## 3 Results Simulation Analysis and Optimization
### Optimization of Multi-Ring Elliptical Core Parameters  
Figure 2 presents the schematic cross-section of the bow-tie muti-ring elliptical multi-mode fiber (BT-MREC-MMF). In this study, we first optimized the parameters of the optical fiber without adding bow-tie air holes. That is, we optimized the parameters of the bow-tie type multi-ring elliptical core optical fiber. We first optimized the parameters of the fiber high-index ring structure. We fixed \(a_{1}\) = 3 \(\upmu\)m, \(a_{3}\) = 9 \(\upmu\)m, \(a_{4}\) = 12 \(\upmu\)m, \(\Delta n_{1}\) = 0.012, \(\Delta n_{2}\) = 0.018, \(\Delta n_{3}\) = 0.005, \(r_{1}\) = 20 \(\upmu\)m, \(h\) = 8.9 \(\upmu\)m, \(\theta\) = 90\({}^{\circ}\), and \(r_{2}\) = 28.9 \(\upmu\)m. The parameters \(a_{2}\) and \(e\) were adjusted to optimize the \(\Delta n_{\mathrm{eff}}\) between adjacent eigenmodes. The influence of \(a_{2}\) and \(e\) on min \(\Delta n_{\mathrm{eff}}\) and the number of supported eigenmodes are shown in Figure 3. It can be seen that the min \(\Delta n_{\mathrm{eff}}\) first increases, and then decreases, with increasing \(a_{2}\) and \(e\), while the number of supported eigenmodes gradually increases with increasing \(a_{2}\) and decreasing \(e\). Within the region of 5.7 \(\upmu\)m \(\leq\)\(a_{2}\)\(\leq\) 6.3 \(\upmu\)m and 1.6 \(\leq\)\(e\)\(\leq\) 1.71, the proposed fiber supports 22 eigenmodes, and min \(\Delta n_{\mathrm{eff}}\) between adjacent eigenmodes is larger than 2.3 \(\times\) 10\({}^{-4}\). To obtain a larger min \(\Delta n_{\mathrm{eff}}\) and better-designed tolerances, we selected the middle point \(a_{2}\) = 6.1 \(\upmu\)m and \(e\) = 1.65 (i.e., \(b_{2}\) = 3.7 \(\upmu\)m) in the region as the parameters of the high-index ring structure,  
Figure 1: Schematic cross-section and refractive index profile of the DBT-MREC-MMF. which is marked by the red dot in Figure 3a. The preparation error ranges are \(\pm\) 0.2 \(\upmu\)m and \(\pm\) 0.5 \(\upmu\)m, respectively. The proposed MMF can support 22 eigenmodes and the effective refractive index difference between adjacent eigenmodes is greater than 2.58 \(\times\) 10\({}^{-4}\). We continued to optimize the parameters of the high-index ring; here, we fixed \(a_{1}\) = 3 \(\upmu\)m, \(a_{2}\) = 6.1 \(\upmu\)m, \(a_{3}\) = 9 \(\upmu\)m, \(a_{4}\) = 12 \(\upmu\)m, \(\Delta n_{1}\) = 0.012, \(\Delta n_{3}\) = 0.005, \(r_{1}\) = 20 \(\upmu\)m, \(h\) = 8.9 \(\upmu\)m, \(\theta\) = 90\({}^{\circ}\), and \(e\) = 1.65. Figure 4a,b shows variations of \(n_{\text{eff}}\) and the min \(\Delta n_{\text{eff}}\) as a function of the relative refractive index \(\Delta n_{2}\) in the high-index ring. Within the range of 1.6% \(\leq\Delta n_{2}\leq\) 2%, the min \(\Delta n_{\text{eff}}\) is related to the \(\Delta n_{\text{eff}}\) of the H\({}_{12}^{\text{G}}\) and H\({}_{12}^{\text{G}}\) modes. As shown in Figure 4b, with increasing \(\Delta n_{2}\), min \(\Delta n_{\text{eff}}\) decreases. To obtain a larger min \(\Delta n_{\text{eff}}\), we took \(\Delta n_{2}\) = 1.8%, such that the min \(\Delta n_{\text{eff}}\) could reach 2.5 \(\times\) 10\({}^{-4}\). As shown in Figure 5, we investigated the influence of \(a_{1}\) and \(a_{3}\) on min \(\Delta n_{\text{eff}}\) and the number of supported eigenmodes by the fiber at \(a_{2}\) = 6.1 \(\upmu\)m, \(a_{4}\) = 12 \(\upmu\)m, \(\Delta n_{1}\) = 0.012, \(\Delta n_{3}\) = 0.005, \(r_{1}\) = 20 \(\upmu\)m, \(h\) = 8.9 \(\upmu\)m, \(\theta\) = 90\({}^{\circ}\), and \(e\) = 1.65. From the rectangular area of Figure 5, within the region of 2.5 \(\upmu\)m \(\leq\)\(a_{1}\)\(\leq\) 3.6 \(\upmu\)m and 8.4 \(\upmu\)m \(\leq\)\(a_{3}\)\(\leq\) 9.2 \(\upmu\)m, the proposed fiber supports 22 eigenmodes, and the min \(\Delta n_{\text{eff}}\) between adjacent eigenmodes is larger than 2 \(\times\) 10\({}^{-4}\).