are investigated in section 3. Finally, we clarify the fabrication tolerance and fiber properties. Through numerical simulations, we confirm that our proposed fiber shows great potential in the short-range large capacity simpler MIMO SDM systems.

II. PRINCIPLE AND NANOPORE-ASSISTED CORE DESIGN A. DESIGN PRINCIPLE OF WEAKLY COUPLED FIBER
According to the coupled mode theory and power coupling theory, the mode coupling coefficient is inversely proportional to the n eff [13], [22]. In the experiment of [23], there is a threshold of n eff to describe the relationship between n eff and the power coupling coefficient. When the n eff is smaller than the threshold value, the power coupling coefficient decreases with increasing of n eff . Yet, it changes insignificantly if n eff is larger than that value.
It is found that the existing of lateral pressure and environmental fluctuation may lead to a larger threshold value of n eff . Therefore, a fiber with large enough n eff is expected to have better robustness to external fluctuation [6]. Researches show that to design a weakly-coupled FMF, sufficiently large n eff between LP modes is essentially required.
The minimum n eff of larger than 1 * 10 −3 is the usual threshold for almost no modal-crosstalk [22]. Generally, the sweptwavelength interferometry technology is adopted to evaluate the inter-mode crosstalk. A former research indicate that when n eff ≥ 0.8 * 10 −3 , the power coupling coefficient for different modal vary from −29.8 dB/km to −22.7 dB/km in a ring-core 4-LP mode fiber, which is low enough to achieve weak coupling [6]. Thus, a FMF with the minimum n eff ≥ 1 * 10 −3 can be considered as a weakly-coupled FMF.
Step-index core profiles have proven to be very well adapted to weakly coupled fibers. Fig. 1 shows, for this profile type, normalized propagation constant b as functions of normalized frequency V , where n eff is the effective index of the LP mode, n co is the index of core, n cl is the index of cladding, r is the core radius, λ is the operating wavelength, taken here as 1550 nm. V and b are defined as (1) and (2), respectively [6]. The V number of 6.4 ensures robustness, supporting 6-LP modes and cutting off the next higher order modes. We can see that b of LP 21 and LP 02 is closest to each other, which means the minimum n eff between them leads to the strongest mode crosstalk. To alleviate mode coupling, it is necessary to analyze the modes power distribution in step-index core. Figure 2 shows the normalized power distribution of 6 supporting modes along the normalized radial distance when V = 6.4. In theory, the n eff of a LP mode can be manipulated by varying the refractive index of the corresponding core region in where the LP modes' most power distribute [13]. From Fig. 2, the field profiles of LP 01 mode and LP 02 mode have a maximum intensity at the center of core, while LP 21 mode has its peak intensity in the middle area of the core. For a better understanding, any change of one's n eff will have less of an effect on the other's. Therefore, adjusting the n eff of a certain 76174 VOLUME 8, 2020 mode (or changing the position of its maximum power area) based on the step-index core profile is a simple and effective method for designing weakly-coupled FMFs. A large number of achievements have been obtained through this design theory. For instance, a modified ring-core design based on step-index core reduces the n eff of LP 02 through the low index region in the center of core [16] and another ring-assisted FMF design based on step-index core increase the n eff of LP 21 by inserting a high index ring [6]. In our consideration, the strategies of abandoning any mode or introducing additional ring structures are not adopted, since either of them will sacrifice mode amount or significantly change the core index profile and increase the manufacturing difficulty.

B. NANOPORE-ASSISTED CORE DESIGN
According to the design theory mentioned above, we demonstrate the design of an air hole with a nanometer width, which called nanopore, placed in the center of step-index core to modify overall mode spacing, since the nanopore has the lowest index so that it can sufficiently decrease the n eff of LP 02 and leaves no significant impact on other modes. The nanopore extends through the entire fiber lengths. Here, we set up a two-dimensional model through COMSOL Multiphysics to analyze the mode properties in FMFs. The frequency domain physics field under the wave optics module is selected. Figure 3 shows the index profile of the nanopore core design, where r 1 , d, W are core radius, radius of nanopore and the width of 1 st cladding, respectively. The influence of double-cladding structure will be discussed in section 3. The index contrast of core and cladding is defined as 1 = (n 2 co − n 2 cl )/(2n 2 co ). First, we need to select appropriate core parameters as the basic model. Considering 6 LP-mode transmission and the large space between the n eff of the highest order mode and cladding index, r 1 = 7.5 µm and 1 = 1.0% are set as the core parameters in terms of Eq. (2). In order to ensure the rigor and reliability of the calculation results, some preset parameters need to be arranged. To model the air hole with nano-scale sizes, the extremely-refined free-triangular divided meshes is used to improve the spatial resolution of the model. In addition, the amount of supporting modes as well as the operating wavelength are also should be considered in the modeling. To reveal the impact of the nanopore, the modal properties relative to nanopore parameters are calculated. Fig. 4 shows variation of the n eff with d/r 1 at the wavelength λ = 1550 nm. The n eff of LP 01 and LP 02 becomes small by the influence of nanopore, and it is reduced noticeably as d/r 1 grows large. Compared with LP 01 's, the n eff of LP 02 exhibits a more pronounced curvature. When d/r 1 is larger than 0.03, the n eff of LP 02 is approaching to that of LP 31 sharply. In order to keep n eff among all the modes large, the ratio of d/r 1 should range from 0.01 to 0.03.   that is, d = 170 nm, as the nanopore radius. The mode field distributions of LP modes in the nanopore-assisted core are shown in Fig. 6. It can be found that the nanopore is located at the most power area of LP 01 and LP 02 , which means the properties of other modes are hardly affected by this structure. Fig. 7 shows the normalized power distribution of nanoporeassisted core along the normalized radial distance. We find that the power of LP modes (except for LP 01 and LP 02 ) in the nanopore-assisted core is almost unchanged in comparison to the normal step-index core. The most power area of LP 01 FIGURE 6. The mode field distributions of the LP modes in nanopore-assisted core profile. and LP 02 shift to the right with the existing of the nanopore. The results imply that the nanopore makes LP 02 faces more power loss than LP 01 , which is consistent with the change happened to their n eff values in Fig. 4. The power change of some modes caused by nanopore is similar to the negative impact of ring-core or elliptical core structures. But the role that nanocore plays in suppressing mode coupling is what we should pay more attention to.
The n eff and n eff of step-index core and nanoporeassisted core dependence on wavelength are calculated in Fig. 8. Fig. 8 (a) and (b) show the n eff and n eff of step-index core without nanopore corresponding to wavelength. The minimum n eff of lower than 1 * 10 −3 between LP 21 and LP 02 implies that the mode coupling may not be controlled into a desired value [22]. Furthermore, the n eff and n eff of nanopore-assisted core profile dependence on wavelength are presented in Fig. 8 (c) and (d). The designing core profile makes the n eff of any neighbor mode of larger than 1.8 * 10 −3 over the entire C+L band. Thus, if all the mode-groups are simultaneously used for dual-polarization SDM transmission, a relatively simple 2 * 2 and 4 * 4 MIMO is needed to recover the signals.

III. CHARACTERIZATION OF NANOPORE-ASSISTED FMF WITH DOUBLE-CLADDING STRUCTURE A. FIBER PROPERTIES
The introduction of nanopore may change the fiber sensitivity on bending. To control this trend, a large piece of 1 st cladding and a small piece of 2 nd cladding with lower core-cladding index contrast are applied. The purpose of double-cladding design is to suppress the bending loss of guiding modes and increase the leakage loss of the unwanted higher-order mode in a limited cladding diameter. We evaluate the broadband characteristics of our proposed fiber over C and L band. Fig. 9 shows the schematic cross section of nanopore-assisted FMF with double-cladding structure.  The 6-LP modes should be guaranteed to operate over C and L band. In reference to ITU-T recommendations G. 654, the bending loss should be lower than 0.5 dB/100 turns (R b = 30 mm) for the highest-order mode (LP 12 ) at 1625 nm [24]. Furthermore, the bending loss of unwanted mode (LP 41 ) at 1530 nm (R b = 140 mm) should be larger than 1 dB/m [24], [25]. In our modeling, a bent fiber is transformed into a straight fiber with equivalent refractive index method. Here, the perfect matching layer (PML) is necessary for calculating the characteristics of the bend fiber. Also, the bending radius and direction should be considered. The left bending direction along the x-axis is chosen in our simulation. The bending loss can be expressed by [21]: where imag(n eff ) means the imaginary part of the n eff . The bending loss dependence on double-cladding parameters are calculated in Fig. 10. The solid and dashed lines correspond to the bending loss of LP 12 after 100 turns at 1625 nm (R b = 30 mm) and the bending loss of LP 41 after 1-m transmission at 1530 nm (R b = 140 mm), respectively. From Fig. 10 (a), W > 18.2 µm is needed to satisfy the demand of BL < 0.5 dB/100 turns of LP 12 , while it should be smaller than 38 µm to keep bending loss of unwanted mode larger than 1 dB/m. In Fig. 10 (b), the 2 of larger than 0.86% no longer meets the bending loss condition. Thus, the parameters of cladding can be determined, and the W = 30 µm and 2 = 0.8% is acceptable.
Specifically, the influence of fiber structure on chromatic dispersion should be noted. Chromatic dispersion of the transmission fibers is a major factor causing optical pulse VOLUME 8, 2020 broadening [26]. The dispersion is composed of two components: material dispersion D m and waveguide dispersion D w , which can be written as Eq. (4), where c is the velocity of light in a vacuum, n M is dependent on λ in dispersive media, Re(n eff ) is the real part of the n eff , respectively.
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 Refs. [26]. Waveguide dispersion depends among others on the fiber structure. The degree of effect of nanopore and double cladding on waveguide dispersion is an important part in determining the total dispersion. For example, we calculate the dispersion of LP 01 mode in the proposed fiber dependence on wavelength in Fig. 11. As one can see, the slope of the dispersion curve of LP 01 mode changes insignificantly with the added nanopore and double  cladding structure. Definitely, we also calculate the dispersion of other higher-order modes along the wavelength. The simulation results indicate that the dispersion of higher-order modes follows the same trend as the fundamental mode. They are not varying too much on the entire C and L band in comparison to the dispersion values of the fiber without the nanopore and double-cladding structure. The chromatic dispersion values of these LP modes in our proposed fiber are summarized in Table 2.
Furthermore, the relationship between the A eff and doublecladding parameters in nanopore-assisted core FMF is simulated in Fig. 12 (a) and (b), respectively. The A eff of the propagation modes are approximately constant even if W and 2 change, and they are larger than 110 µm 2 . The DMD characteristics is important for the reduction of complexity and power consumption of MIMO processing. For example, the DMD between LP 01 and LP 11 can be defined as value that subtracted the group delay of LP 01 from that of LP 11 [16]. Figure 13 shows DMD of the proposed fiber dependence on wavelength. We find that the DMD increases slightly from short wavelength to the long wavelength. The broadband characteristics of the proposed fiber can reach the requirements of working in the C and L band.

B. FABRICATION TOLERANCE ANALYSIS
It is very necessary to analyze the feasibility and manufacturing tolerance of the nanopore-assisted FMF with double-cladding structure in the actual fabricating process. The nano-level air holes are usually used in the development of micro-structured sensors [27], [28], probes [29] and the design of photonic crystal fibers [26], [30]- [32]. To the best of our knowledge, it is the first time that the nanopore has been applied to the FMFs in the transmission field.
As demonstrated above, the nanopore-assisted FMF with double-cladding structure features low mode coupling, low bending loss and large mode field area. However, it also has certain downsides. The obvious shortcoming is that the proposed fiber is not an all-solid structure, which means it has greater requirements on fabricating accuracy and difficulty than traditional fibers. Since the fiber structure is a simple combination of nanopore-assisted core and double-cladding, the main issues that are very likely to occur in the production are concentrated on the fabrication of the nanopore. One of them is the occlusion of the nanopore, and the other is the excessive nanopore diameter.
The problem of air holes occlusion sometimes appears in the fiber design of hole-assisted structure. The occlusion of air holes will affect the fiber performance to a certain extent. However, in our proposed fiber, the nanopore occlusion will only partially affect the modes coupling characteristics. Nonetheless, due to the pre-design of the high core-cladding contrast, the all-solid FMF caused by nanopore occlusion will make a limited effect on n eff and it is acceptable, as shown in Fig. 8 (a) and (b).
On the other hand, an excessive nanopore diameter will also affect the fiber performance. As shown in Fig. 5,  as d/r 1 continues to increase, the minimum n eff between LP modes will reduce. When d/r 1 = 0.03, the lowest n eff will become 1.6 * 10 −3 , which will sacrifice the weak coupling performance to some extent. Therefore, the tolerance of the d/r 1 should be less than 0.03, that is, d ≤ 225 nm. If we aimed at the best weak coupling performance, d should be set between 160 nm and 180 nm.
Choosing an appropriate fabricating method helps reduce the failure probability that may occur during the production. The stack-and-draw, which is a unique technology that can be used to fabricate micro-structured optical fiber via multiple glass rods and capillaries. This method is relatively mature and is widely used in making hole-assisted MCFs [33], photonic crystal fibers [30], [31] and segmented cladding fibers [34]. Figure 14 shows the schematic diagram of a stacked nanopore-assisted double-cladding structure FMF preform, where the GeO 2 -doped preform is used to form the core, the glass rod can become the cladding and the glass capillary can shape the nanopore structure. After the above discussion, we believe that the design scheme of the nanopore-assisted double-cladding structure FMF is feasible, and the tolerance can meet the demand of the target broadband characteristics of the proposed fiber. Finally, we summarize the optimal fiber parameters with fabrication tolerance and performance of our proposed fiber in Table 1 and Table 2, respectively. Simpler MIMO process will be adopted due to the ultra-high n eff of larger than 1.8 * 10 −3 with nanopore-assisted core profile. The broadband characteristics have been optimized to maintain stable operation over the entire C and L band.

IV. CONCLUSION
We have proposed a nanopore-assisted core double-cladding profile 6-LP-mode fiber with ultra-high n eff and optimized bending loss. By introducing a 170 nm-radius nanopore through the fiber length, the minimum n eff between any two LP modes of larger than 1.8 * 10 −3 over C and L band can be achieved. The mode coupling and the complexity of MIMO process can be greatly reduced via this simple core structure. The broadband performances, including bending loss, dispersion and mode field area dependences on fiber parameters and structures are investigated. Through numerical simulations, we show that the nanopore has limited contribution to the changing of dispersion and the mode field area. Moreover, the bending loss are efficiently reduced by the double-cladding structure. The fabrication method and tolerance of the proposed fiber are illustrated. With the viewpoints of these properties, we believe that our designed fiber can be applied into simpler MIMO SDM systems in scenarios such as data center and computer rooms.
YUHENG XIE was born in Shenyang, China, in 1993. He is currently pursuing the Ph.D. degree in communication and information system with Beijing Jiaotong University, Beijing, China. His current research interests include special fiber design, optical fiber sensor, and novel optical devices.
LI PEI was born in Shanxi, China. She is currently a Professor with the School of Electronics and Information Engineering, Beijing Jiaotong University, Beijing, China. Her current research interests include high-speed optical telecommunication networks, optical fiber sensor, radio-over-fiber communication systems, and novel optical devices.
JINGJING ZHENG was born in Henan, China. She is currently an Associate Professor with the School of Electronics and Information Engineering, Beijing Jiaotong University, Beijing, China. Her current research interests include specialty fiber and components, optical fiber sensor, and novel optical devices.