Design of Air-Trenches/Holes Assisted Depressed-Core 12-Mode Fiber for Less MIMO Space Division Multiplexing

An air-trenches/holes assisted depressed-core few-mode fiber (AT-AH-DC-FMF) that features some asymmetrical air trenches, air holes and a depressed step-index core is proposed in this paper. The AT and DC can produce great contribution in separating the non-degenerated LP modes and spatial modes, thus suppressing the crosstalk efficiently. Based on AT and AH, the bending loss values of AT-AH-DC-FMF can be satisfied with the ITU-T recommendations of G. 654. The simulation results indicate that our proposed fiber can support 12 spatial modes with the effective index difference between adjacent LP modes (Δneff) larger than 1.03×10-3 and the effective index difference between adjacent spatial modes (δneff) larger than 1.24×10-4. The broadband characteristics such as Δneff, δneff, chromatic dispersion and effective mode field area (Aeff) over the whole C and L band are comprehensively investigated. Moreover, fabrication methods and birefringence are detailed discussed. Our proposed AT-AH-DC-FMF possesses great potential to be applied to less multiple-input multiple-output (MIMO-less) space-division multiplexing (SDM) enlarging optical communication capacity and simplifying the system complexity.


I. INTRODUCTION
With the widespread of big data and cloud computing, the amount of data tends to an explosive growth [1]. The current data centers and optical networks are rapidly reaching their ultimate capacity due to the nonlinear Shannon limit of single-mode fibers (SMFs) [2]- [4]. Space division multiplexing (SDM) systems based on multi-core fiber (MCF) and few-mode fiber (FMF) have been extensively investigated for their potential to overcome these looming communication bottlenecks [5]- [7]. However, the high-performance MCF with some characteristics such as high core density, low crosstalk and low loss simultaneously faces a large number of technical challenges in the design and manufacturing process. It is also challenging to efficient couple signals in and out after all cores are tightly packed in a MCF with a limited cladding size [8]. FMF with its relatively simple structure produces unique advantages in terms of manufacturing methods and connectivity with standard SMFs. In addition, FMF with broadband characteristics over a wide wavelength range can provide the opportunities for SDM compatible with the mature wavelength-division multiplexing (WDM) technique [9]. The most crucial issue encountered in FMF is the crosstalk among different modes [10]. Employing the multiple input multiple output (MIMO) processing at the receivers is one of the approaches to settle this issue [11]- [13]. But the complexity of MIMO increases nonlinearly with the number of propagation modes, thus resulting in the depleting of the digital signal processing (DSP) capacity and the consumption of electric power [14]. To simplify MIMO, it is necessary to design special weakly-coupled FMF whose effective index difference between nondegenerated propagate modes (Δneff) is larger than 10 -3 . Moreover, MIMO can be eliminated by enlarging the effective index difference between adjacent degeneracy modes (δneff) to larger than 10 -4 . This method would be well satisfied with short-reach transmission such as data centers and computer rooms as its advantages of simplify systems [15]- [17].
Recently, researchers have made a large number of attempts in terms of adjusting the effective index (neff) of higher-order mode to enlarge the effective index difference between adjacent non-degenerated and degeneracy modes [18]- [21]. For example, in [20], a hole-assisted gradedindex 4 LP-modes fiber with Min Δneff as high as 1.94×10 -3 is proposed. In [21], a 3 LP-mode ring core fiber with Min Δneff as high as 2.5×10 -3 is designed. However, 4×4 MIMO facilities are still needed to recover their fourfold degeneracy modes. It has been reported in [2], [15], [22], [23] that elliptical core fibers can act as a polarization maintaining fiber (PMF) used for MIMO-free SDM systems. For example, in [15], a 10 polarizationmaintaining PANDA ring-core fiber with min δneff of 1.29×10 -4 has been proposed. In [2], 10 distinctive polarization modes with Min δneff of larger than 1.32×10 -4 have been achieved in a FMF composed of a central circular-hole and an elliptical-ring core. However, drawback of high doping difficulty is induced in those PMF, due to its large core and cladding relative refractive index difference of up to 3%. In addition, the core region of most of the PMF stays at a small value, resulting in a smaller effective mode field area than other FMF. What's more, PMF with the minimum effective refractive index difference of larger than 1.32×10 -4 can only be applied in a shorter distance transmission than that of other FMF. Therefore, designing a weakly-coupled FMF with low doping, enough modes, large effective mode field area, sufficient Δneff, sufficient δneff and stable comprehensive broadband characteristics for less MIMO SDM-WDM systems is urgent.
In this paper, we proposed an air-trenches/holes assisted depressed-core 12-mode fiber for 2×2 MIMO SDM systems. COMSOL Multiphysics and MATLAB are utilized for investigating the fiber performance. The main contents of this paper are as follows: (Ⅰ) In section 2, the schematic topology and particularities of our proposed airtrenches/holes assisted depressed-core few-mode fiber (AT-AH-DC-FMF) are introduced. (Ⅱ) In section 3, the δneff, Δneff dependence on the parameters of DC and AT are investigated to explain the roles of DC and AT for AT-AH-DC-FMF. Then the bending loss as a function of the parameters of AT and AH is calculated to help the bending loss values of AT-AH-DC-FMF to satisfy the ITU-T recommendations of G. 654. (Ⅲ) In section 4, the geometrical birefringence and stress birefringence are analyzed. (Ⅳ) In section 5, to confirm the compatibility of our proposed AT-AH-DC-FMF with mature WDM technology, the broadband characteristics such as δneff, Δneff, Aeff and chromatic dispersion are comprehensively researched over the whole C and L band. (Ⅴ) In section 6, the fabrication methods and feasibility of our proposed AT-AH-DC-FMF are briefly discussed. Fig. 1 is the schematic of our proposed AT-AH-DC-FMF which composes of a step-index core in the center with some surrounded asymmetric air holes and air trenches, and a circular pure SiO2 cladding. A list of parameters are used to describe the geometric structure of AT-AH-DC-FMF, where r, d, θ, β, w and p are the core radius, the DC radius, the angles of air trenches along y-axis and x-axis, the width of air trenches and the radius of air holes, respectively. l is the distance between air holes and core. Three particularities of our design should be noted: (1) asymmetric air trenches lie closely to the core, which affects the symmetry of the mode fields and the effective index of guided modes directly, producing great contribution in separating the degenerated spatial modes and non-degenerated LP modes; (2) the AT and AH has the maximum refractive index difference to the core, thus it exhibits better capability in reducing the bending loss of higher-order LP modes; (3) compared with bow-tie and steering wheel-type ring structure FMF, our design without complex doping of multi-clad structures and the extreme short gap between special structure and core, thus avoiding the increase of manufacturing complexity. The refractive index profile of AT-AH-DC-FMF is shown in Fig. 1(c), where nco, nde, ncl and nair represent the refractive index of core, DC, cladding and air holes/trenches. The refractive index profiles in x-axis and k-direction are different as the exist of air trenches and air holes. We denote the refractive index difference Δn1 = ndenco. Correspondingly, Δn2 = ncl -nco. In this paper, the core parameters with r = 8.8 μm, Δn1 = -0.2% and Δn2 = -1.1% support 7-LP-mode (LP01, LP11, LP21, LP02, LP31, LP12 and LP41) are employed. Then the detailed numerical simulations are carried out with a finite element mode solver (COMSOL Multiphysics) in the following sections. It has been pointed out in [24] and [25] that a detailed explanation of the analysis method is very important to prove the correctness of a study. Therefore, we list the main simulation steps as follows: Firstly, establishing the model of our proposed AT-AH-DC-FMF with the "Wave optics" module of COMSOL Multiphysics. Secondly, setting up different materials for different regions of our proposed AT-AH-DC-FMF. Thirdly, setting up a perfectly matched layer (PML) at outmost region of the model to absorb the light escaping out the concerned region. Fourthly, dividing the grid with normal size. Finally, the simulation results can be achieved after the mode analysis.

A. DC PARAMETERS
To highlight the importance of DC for our proposed AT-AH-DC-FMF, the min Δneff of the AT-AH-FMF is calculated and presented in Fig. 2 (Δn1 = 0). As can be seen from this region, without DC, the min Δneff with its value of 0.86×10 -3 appears between LP21 and LP02, indicating that there is a higher possibility of coupling between LP21 and LP02 during the transmission. This phenomenon is consistent with the results in [5], indicating the correctness of our simulation method. Previous studies have shown that the neff of LP modes can be adjusted by changing the refractive index of the core region [5]. Therefore, a DC is added to the center of core to manipulate the neff between LP modes, to keep the min Δneff larger than the threshold value of 1×10 -3 . The colormap of min ∆neff between adjacent LP modes dependence on d and ∆n1 is presented in the other regions of Fig. 2.   Fig. 2, two black dotted lines indicate that with the variation of d and ∆n1, the min ∆neff of our proposed AT-AH-DC-FMF appears between different adjacent LP modes. We circle the region that produces great contribution in separating the non-degenerated LP modes with red dotted line. In this region, the min ∆neff between adjacent LP modes can be increased from initial 0.86×10 -3 to 1.06×10 -3 .
In addition, the region with ∆n1 ranging from -1.0% to -0.1% and d ranging from 0.6 to 1.6 μm shows enough fabrication tolerances. Therefore, to apply our proposed AT-AH-DC-FMF to MIMO-less SDM systems, the parameters of DC (d and ∆n1) should be selected strictly according to our simulation results. Here, we determine parameters of the DC as d = 1 µm and ∆n1 = -0.2%. At this time, the min ∆neff is 1.02×10 -3 .

B. AT/AH PARAMETERS
In this section, we aim to find out the optimum parameters of the air trenches, air holes and cladding, so as to achieve the 12-mode AT-AH-DC-FMF. Before setting out to do this work, the first thing we should notice is that the number of LP modes may be decreased after proper air trenches are added to our fiber. The main reason is that the air trenches affect the neff of LP mode, resulting in some neff of higherorder LP modes (LP12 and LP41) being less than ncl, which leads to the leakage of some higher-order LP modes. Traditionally, to increase the number of propagation modes, either the radius of core or the refractive index difference between the core and the cladding |∆n2| should be increased. However, the bending loss of the higher-order LP modes may still stay at a high value through this way, and the doping concentration will be increased. Considering the particular structure of our design, four air holes are added to the cladding to reduce the equivalent index of the cladding, so that the effective refractive index of LP12 and LP41 are larger than the equivalent index of the cladding. So the LP12 and LP41 can be bound in the core. Then the following steps are taken to select the optimum parameters of the air trenches, air holes and cladding: (1)   To simultaneously keep Δneff larger than 1×10 -3 and δneff larger than 1×10 -4 , the θ and β should be selected strictly. We present the min Δneff between adjacent LP modes and the min δneff between adjacent spatial modes as a function of θ and β at the wavelength of 1550 nm in Fig. 3. In our simulation, we increase the angle β from 28 deg to 41 deg with a step size of 1 deg and the angles θ from 81 deg to 105 deg with a step size of 1 deg. From Fig. 3, it is obvious that the min Δneff increases with the increase of θ and β. In the region of 38 deg ≤ β ≤ 41 deg and 96 deg ≤ θ ≤ 105 deg, Δneff is larger than 1×10 -3 , however, δneff is smaller than 1×10 -4 . In the region of 28 deg ≤ β ≤ 33 deg and 82 deg ≤ θ ≤ 95 deg, δneff is larger than 1×10 -4 , however, Δneff is smaller than 1×10 -3 . Both regions should be excluded because they are inconsistent with the above requirements for Δneff and δneff. We circle the selected region that well satisfies the requirements for Δneff and δneff with a black dotted box. The selected region with θ ranging from 86 deg to 102 deg and β ranging from 31 deg to 40 deg indicates that our proposed AT-AH-DC-FMF possesses feasible fabrication tolerance during manufacturing air trenches. Finally, the parameters of air trench are fixed as θ = 88 deg and β = 36 deg. At this time, the min Δneff = 1.03×10 -3 and the min δneff = 1.24×10 -4 .
After the θ and β have been determined, the bending loss of our proposed AT-AH-DC-FMF can be controlled by adjusting the radius of air holes, the width of air trenches and the refractive index of cladding. To guarantee our proposed AT-AH-DC-FMF working over the C and L band, according to ITU-T recommendations G. 654, the bending loss of the highest-order mode (LP41) should be lower than 0.5 dB/100 turns (the bending radius R = 30 mm) at 1625 nm [26]. In addition, the bending loss of the redundant mode (LP22) (R = 140 mm) should be greater than 1 dB/m at 1530 nm [26]. Before setting out to get proper parameters for satisfying the requirements of bending loss, it is necessary for us to figure out the influence of the parameters of air holes, cladding and the width of air trenches on Δneff, δneff. Therefore, the influence of l, w and Δn2 on the number of modes, min Δneff and min δneff are also explored in Fig. 4(a-c). Then the bending loss of AT-AH-DC-FMF as a function of the width of air trenches and the radius of air holes is investigated and the results are plotted in Fig. 4 (d). VOLUME XX, 2017 1  Fig. 4(a,b) shows the min Δneff for adjacent LP modes and the min δneff for adjacent spatial modes as a function of l and w at the wavelength of 1550 nm, respectively. As can be seen from Fig. 4(a), the requirements of min Δneff and min δneff can be satisfied when l is larger than 1.6 μm, and the min Δneff and min δneff change slightly with the increase of l. In order to facilitate processing, we can choose l with a relatively larger value. Fig. 4(b) illustrates that the requirements of min Δneff and min δneff can be satisfied when w is ranging from 6.2 μm to 12.2 μm, and the min Δneff and min δneff change slightly with the increase of w. In Fig. 4(c), the number of modes can be increased with the increase of doping concentration. Five-modes fiber with min Δneff = 1.98×10 -3 can be realized when -0.5% ≤ Δn2 ≤ -0.1%. When Δn2 = -1.1%, 12 spatial modes are supported in our proposed AT-AH-DC-FMF with the min Δneff larger than 1.03×10 -3 and the min δneff larger than 1.24×10 -4 , respectively. When Δn2 ≤ -1.2%, 14 spatial modes can be supported in our proposed AT-AH-DC-FMF. In this paper, we determine Δn2 to -1.1%. Fig. 4(d) shows the bending loss of AT-AH-DC-FMF as a function of the width of air trenches and the radius of air holes. The orange solid line and orange dashed lines with circles correspond to the bending loss as a function of p at w = 10.2 μm. According to the orange dashed lines with circles, 7.9 μm ≤ p ≤ 8.0 μm is required to keep the bending loss of unwanted mode (LP22) larger than 1 dB/m (the bending radius R = 140 mm) at 1530 nm. The orange solid line with circles indicate that 7.9 μm ≤ p ≤ 8.0 μm is needed to keep the bending loss of the highest mode(LP41) lower than 0.5 dB/100 turns (R = 30 mm) at 1625 nm. The blue solid and dashed lines with circle represent the bending loss as a function of w at p = 7.9 μm. It can be seen in these two lines that 10.2 μm ≤ w ≤ 10.4 μm is needed to keep the bending loss of the highest mode(LP41) is lower than 0.5 dB/100 turns (the bending radius R = 30 mm) at 1625 nm. Therefore, 7.9 μm ≤ p ≤ 8.0 μm and 10.2 μm ≤ w ≤ 10.4 μm is acceptable to help the bending loss values of AT-AH-DC-FMF to satisfy the ITU-T recommendations of G. 654.

FIGURE 5. The mode field distributions and electric field polarization directions of 12 spatial modes in our optimized AT-AH-DC-FMF at 1550 nm.
It is worth noting that in the core regions without air trenches around, its ability to bound light is weaker than that of the core regions with air trenches around, thus resulting in a little energy leakage into the cladding. But it does not affect the performance of our proposed AT-AH-DC-FMF.

IV. BIREFRINGENCE
Birefringence analysis of an optical fiber with an asymmetric refractive index distribution is necessary [5], [19]. Our proposed AT-AH-DC-FMF with asymmetrical air holes/trenches in the x and y axis, so that the residual thermal stress components in the x and y directions are different, which may result in birefringence through the elasto-optical effect. After the material constants such as the Yong's modulus E, the Poisson's ratio ν, the thermal expansion coefficients α and material density ρ are determined, the birefringence analysis can be conducted with the "Structural Mechanics Module" of COMSOL Multiphysics. α of GeO2 and SiO2 are 7×10 -6 (1/K) and 5.4×10 -7 (1/K), respectively. α of a doped material can be expressed by mixture model shown as α = (1-m)α0 + mα1 [27], where α0 and α1 are thermal expanding coefficient of the two kinds of dopants, 1−m and m denote the mole percentage of each dopant. The used elastic material parameters for modeling are calculated with the above method and listed in Table 1. Drawing temperature T0 (℃) 1000 Operating temperature T1 (℃) 20 FIGURE 6.

The Von Mises stress distribution and stress birefringence distribution on the transverse cross section of AT-AH-DC-FMF.
Birefringence consists of two components: geometrical birefringence (BG) that is induced by irregular geometry and stress birefringence (Bs) that is related to thermal stress. The geometrical birefringence is defined as the difference between the neff of x-axis and y-axis, namely, B G = n eff x − n eff y . The BG of the eigenmodes LP01 is only 2.4×10 -6 . The stress birefringence is expressed as Bs =Δc(δx -δy), where Δc = 3.43×10 −12 m 2 /N, δx and δy are the stresses along the x-axis and y-axis, respectively. Fig. 6 shows the Von Mises stress distribution and the stress birefringence on the transverse cross section of our proposed AT-AH-DC-FMF. The maximum Bs in the core region is only 5.6×10 -5 . The average Bs is only 5.55×10 -6 through calculating with integral formula [28]. Note that the Birefringence in our proposed AT-AH-DC-FMF is relatively small.

V. BROADBAND CHARACTERISTICS
In this section, the performance of AT-AH-DC-FMF (neff, Δneff, δneff, chromatic dispersion D and effective mode field area) covering the whole C and L band is investigated. The chromatic dispersion of AT-AH-DC-FMF is worth noting as it is a key factor resulting in optical pulse broadening. The chromatic dispersion D is composed of material dispersion Dm and waveguide dispersion Dw, both of which can be expressed as the following equations [13].
Where c is the velocity of light in a vacuum, λ is the wavelength of light, nm is the refraction index of material as a function of λ, Re(neff) is the real part of the neff, respectively. The hybrid Sellmeier equation [29] is applied to calculate the refraction index of SiO2 and GeO2-SiO2 at different wavelengths. The differential mode delay (DMD) that is crucial for reduction of power consumption and complexity of MIMO processing is defined as follows (take LPnm and LP01 as an example) [13].
The neff, Δneff, δneff, chromatic dispersion and effective mode field area (Aeff) as a function of wavelengths are calculated and represented in Fig. 7. VOLUME XX, 2017 1 The effective refraction index, min Δneff and min δneff as functions of wavelengths are investigated in Fig. 7(a) and 7(b). As can be seen from Fig. 7(a), asymmetric air trenches lie closely to the core, which affects the symmetry of the mode fields and the effective index of guided modes directly, thus leading to the neff of LP12 and LP41 is lower than the refraction index of cladding ncl. But the AT and AH with enough size has the maximum refractive index difference to the core, exhibiting better capability in reducing the bending loss of higher-order LP modes and then transferring the leakage modes LP12 and LP41 into guided modes. In Fig. 7(b), the min Δneff of LP modes is larger than 1×10 -3 . In particular, as the min δneff larger than 1×10 -4 over the whole C and L bands, the fourfold degenerate LP modes can be effectively separated into twofold modes, indicating our proposed AT-AH-DC-FMF has great potential to be applied to MIMO-less SDM. Fig.  7(c) illustrates that the chromatic dispersions of all modes (except LP41) increase slightly from conventional band to long-wavelength band. The difference among chromatic dispersion of different modes main caused by waveguide dispersion. That is, due to the particularity geometric structure of our proposed few mode fiber, the transmission path of C-L band light pulse in the few mode fiber is different, which affects the effective refractive index of the mode, and the effective refractive index of LP41 is the most affected. In the whole C and L band, the chromatic dispersion of all modes are lower than |60| ps/nm/km, which does not affect the normal use of our proposed AT-AH-DC-FMF in short-distance transmission. Even if we want to achieve dispersion-free propagation, dispersioncompensation techniques can be employed [30]. Fig. 7(d) shows the Aeff of spatial modes dependence on wavelengths. The Aeff of all spatial modes (except LP12b) are larger than 100 µm 2 and the Aeff of all spatial modes (except LP41) slightly changes with wavelength. The broadband characteristics over a wide wavelength range of covering the whole C and L bands indicate the compatibility of our proposed AT-AH-DC-FMF with the mature WDM technique.
The performance of our depends on systematic variations of the fiber parameters that occur during the production process. So, we summarize the tolerance for Δneff and δneff of AT-AH-DC-FMF in Table 2. In this range, the effective index difference between the effective index difference between adjacent LP modes (Δneff) is larger than 1×10 -3 and the effective index difference between adjacent spatial modes (δneff) is larger than 1×10 -4 , indicating that our proposed few mode fiber has enough device robustness. The optimal fiber performance of AT-AH-DC-FMF is exhibited in Table 3, and it can be seen in Table 3 that the bending loss of our proposed AT-AH-DC-FMF is lower than the bending loss in [5]. The confinement losses (CL) of all modes that are listed in Table 3 where λ is the free-space wavelength and Im(neff) is the imaginary part of the effective refractive index obtained by mode analysis. CL (dB/m) <<10 -9 (LP01), <<10 -9 (LP11a), <<10 -9 (LP11b), <<10 -9 (LP21a), <<10 -9 (LP21b), <<10 -9 (LP02), <<10 -9 (LP31a), <<10 -9 (LP31b), 1×10 -6 (LP12a), 3

VI. RECOMMENDATIONS FOR FABRICATION
Selecting an appropriate fabricating method can improve the probability of success during fiber fabrication. D. Chen et al. [31] used uniform air holes to form different basic cell structures for manufacturing triangular-hole photonic crystal fiber (PCF) and rectangular-hole PCF in practice, which provided an idea for manufacturing the AT in our proposed AT-AH-DC-FMF. Through this method, the structural distortion can be minimized during fabricating fiber with different shaped holes. According to mature technology in fabricating hole-assisted fibers [32], we believe the Stack-And-Draw technique is one of the ways that are suitable for the fabrication of our proposed AT-AH-DC-FMF. There are three steps that need to be followed to manufacture the AT-AH-DC-FMF. Firstly, some glass capillaries and glass rods are used to form AT and cladding. Secondly, 6% and 7.3% mole fraction of GeO2 are needed to dope in SiO2 to form the DC and core. Finally, our optimized AT-AH-DC-FMF can be manufactured after the melting and drawing process. The low doping during the manufacturing of DC and core indicates a relatively simpler fabrication process of our proposed AT-AH-DC-FMF than common elliptical core fibers and PMF. There is another method to fabricate our proposed AT-AH-DC-FMF. The main processes are as follows: firstly, fabricating a stepindex core fiber preform with a circular pure SiO2 cladding by utilizing the normal MCVD technology; secondly, drilling the uniform air holes in particular positions of our fabricated preform to form the preform of our proposed AT-AH-DC-FMF; finally, our proposed few mode fiber can be fabricated after the drawing process. Previous study [33] provides us a method of filling nitrogen gas in the uniform air holes to against the ambient pressure, thus stopping the holes collapse during the fiber drawing. The discussion in this section proved that the design scheme of our proposed AT-AH-DC-FMF is feasible.

VII. CONCLUSION
In this paper, an AT-AH-DC-FMF features some asymmetrical air trenches, air holes and a depressed stepindex core, supporting 12 spatial modes is proposed for MIMO-less SDM. The fiber performance dependence on the parameters of DC, AT and AH are investigated with FEM. According to our simulation results, the minimum Δneff of larger than 1.03×10 -3 and minimum δneff of greater than 1.24×10 -4 can be achieved with the help of AT and DC. The AT and AH plays an important role in helping the bending loss of AT-AH-DC-FMF satisfying the ITU-T recommendations of G. 654. Our proposed AT-AH-DC-FMF shows small birefringence effect and it is feasible to be fabricated with the Stack-And-Draw technique. In addition, stable comprehensive broadband characteristics such as small chromatic dispersion, large Δneff, large δneff and large effective mode area over the whole C and L band provide an opportunity for the MIMO-less SDM based on AT-AH-DC-FMF to be compatible with mature WDM technology, thus further enlarging optical communication capacity and simplifying the system complexity.