Design of Weakly-Coupled 16-Vector-Mode Coaxial Bragg FMF for Short-Haul Communication

Utilizing weakly-coupled few-mode-fibers (FMFs) for MIMO-DSP eliminating in short-haul mode division multiplexing(MDM) systems has attracted much attention recently, as an effective solution to suppress modal crosstalk. In this article, we propose a weakly-coupled 16-vector-mode coaxial Bragg FMF supporting 16 vector modes(HE<sub>11a</sub>, HE<sub>11b</sub>, TM<sub>01</sub>, HE<sub>21a</sub>, HE<sub>21b</sub>, TE<sub>01</sub>, HE<sub>31a</sub>, HE<sub>31b</sub>, EH<sub>11a</sub>, EH<sub>11b</sub>, HE<sub>12a</sub>, HE<sub>12b</sub>, EH<sub>21a</sub>, EH<sub>21b</sub>, HE<sub>41a</sub>, HE<inline-formula> <tex-math notation="LaTeX">$_{\mathrm {41b}}$ </tex-math></inline-formula>) which works across the C-band and explore the possibility for weakly-coupled Bragg FMFs. We optimize the geometric parameters and evaluate their effects on modal <inline-formula> <tex-math notation="LaTeX">$n_{eff}$ </tex-math></inline-formula>, <italic>min</italic> <inline-formula> <tex-math notation="LaTeX">$\Delta n_{eff}$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$A_{eff}$ </tex-math></inline-formula>. The DMD, modal confinement loss and bending loss are also investigated. The simulation results indicate that the proposed fiber can obtain large <italic>min</italic> <inline-formula> <tex-math notation="LaTeX">$\Delta n_{eff}$ </tex-math></inline-formula> between adjacent vector modes (<inline-formula> <tex-math notation="LaTeX">$\sim 2\times 0.0001$ </tex-math></inline-formula>), and can be a promising candidate for MIMO-DSP simplified short-haul MDM transmission systems.


I. INTRODUCTION
As the single-mode fiber (SMF) transmission system is facing the approaching of Shannon limit, the mode division multiplexing (MDM) transmission with few-mode fibers (FMFs) is considered to be a promising future solution of the capacity crunch. Dealing with modal crosstalk caused by modal coupling is a core issue for MDM systems. Using the multiinput-multi-output digital signal processing (MIMO-DSP) for compensating crosstalk is suitable for long-haul scenarios, however the computation complexity and cost are burdensome and not acceptable for the rapidly developing short-haul MDM transmission systems [1]. Therefore, weakly-coupled FMFs with large effective index difference ( n eff ) between adjacent modes have been suggested for modal coupling suppression in short-haul MDM systems, in which the signals of adjacent modes can be transmitted independently. Among weakly-coupled FMFs, several design schemes such as FMFs with ring-core [2], [3], elliptical core [4], or stress region for polarization maintaining [5] have been carried out. These The associate editor coordinating the review of this manuscript and approving it for publication was Barbara Masini . fibers mainly work at 1550nm and require little or none MIMO-DSP in the transmission systems.
The Bragg fibers achieve light confinement through the photonic bandgap by its periodic cladding of alternating high and low refractive indices. This concept was first proposed by Yeh et al [6]. Theoretical analyses using other methods such as the asymptotic analysis were carried out after that [7]- [11]. Bragg fibers have the appealing features of low dispersion [9], low radiation loss [10], and the possibility of guiding light in an air core [14]- [17]. Adjusting the properties of such fibers can obtain various characteristics for various scenarios, such as dispersion management [18] or mode selection [19] for transmission, and fiber sensor [20]- [22] or laser [23] for fiber optical devices. Several reports on Bragg fibers as transmission waveguides have been carried out recently. Y. J. Yehouessi et al. reported an all-solid Bragg fiber with the effective mode area of about 3700 µm 2 at 1035 nm by using a pixelated Bragg fiber geometry [24]; M. Shang et al. introduced two air holes into the core of an all-solid Bragg fiber to produce high birefringence [25]; X. Bai et al. proposed a Bragg fiber with dual claddings, which can support 50 OAM modes [26]. The study of Bragg fibers as weakly-coupled FMFs is still a shortage, to our knowledge.
Bragg fibers with a high index dielectric medium core are called the coaxial Bragg fibers. The concept of coaxial Bragg fibers came from the analogy between dielectric coaxial fibers and metallic coaxial cables. Coaxial fibers have better performance of light confinement while possessing the same light-guiding mechanism as ordinary Bragg fibers. The alldielectric single-mode coaxial Bragg fiber was first proposed by Ihanescu et al [27] to study its dispersion and polarization rotation performance in high-data-rate scenarios, while later researches have not given full consideration to coaxial Bragg fibers for few-mode applications. Over all, the Bragg fiber has the potential of applying to few-mode short-haul transmission, owing to its large effective index differences between adjacent propagation modes, yet few studies have been carried out on few-mode Bragg fibers, let alone coaxial Bragg fibers.
In this article, we mainly propose a weakly-coupled 16-vector-mode coaxial Bragg FMF supporting 16 vector modes. We carry out numerical simulations to investigate the influence of fiber parameters on its effective refractive index (n eff ), minimal n eff (min n eff ), and modal effective area (A eff ). We also evaluate the differential mode delay, modal confinement loss and bending effect of propagation modes. Simulation results show that by optimization, the weaklycoupled few-mode Bragg fiber can be achieved with large min n eff of around 0.0002 between adjacent spatial modes at 1550nm. The proposed fiber has potential for applying to short-haul MDM communication systems such as passive optical network (PON), datacenter network and optical interconnections for MIMO-DSP simplifying and optical sensing devices.

A. FIBER STRUCTURE AND PARAMETERS OF THE COAXIAL BRAGG FMF
We design a weakly-coupled coaxial Bragg FMF supporting 16 vector modes with the parameters shown in Fig. 1. The fiber structure includes several periodic layers of alternating refractive indices with width l, an air core with radius r and a central circular dielectric medium with radius d and refractive index n c , and the refractive index (RI) difference n 1 between the high-index and low-index layers. The width of each high-index layer l 1 makes up 0.33 of l in terms of proportion, and the refractive indices of high and low index parts are set to be 4.58 and 1.55 (at 1550nm), by which a single-mode Bragg fiber has been fabricated, according to previous works [27]- [30]. The fiber characteristics are simulated with finite element method (FEM) methods. The large refractive index difference between cladding layers is large enough to form rather wide photonic bandgap and therefore is beneficial for light guiding [29]. Schematics of cross section and refractive index profile for the proposed fiber are shown in Fig. 1(a), (b).
The Bragg FMF is added with the central dielectric medium to confine light in the air core better [31], and what's more to further separate vector modes. An ordinary hollow  Bragg FMF is designed to contrast the min n eff between adjacent modes with the proposed fiber. Schematics of cross section and refractive index profile for the hollow Bragg FMF are shown in Fig. 1(c) and (d), with parameters of l, l 1 , l 2 , r and n 1 . Where l stands for the width of the high(l 1 ) and low(l 2 ) index layers of the cladding, r is the radius of the air core, and n 1 represents the RI difference between the layer refractive indices. Fig. 2 shows the curves of min n eff as a function of r in the proposed Bragg FMF and an ordinary hollow Bragg FMF when material, air core size r, Bragg layer numbers and layer width l of these two kinds of fibers are set as the same. The results in Fig.2 indicate that for the hollow fiber, the min n eff of the desired modes is around 5 × 10 −5 , while that of the proposed FMF exceeds 10 −4 , which means the latter has a better performance in terms of separating adjacent eigenmodes. Therefore, the coaxial center structure is better for air-core Bragg FMFs to more fully utilize eigenmodes and improve the transmission capacity.

B. ELECTROMAGNETIC FIELD DISTRIBUTIONS OF THE COAXIAL BRAGG FMF
As has been studied in [10] by asymptotic matrix formalism, the electromagnetic fields in a Bragg fiber can be separated into two parts: the core region and the cladding region. In this In the cladding region, the high-index layer has the thickness of l 1 and refractive index n 1 , while the low-index layer has the thickness of l 2 and refractive index n 2 .
article, the core region is composed of the central dielectric medium cylinder, and the air core. Schematic of the cross section of the proposed Bragg fiber is shown in Fig. 3. The electromagnetic field intensity in the central dielectric medium is supposed to be low. Using the Maxwell equations, the transverse field components in the air core region can be represented by H z and E z in formula (1) -(4).
The general solutions of Ez(θ) and H z (θ) can be given by the superposition of Bessel functions. Here we use the Bessel functions Jm(kr) and Ym(kr), where r means the distance from the center of the circle to the edge of core (r ≥ d) for each m.  where n air = 1 and k air = For the cladding region, the high-index layer has the thickness of l 1 and refractive index n 1 , while the low-index layer has the thickness of l 2 and refractive index n 2 . As shown in Fig. 3, d n h ≤ r < d n h + l 1 denotes the high-index region and d n l ≤ r < d n l + l 2 denotes the low-index region. When the TM and TE components are regarded as decoupled with each other, with amplitudes of A TM and A TE respectively, the field distributions can be described as: Using the boundary conditions, as when E z (θ) and H z (θ ) are continuous at the edge of each layer, the fields in both core and cladding region can be described by the asymptotic approximation.

C. OPTIMIZATION OF THE COAXIAL BRAGG FMF
The proposed fiber is supposed to support 16 vector modes (HE 11a , HE 11b , TM 01 , HE 21a , HE 21b , TE 01 , HE 31a , HE 31b , EH 11a , EH 11b , HE 12a , HE 12b , EH 21a , EH 21b , HE 41a , HE 41b ). The min n eff between two adjacent modes must be larger than 10 −4 to suppress modal coupling [32]. The modal  properties of the proposed fiber can be adjusted by operating the geometrical parameters shown in Fig. 1.
Bragg fibers confine light in the low-index core by generating bandgap effects in the periodic cladding rather than total internal reflection [9]. In order to support more higher-order modes, the confinement losses of several modes are larger than that of a single-mode Bragg fiber. Figure 4 shows the first few TE and TM bandgaps of the proposed Bragg fiber when the high-index layer width l 1 accounts for 0.33 of each alternating layer l, and the effective refractive indices of the layers are fixed to be 4.58 for high-index regions and 1.55 for low-index regions at 1550nm. As is shown in Fig. 4, the TE mode bandgap is wider than that of the TM mode, so we only consider the latter during optimization. The surface-parallel wavevector component β and the frequency ω are expressed in the unit of 2π/ and 2πc/ , respectively, where represents the width of each alternating layer (1 high-index layer and 1 low-index layer) l. The white regions are the bandgap regions, among which light can propagate in the air. We choose the 6th bandgap in Fig. 4(b), where the vertical axis ω/(2π c/ ) ≈1.2, and sets at (1.7-1.95) µm. For the convenience of fabrication, we keep one decimal fraction and choose l to be 1.8 or 1.9µm. It needs to be noticed that the center high-index core plays the vital role when the guided modes pass the air light line [31].
For Bragg fibers, high-order modes may also exist in the fiber until they fade away after several miles of transmission. Our aim is to make the proposed fiber support up to 16 vector modes, so we have to sacrifice some loss performance, therefore the maximum confinement loss of the proposed fiber is not as good compared with a single-mode Bragg fiber of the previous work [33], which also means that the confinement loss of the proposed fiber is not suitable for long-haul transmission scenarios. In terms of the proposed 16-mode FMF, the modes of more than 17th have the confinement loss of exceeding 10 3 dB/m, while the confinement loss of the desired 16 modes can be less than 10 2 dB/m as will be shown later in Fig. 10, so we suppose the loss of the former to be too large to satisfy the propagation condition, or in other words, to be cut-off.
The confinement loss L c (dB/m) of each mode can be calculated through the imaginary part of each mode's effective mode index Im(n eff ) [34].     be 0.33. The curves of adjacent modes with min n eff are enlarged and inserted at the bottom of Fig. 5. From Fig. 5, the min n eff changes little when r goes from 2.4 to 2.45, while the 16-mode interval narrows. Among the interval, the min n eff can always be > 10 −4 . Taking the above into account, we optimize the periodic layer width l to be 1.8(l 1 ≈ 0.59) as the final design. In this case, r should be less than 2.43µm to avoid the appearance of the 17th propagation mode (TE 02 ). Figure 6 indicates the variation of n eff as a function of circular dielectric medium radius/air core radius (d/r) with r = 2.4µm, 2.42µm and 2.44µm, when each periodic layer width l is fixed to be 1.8µm. Figure 7 indicates the variation of n eff as a function of air core radius r with d/r = 0.08, 0.1 and 0.12. From Fig. 6, the min n eff of the required modes increases with d/r increasing, which means the dielectric medium core can effectively separate adjacent modes, also be consistent with the results in Section A. From Fig. 7, as r increases, the overall n eff as same as the min n eff will increase, and the number of propagation modes also increases. Figure 7 indicates that the enlarging of air core size within certain range contributes to the increasing of propagation mode numbers, which is the same as ordinary total reflection fibers. However the rising range is not much for n eff when d/r rises from 0.1 to 0.12, and in order to keep 16 modes, either r or d cannot be too large. Taking the fabricating complexity into consideration, we reserve two decimal fractions and choose d/r to be 0.1. In this case, r is chosen to be ≈ 2.4µm. The overall result shows that minor manufacturing differences of the core parameters are permitted among the fault tolerance requirements, and will not influence the modal distance greatly. Both the core and the inner radius of cladding can be made a little larger for larger min n eff as long as the 16-mode condition is met.
We further study the joint influence of d and r on the values of min n eff . Figure 8 shows the distribution of min n eff between every two of the desired 16 modes. The vertical axis stands for the air core radius r, and the horizontal axis stands for the proportion of central dielectric medium in the core region d/r. The solid curves correspond to the threshold lines when the number of propagation modes is 16 and ≥ 17(or in other words, mode 'cut-off' lines), thereby forming a 16vector-mode region circled by these curves at the wavelength 1550nm. When the number of propagation spatial modes exceeds 16 with the variation of parameters, n eff related to higher-order modes (e.g. n eff between HE 41b and TE 02 ) are not considered. Also, when some of the desired 16 modes are cut-off, n eff related to the certain modes are not considered. So the comparatively large min n eff can be found in this region. From Fig. 8, when the other parameters are fixed to be constant, the number of modes goes up as r or d/r increases. Also, the overall min n eff will increase with the increasing of r or d. Compared with r, d/r effects min n eff more obviously. Though the min n eff in Fig. 8 all exceed 10 −4 , the min n eff change is not evident in the circled region. It's better to choose geometrical parameters set at the right side to gain larger min n eff . For all of the above reasons, we optimize r and d to be 2.4µm and 0.24µm(d/r ≈ 0.1) as the final design. Figure 9 indicates the variation of n eff as a function of air core radius r with n c = 4.2, 4.4, and 4.58, when d/r is fixed to be 0.1. The min n eff of the required modes increases when n c gets larger according to Fig. 9, and the appearance of higher-order modes gets earlier. So one can make a high refractive index central core by doping high-index materials to obtain high n eff between adjacent modes. We choose the central dielectric medium to be the same material as the high-index cladding layer in our research.
For Bragg fibers, the number of periodic layers determines the capability of light confinement, while the modal confinement loss of the Bragg fiber mainly decides the transmission haul of a certain mode before it fades away. During the above optimization, we use 15 layers. Figure 10(a) indicates the variation of confinement loss as a function of periodic layer numbers when each periodic layer l is fixed to be 1.8µm, and Figure 10(b) shows the enlarged partial of the left bottom in Fig. 10(a). From Fig. 10, the confinement losses of several modes are quite large when the layer numbers ≤ 15. When the layer number exceeds 18, the confinement losses of most modes drop to under 10 dB/m and the curves become steady. The maximum loss declines below 1 dB/m when the layer number is more than 25. We optimize the number to be 18 preliminarily. VOLUME 8, 2020  In air-core Bragg fibers, the transmission loss L t can be defined as the sum of confinement loss L c and absorption loss L a under relatively ideal conditions (no crosstalk between channels or manufacturing errors): The absorption loss coefficient can be obtained by: [35] where R is the normalized core radius, A core and A layers are the amplitudes of the electric field in the core and cladding layers. Air-core Bragg fibers are supposed to have much smaller absorption loss than ordinary silica fibers [28]. The calculated absorption losses of the proposed fiber are presented in Table 1. From Table 1, the absorption loss L a (< 2.5dB/km) is far smaller than the confinement loss L c (< 1dB/m), so for the proposed fiber, the transmission loss L t is mainly determined  by L c . When the number of layers is enough, L t can be less than 1dB/m.

III. RESULTS
Based on the above design, we choose the parameter of r = 2.4µm, d = 0.24µm, l = 1.8µm and n c = 4.58. We study some of the performance parameters of the proposed fiber, and evaluate the results.

A. WORKING WAVELENGTH
The variation of n eff as a function of wavelength is shown in Fig. 11. Results illustrate that the designed fiber works over a range of wavelength from 1520nm to 1570nm, across the whole C-band. Both n eff and n eff fall with the wavelength going up, and the number of supporting modes reduces. At 1570nm, min n eff turns to be 0.00012, which is also acceptable.

B. MODE EFFECITVE AREA
The mode effective area(A eff ) of fibers determines their properties of nonlinearity impairment limitation. As for the proposed fiber, the A eff is not supposed to be too large as the geometrical size of the fiber is small. We compute A eff by [35]: The variation tendency of the minimal and maximal A eff under the influence of varying r and d/r are shown in Fig. 12(a) and 12(b) separately. Based on Fig. 12(a) and 12(b), the A eff among the 16 propagation modes can be varied from 10 to 130µm 2 . The min(A eff ) of the proposed fiber is around 11µm 2 , and its value tends to increase a little with the enlarging of r and d. On the other hand, when r or d gets higher, the max(A eff ) gets smaller in the given interval. Of all the 16 vector modes, when the air core radius r is set to be 2.4µm and d/r is set to be 0.1, TE 01 has the lowest A eff of 10.97 µm 2 and HE 31a has the highest A eff of 130.08 µm 2 . Other results are shown in Table 2 at the bottom.

C. DIFFERENTIAL MODE DELAY
For short-haul transmission systems, the computation complexity and power consumption can be measured by the differential mode delay(DMD), which is critical for simplifying or eliminating the MIMO DSP in the receiver end. DMD between two adjacent modes can be defined as the value of subtracting the group delay τ A of τ B , which is expressed as follows [2]: where τ i represents the group delay of two modes, n gi represents the group indices, c is the light velocity in a vacuum and λ means the free space wavelength. Figure 13 shows the variation of DMD as a function of wavelength, with all the other design parameters set as the values above. As shown in Fig. 13, the DMD between all adjacent modes < 5ns/m. The actual calculated results of DMD for the designed Bragg fiber are shown in Table 2, with all the other characteristics.

D. BENDING LOSS ANALYSIS
The bending effect can be determined due to different modes' field distributions and propagation constants. In order to contain more high-order modes in the proposed FMF, the bending loss performance is influenced by the confinement loss to some content. By calculating the imaginary part of fiber effective mode index Im(n eff ), fiber bending loss of each mode can be obtained by formula (1) in 3.1. In calculations,  the bending equivalent refractive index is computed by [37]: n(x, y) = n 0 (x, y)(1 + x/R) The variation of loss as a function of fiber bending radius R is shown in Fig. 14. The fiber is bent towards the horizontal axis of the cross section. It can be acquired that modes with higher confinement loss are easier to fade out while bending. According to Fig. 14, when the bending radius is more than 10cm, the bending losses of each mode are below 10 dB/m. At the bending radius of 1mm, the bending losses of 10 modes are less than 10dB/m. When bending radius is small, losses of higher-order modes are more sensitive to the bending radius, while also influenced by the confinement ability. The bending loss as well as the confinement loss will decrease when the number of cladding layers is large enough.
The final optimized modal properties of the 16 desired modes are shown in Table 2. We achieve large min n eff of ∼2 × 0.0001 as the final design. Comparing with the conventional FMFs, the proposed fiber structure can greatly improve the transmission capacity of the communication system by making full use of eigenmodes, as the conventional FMFs mainly support LP modes [37] and their mode groups are not fully utilized [38]. Though the few-mode Bragg fibers are not as easily to be fabricated as ordinary FMFs, they have a good prospect of being applied in MDM systems.

IV. CONCLUSION
In conclusion, the proposed weakly-coupled coaxial Bragg fiber supporting 16 vector modes(HE 11a , HE 11b , TM 01 , HE 21a , HE 21b , TE 01 , HE 31a , HE 31b , EH 11a , EH 11b , HE 12a , HE 12b , EH 21a , EH 21b , HE 41a , HE 41b ) can achieve large min n eff of ∼2×0.0001 between adjacent modes at 1550nm, and works across the C-band. Numerical simulations are carried out on n eff , min n eff and A eff to investigate the influence of geometric parameters within the regions that meet the 16-vector-mode design condition. The DMD, modal confinement loss and bending effect are also evaluated. Simulation results show that the coaxial structure can effectively split adjacent eigenmodes, and reveal the possibility for weakly-coupled Bragg FMFs for short-haul communications.