## 2 Fiber Design  
Figure 1 presents the schematic cross-section and refractive index profile of the proposed DBT-MREC-MMF. The fiber comprises a cladding made of pure SiO\({}_{2}\), two bow-tie air holes, two bow-tie stress-applying areas doped with B\({}_{2}\)O\({}_{3}\)-doped silica, and three different step-index cores doped with GeO\({}_{2}\)-doped silica. The refractive index difference between the elliptical core and the cladding of the fiber is denoted as \(\Delta n_{1}\), and the long and short semi-axes of the elliptical core are \(a_{3}\) and \(b_{3}\), respectively. A high-index ring with the same ellipticity as the elliptical core is introduced by addition to the elliptical core. The high-index ring has an inner radius of long semi-axis \(a_{1}\) and short semi-axis \(b_{1}\), and an outer radius of long semi-axis \(a_{2}\) and short semi-axis \(b_{2}\), with a relative refractive index difference between the high-index-ring and cladding of \(\Delta n_{2}\). Moreover, a trench layer with the same ellipticity as the core is added close to the outer side of the elliptical core, with long semi-axis \(a_{4}\) and short semi-axis \(b_{4}\). The relative refractive index difference between the trench and the cladding is \(\Delta n_{3}\). The ellipticity of the core, high-index ring, and trench is defined as \(e=a_{1}/b_{1}=a_{2}/b_{2}=a_{3}/b_{3}=a_{4}/b_{4}\). The bow-tie stress-applying areas have parameters of an inner radius of \(r_{1}\), width \(h\), and angle \(\theta\), with an outer radius of \(r_{2}=r_{1}+h\). The bow-tie air hole parameters have the long semi-axis \(a_{5}\) and short semi-axis \(b_{5}\), angle \(\theta_{0}\), and ellipticity \(e_{0}=a_{5}/b_{5}\). The refractive index of air is 1. The diameter of the fiber cladding is 125 \(\upmu\)m.  
The relative refractive index differences between the cores and cladding are denoted as \(\Delta\)= \((n_{\text{core}}^{2}-n_{\text{clad}}^{2})/2n_{\text{core}}^{2}\). The cladding material is SiO\({}_{2}\) with \(n_{\text{eff}}\) of 1.444 at a wavelength of 1550 nm according to the Sellmeier equation [29]. The mole fraction (mol%) of GeO\({}_{2}\) and B\({}_{2}\)O\({}_{3}\), corresponding to \(\Delta n_{1}\), \(\Delta n_{2}\), and \(\Delta n_{3}\), and the stress-applying area refractive index can be theoretically calculated using the hybrid Sellmeier equation [29; 30]. The hybrid Sellmeier equation describes the relationship between the refractive index, wavelength, and concentration of dopants.  
The thermal expansion coefficient (\(\alpha\)) of the doped material is defined as \(\alpha\) = (1 \(-\) m) \(\alpha_{0}\) + \(\Delta\alpha_{1}\)[31], where m and \(\alpha_{1}\) represent the mole percentage and thermal expanding coefficient of the doped material, and 1 \(-\) m and \(\alpha_{0}\), respectively, are the mole percentage and thermal expanding coefficient of raw material. Thermal expansion coefficients of SiO\({}_{2}\), GeO\({}_{2}\), and B\({}_{2}\)O\({}_{3}\) are 5.4 \(\times\) 10\({}^{-7}\) (1/K), 7 \(\times\) 10\({}^{-6}\) (1/K), and 10 \(\times\) 10\({}^{-6}\) (1/K), respectively. In practical fiber fabrication, a 30% mol-doped fraction of B\({}_{2}\)O\({}_{3}\) in the stress-applying area has been utilized [31]. Based on this calculation, the refractive index of the stress-applying area is 1.468 at 1550 nm, and the thermal expansion coefficient is 3.378 \(\times\) 10\({}^{-6}\) (1/K). Further details regarding the fiber elastic parameters are summarized in [31], including the thermal expansion coefficient (\(\alpha\)), Young's modulus (E), Poisson's ratio (v), the first and second stress optical coefficients (C\({}_{1}\), C\({}_{2}\)), and operating and reference temperature. All simulations in this work were implemented using the finite element method at a wavelength of 1550 nm.