## Abstract

We present a steering wheel-type ring depressed-core few-mode fiber (SWTR-DC-FMF) that features a central depressed step-index core and a novel SWTR structure consisted of two symmetrical high-index parts and low-index parts, respectively. The DC and SWTR make great contribution to separate the non-degenerated LP modes and spatial modes in the circular symmetry core, resulting in fully improved mode spacing. The designed fiber is able to support 10 spatial modes with the minimum effective index difference (Min Δ*n _{eff}*) between adjacent spatial modes larger than 1.93 × 10

^{−4}and the Min Δ

*n*between adjacent LP modes above 1.51 × 10

_{eff}^{−3}at the same time, facilitating potential fiber spatial mode multiplexing transmission with less multiple-input multiple-output (MIMO-less) digital signal processing technique. The broadband performance including

*n*

_{eff}, Δ

*n*, effective mode area (

_{eff}*A*

_{eff}) and differential mode delay (

*DMD*) is comprehensively investigated over the whole C and L band. Moreover, the birefringence and fabrication tolerance are discussed. The designed fiber targets emerging applications in short-reach weakly coupled space-division multiplexing (SDM) optical networking to increase transmission capacity and spectral efficiency and further reduce the system complexity effectively.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the coming era of 5G, cloud computing and its applications, data amount is gradually showing explosive growth [1]. It is reported that the current optical networks and data centers based on traditional single-mode fibers (SMFs) are rapidly reaching their maximum capacity following the nonlinear Shannon limit [2]. To solve this bottleneck as soon as possible, space-division multiplexing (SDM) transmission has been recently studied in depth as a feasible measure [3]. As one of the channel multiplexing technologies, multi-core fiber (MCF) is the most direct solution in SDM [4]. Inevitably, there are a great deal of scientific challenges in the design and manufacture of MCFs, which need to ensure high core density, low attenuation loss and inter-core crosstalk simultaneously. It is also increasingly difficult to couple signals in and out of each core without causing large coupling loss when the cores have to be closely packed in MCFs with a limited cladding range. The few-mode fiber (FMF), acts as another practical approach in SDM, due to its relatively simple manufacturing method and low loss connectivity with traditional SMFs [5]. The main difficulty encountered in FMFs is the crosstalk induced by mode coupling [4,5]. In conventional FMFs, the eigenmodes in each mode group are almost degenerate, and their optical power easily coupled to each other under external perturbations, resulting in serious crosstalk among signal channels [6,7]. The overall cost increases due to the multiple-input multiple-output (MIMO) processing must be used at the receivers. In addition, the complexity of MIMO scales linearly with the number of supporting transmission modes, the power usage is likely to burst out in SDM systems with complex MIMO facilities. A competitive approach to simplify or eliminate MIMO, is to lift the degeneracy between the adjacent modes and achieve effective index difference, $\Delta {n_{eff}}$, values larger than ${10^{ - 4}}$, which is also the typical value of birefringence in polarization-maintaining fibers (PMFs) [3,8]. Moreover, when the $\Delta {n_{eff}}$ between non-degenerated modes is larger than 10^{−3}, it is considered to be a weakly-coupled FMF [9]. This method would have a clear advantage in system simplification; therefore, it would be especially suitable for the short-reach transmission in the scenarios such as data centers and computer rooms.

Recently, attempts to use special core structure as a common method has been reported to improve mode spacing. The ring-core, added to the specific area of the higher-order mode so that the ${n_{eff}}$ of this modal can be adjusted significantly. For example, in Ref. [10], a ring-assisted 4-LP-mode fiber has been fabricated with the Min $\Delta {n_{eff}}$ as high as $1.8 \times {10^{ - 3}}$. Other 6-LP-mode fiber design strategies such as combined-ring core and nanopore-assisted core, the Min $\Delta {n_{eff}}$ between adjacent modal is greater than $1.8 \times {10^{ - 3}}$ in [11] and [12], respectively. However, only non-degenerated LP modes are split effectively, $4 \times 4$ MIMO facilities are also needed to recover the fourfold degenerate LP modes. In order to realize MIMO-free, PMFs have been applied [13]. The stress-applying parts (SAPs) and specifically designed noncircular core (e.g., elliptical core) in PMFs can induce enough birefringence to separate each eigenmodes. For example, in [1], a 10 distinctive polarization modes fiber has been achieved with Min $\Delta {n_{eff}}$ of larger than $1.32 \times {10^{ - 4}}$ by air hole and elliptical-ring core. A PANDA-type elliptical core fiber supporting 24 eigenmodes has been proposed with Min $\Delta {n_{eff}}$ of larger than $1.35 \times {10^{ - 4}}$ in [4]. However, in most of PMFs, the relative refractive index difference between core and cladding is up to 3%, resulting in high doping difficulty and high loss. The perturbation of ellipticity also has a great impact on fiber performance. Moreover, the high doping concentration of B_{2}O_{3} in the SAPs and narrow distance between core and SAPs bring huge challenges to the current manufacturing technologies. Therefore, it is urgent to design a weakly-coupled FMF that features enough propagation modes, sufficient mode spacing, low doping and high fabrication tolerance to meet the requirements of MIMO-less SDM systems.

In this paper, we present a novel steering wheel-type ring (SWTR) in depressed-core (DC) FMF to significantly separate the non-degenerated LP modes and spatial modes, thereby fully improving mode spacing. COMSOL Multiphysics are used to simulate the modes and then the data are integrated to MATLAB for analyzing. Firstly, the fiber schematic topology and theory are introduced in section 2 to explain our designing ideas, the fiber parameters are selected to enable support of 10 spatial modes. Then, in order to understand the influence of DC and SWTR, the ${n_{eff}}$ and $\Delta {n_{eff}}$ of propagation modes dependence on fiber parameters are investigated by finite element method (FEM). From section 3, the Min $\Delta {n_{eff}}$ between adjacent spatial modes is able to larger than $1.93 \times {10^{ - 4}}$, and the Min $\Delta {n_{eff}}$ between adjacent LP modes can be above $1.51 \times {10^{ - 3}}$ at 1550 nm. Further, the broadband characteristics over C and L band are also realizable in section 4. Through numerical simulations, we confirm that the designed structure effects the geometric symmetry of the refractive index and mode field distribution very well. The fourfold degenerate LP modes can be sufficiently divided into twofold modes, thus only $2 \times 2$ MIMO facilities are needed at the receiver end. Finally, we briefly discuss the birefringence and fabrication tolerance of the proposed SWTR-DC structure and the results are feasible.

## 2. Schematic topology and theory

The cross section of the proposed 10-spatial-mode SWTR-DC-FMF is shown in Fig. 1, which comprises a DC step-index core in the center with a surrounded SWTR, and a circular pure silica cladding. The SWTR is created by means of two symmetrical sector high-index parts and low-index parts, respectively. The particularity of our design is that the SWTR lies closely enough to the core, which can directly affect the refractive index of the guided modes and change the geometric symmetry of the mode fields. More importantly, compared with the scheme of side holes and bow-tie structure, our design avoids the increase of manufacturing complexity caused by the extreme short gap between the core and the special structure (about 1∼2 $\mu \textrm{m}$). Meanwhile, it changes the reduction of robustness due to the multiple independent parameter selection in the design of PMFs.

While the mentioned benefits of the proposed fiber are unquestionable, some downsides of such a structure also need to be considered. First, the SWTR may modify propagation characteristics such as dispersion and differential mode delay (*DMD*). Then, it may also lead birefringence effect in the core area. The discussion of fiber properties and birefringence is necessary and we will extend these analyses in section 4.

A number of parameters are used to describe the fiber geometric structure in Fig. 1(b), where *d, r, θ* and *w* are the DC radius, the core radius, the sector angle of low-index parts and the width of SWTR, respectively. The cladding range is 125 $\mu \textrm{m}$. Figure 2 shows the refractive index profile of the SWTR-DC-FMF, where ${n_{de}}$, ${n_{co}}$, ${n_h}$, ${n_{cl}}$ and ${n_l}$ represent the refractive index of DC, core, high-index sector parts, cladding and low-index sector parts, respectively. The relative refractive index difference between the core and cladding is denoted as $\Delta {n_{co}} = ({{n_{co}} - {n_{cl}}} )/{n_{cl}}$. Correspondingly, $\Delta {n_{de}} = ({{n_{co}} - {n_{de}}} )/{n_{cl}}$, $\Delta {n_h} = ({{n_h} - {n_{cl}}} )/{n_{cl}}$ and $\Delta {n_l} = ({{n_{cl}} - {n_l}} )/{n_{cl}}$. The core parameters supported 6-LP-mode (LP_{01}, LP_{11}, LP_{21}, LP_{02}, LP_{31} and LP_{12}) are deployed, which are $r = 8\; \mu \textrm{m}$ and $\Delta {n_{co}} = 0.93\%$.

## 3. Fiber parameter selection

#### 3.1 DC parameter

The normalized frequency is $V = 2\pi r\sqrt {n_{co}^2 - n_{cl}^2} /\lambda $, where λ is the operating wavelength, taken here as 1550 nm [14]. Figures 3(a) and 3(b) show the normalized propagation constant *b* as a function of *V* and the normalized power distribution as a function of normalized radius in FMF, respectively. The *b* is defined as $b = ({n_{eff}^2 - n_{co}^2} )/({n_{co}^2 - n_{cl}^2} )$ [14]. It can be observed that the *b* of LP_{21} and LP_{02} are closest to each other within the first 6 LP modes, which indicates they have a higher possibility of coupling during the transmission. In Fig. 3(b), the area where most of the LP_{21} mode’s power is distributed is different from that of LP_{02} mode’s, and it is possible to change one’s ${n_{eff}}$ without influencing another’s seriously.

Former researches show that the ${n_{eff}}$ of LP modes can be operated by changing the refractive index of the corresponding core area where the most of the LP modes’ power is distributed in [15]. Therefore, we apply a DC to manipulate the ${n_{eff}}$ of LP_{02}, thereby increasing the mode spacing between LP_{21} and LP_{02}. Figure 4 presents a colormap of the Min $\Delta {n_{eff}}$ between adjacent LP modes dependence on *d* and $\Delta {n_{de}}$. The Min $\Delta {n_{eff}}$ between adjacent LP modes can be increased from the initial $0.8 \times {10^{ - 3}}$ to larger than $1.6 \times {10^{ - 3}}$, with the help of DC. There are mainly two areas of DC parameters combination that can effectively heighten the Min $\Delta {n_{eff}}$, one is a small *d* (less than 1 $\mathrm{\mu }\textrm{m}$) with a high $\Delta {n_{de}}$, and the other is a bigger *d* with a lower $\Delta {n_{de}}$. The latter is chosen as the DC parameters (the selected region) because it has a wider tolerance range. For example, *d* from 1 $\mathrm{\mu }\textrm{m}$ to 4.5 $\mu \textrm{m}$ and $\Delta {n_{de}}$ from 0.1% to 0.3% are able to keep Min $\Delta {n_{eff}}$ much greater than $1 \times {10^{ - 3}}$, which is a recognized threshold for weakly coupled FMFs [9]. Figure 5 shows the normalized power distribution of LP_{02} as a function of normalized radius with different combinations of DC parameters in the selected region. It can be seen that due to the influence of DC, the mode distribution of LP_{02} expands outward, compared with the normal step-index core. The difference in the impact of the three listed DC parameter combinations on the LP_{02} mode is negligible. Here, we fix the structure of DC with $d = 1.5\; \mu \textrm{m}$ and $\Delta {n_{de}} = 0.3\%$.

#### 3.2 SWTR parameter

The SWTR parameters are optimized to alter the ${n_{eff}}$ of spatial modes and modes field distribution in the *x*-axis and *y*-axis directions, as shown in Figs. 6(a)–6(d). The black and blue lines correspond to the Min $\Delta {n_{eff}}$ for spatial modes and the Min $\Delta {n_{eff}}$ for LP modes at 1550 nm, respectively.

In Fig. 6(a), $|{\Delta {n_l}} |> $ 0.003 is needed to satisfy the demand of the Min $\Delta {n_{eff}}$ for spatial modes of larger than ${10^{ - 4}}$, while the Min $\Delta {n_{eff}}$ for LP modes reaches its peak when $|{\Delta {n_l}} |= 0.0045$. The high-index sector parts also lead significant impact on the ${n_{eff}}$ of spatial modes. However, with the increase of $\Delta {n_h}$, the black line and the blue line show opposite trends in Fig. 6(b). To ensure both of the target values, slightly GeO_{2}-doped sector parts are deployed (the crimson region). It should be noted that as the sector index increases, the interval among spatial modes is improved, and the ${n_{eff}}$ of the modes with more energy at the edge of the core is more susceptible to the influence of SWTR parameters. This will definitely change the spacing among LP modes, which is reflected in the appearance of the Min $\Delta {n_{eff}}$ threshold in Figs. 6(a) and (b). From Fig. 6(c), the low-index parts with a larger $\theta $ are beneficial to increase the Min $\Delta {n_{eff}}$ for LP modes, and *θ* from 110$^\circ $ to 135$^\circ $ can get a better Min $\Delta {n_{eff}}$ for spatial modes. Furthermore, the width of SWTR shows limited impact on the $\Delta {n_{eff}}$, as shown Fig. 6(d). According to the numerical results, the SWTR parameters are fixed as $\Delta {n_l} ={-} 0.005$, $\Delta {n_h} = 0.002$, $\theta = 120^\circ $ and $w = 8\; \mu \textrm{m}$. Therefore, the Min $\Delta {n_{eff}}$ between adjacent spatial modes is able to larger than 1.93*10^{−4}, and the Min $\Delta {n_{eff}}$ between adjacent LP modes can be above 1.51×10^{−3} at 1550 nm. In particular, the intensity profiles and electric field polarization directions (red arrow surface) for 10 spatial modes (LP_{01}, LP_{11a}, LP_{11b}, LP_{21a}, LP_{21b}, LP_{02}, LP_{31a}, LP_{31b}, LP_{12a} and LP_{12b}) at 1550 nm are shown in Fig. 7.

## 4. Ultra-band characteristics and other fiber properties

#### 4.1 Birefringence

In accordance with the hybrid Sellmeier equation describing the refractive index dependence on wavelength and concentration (in mol%) of dopants, we can theoretically get the mode fraction of GeO_{2} and CF_{4} in core and SWTR which are denoted as *m* and *n* [16]. Thermal expansion coefficients of pure SiO_{2}, GeO_{2} and CF_{4} are 5.4×10^{−7} (1/K), 7×10^{−6} (1/K) and 61×10^{−6} (1/K), respectively [17]. Specifically, thermal expansion coefficient ($\alpha $) of a doped material can be expressed by a role of mixture model shown below [16]

Where ${\alpha _0}$ and ${\alpha _1}$ are thermal expanding coefficient of the two kind of dopants, $1 - m$ and *m* denote the mole percentage of each dopant. The *m* of each material in the designed fiber structure can be obtained using the formulas mentioned in [17,18]. Table 1 lists the used elastic material parameters for modeling including thermal expansion coefficient ($\alpha $), Yong’s modulus ($E$), Poisson’s ratio, density ($\rho $), first and second stress optical coefficient (${B_1},\; {B_2}$), drawing temperature (${T_0}$) and operating temperature (${T_1}$) [17].

Birefringence of optical fiber is necessary to analyze when the core structure has an asymmetric refractive index distribution. It consists of two components: geometrical birefringence (${B_g}$) and stress birefringence (${B_s}$) [19]. The former is defined as the difference in the ${n_{eff}}$ of *x*-axis and *y*-axis, and it depends not on the pressure and temperature, namely, ${B_g} = n_{eff}^x - n_{eff}^y$. The maximum ${B_g}$ is only $3.68 \times {10^{ - 5}}$ in the eigenmodes of LP_{12a}. The latter is related with thermal stress, which is defined as ${B_s} = \Delta c({{\sigma_x} - {\sigma_y}} )$, where the $\Delta c = 3.43 \times {10^{ - 12}}\; {m^2}/N$, ${\sigma _x}$ and ${\sigma _y}$ are the stresses along the *x* and *y* directions individually. Figure 8 shows the Von Mises stress distribution and the ${B_s}$ in the transverse cross section of our designed SWTR-DC-FMF. Maximum ${B_s}$ around the core is about ${10^{ - 3}}$, but the area of ${B_s} \ge \; {10^{ - 4}}$ is quite small. The average ${B_s}$ is only $6.2 \times {10^{ - 6}}$ using the integral formula [20]. Note that such birefringence is relatively small in our design.

#### 4.2 Broadband characteristics and fabrication tolerance

In this section, we further explore the wavelength dependence of the supported spatial modes in the designed SWTR-DC-FMF over the whole C and L band in Figs. 9(a)–9(d). The refractive indices of SiO_{2}, GeO_{2}-SiO_{2} and F-doped SiO_{2} at different wavelengths can be calculated using the mentioned above hybrid Sellmeier equation, they are also can be found in [21,22].

In Figs. 9(a) and 9(b), the modal spacing between the adjacent LP modes is maintained above $1.5 \times {10^{ - 3}}$. Specially, the fourfold degenerate LP modes separate into twofold modes efficiently and the Min $\Delta {n_{eff}}$ between adjacent spatial modes is higher than $1.84 \times {10^{ - 4}}$ in the C and L bands, indicating sufficient mode spacing improvement for MIMO-less SDM transmission. We find that the ${n_{eff}}$ of LP_{12} is closer to ${n_{cl}}$ in the L band, which means that it is more likely to cut-off during the transmission. The most straightforward solution is to increase the $\Delta {n_{co}}$, although the $\Delta {n_{co}}$ of less than 1% is sufficient to achieve the target properties in our settings. It is also possible to sacrifice a certain Min $\Delta {n_{eff}}$ through lower-doped $\Delta {n_l}$ in exchange for more stable optical performance. Figure 9(c) shows the ${A_{eff}}$ of spatial modes dependence on wavelength. The ${A_{eff}}$ of all spatial modes slightly increase (besides LP_{12a}) with wavelength and are larger than $100\; \mu {\textrm{m}^2}$. The *DMD* characteristic is defined as (take LP_{11a} and LP_{01} as an example) [9]

*c*is the light velocity in a vacuum. The

*DMD*is controlled within the acceptable range of 5 $\textrm{ps}/\textrm{m}$ to 23.6 $\textrm{ps}/\textrm{m}$ in the C and L bands.

Considering the current fiber manufacture facilities, it is preferred that the molecular fraction doping should be as small as possible, while the fiber structure should not be very special. In our design, the mode fraction of GeO_{2} in the core is only about 9.3% and the fraction of F-doped silica in the SWTR is less than 5%, indicating that it is a relatively very little doping in comparison with common PMFs and elliptical core fibers. Furthermore, we believe that the Stack-And-Draw technique is one of the suitable ways to fabricate the proposed SWTR-DC-FMF, according to the practical technology and experience of fabricating hole-assisted fibers [23,24] and segmented cladding fibers [25]. Figure 10 illustrates the main process of fabricating such a fiber by this method. First, the core preform with DC can be fabricated by chemical vapor deposition (CVD) technology. Then, the core preform needs to be embedded in the glass tube. After that, the glass capillaries are arranged to the core preform in a specific ratio based on the design strategy. Finally, the fiber can be fabricated after melting and drawing process.

Note that the proposed fiber structure has no need to take the drilling process of the preform into consideration, which means it can improve the production reliability significantly. From Fig. 4 and Fig. 5, one can see that different combinations of DC parameters in the selected region will not make the Min $\Delta {n_{eff}}$ between adjacent LP modes below $1.4 \times {10^{ - 3}}$. The range of *d* (from 1 $\mathrm{\mu }\textrm{m}$ to 4.5 $\mathrm{\mu }\textrm{m}$) and $\Delta {n_{de}}$ (from 0.1% to 0.3%) are acceptable. Moreover, section 3.2 shows the SWTR with $\Delta {n_l}$ and $\theta $ of around -0.005 and $120^\circ $ is sufficient to separate a fourfold mode into two twofold modes, and the Min $\Delta {n_{eff}}$ between spatial modes is much greater than the standard requirement of $1 \times {10^{ - 4}}.$ In this case, a weakly-coupled FMF for short-reach MIMO-less SDM systems is designed with feasible fabrication tolerance. Finally, we summarize the optimal fiber parameters with fabrication tolerance and performance of SWTR-DC-FMF in Tables 2 and 3, respectively.

## 5. Conclusion

We propose a novel SWTR and DC structure in FMFs to dramatically improve the mode spacing among non-degenerated LP modes and spatial modes. The SWTR composed of two symmetrical high-index and low-index parts with different sector angles and placed close enough to the core, resulting in significantly impact on the refractive index of the guided modes and thus change the geometric distribution of the mode fields. The FEM is applied for numerical simulation. To show the talents of DC and SWTR, we analyze the Min $\Delta {n_{eff}}$ of LP modes and spatial modes dependence on DC and SWTR parameters. The results show that the Min $\Delta {n_{eff}}$ between adjacent spatial modes is able to larger than $1.93 \times {10^{ - 4}}$, and the Min $\Delta {n_{eff}}$ between adjacent LP modes can be above $1.51 \times {10^{ - 3}}$ at the same time. Furthermore, the proposed fiber shows stable comprehensive broadband performance and relatively small birefringence effect. The fabrication tolerance is feasible and the stack-and-draw technique is acceptable as one of the fabrication methods of our design. With the viewpoints of these properties, we believe that the SWTR-DC-FMF can be used in short-reach SDM systems that require less MIMO and multiple independent signal channels.

## Funding

National Key Research and Development Program of China (2018YFB1801003); National Natural Science Foundation of China (61827817); Fundamental Research Funds for the Central Universities (2019YJS001).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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