## 4 Mode Properties and Broadband Characteristics  
### Birefringence  
Figure 12 shows mode field intensity distributions with electric vectors (white arrows) distributions of modes supported by the DBT-MREC-MMF for all 22 eigenmodes at 1550 nm. It is apparent that all polarization directions of eigenmodes are approximately horizontal or vertical, and all the eigenmodes are well confined within the core region, which displays a superior capacity to maintain polarization.  
Birefringence performance is important for studying the polarization properties of fibers. High values of birefringence provide superior polarization retention. Modal birefringence is  
Figure 11: Variations of \(B_{\text{m}}\) (**left side**) and bending loss (**right side**) along the \(x\)-axis or \(y\)-axis direction at bending radius \(R\) = 30 mm as a function of (**a**) \(e_{0}\), (**b**) \(a_{5}\), and (**c**) \(\theta_{0}\).  
Figure 12: Model field distributions with electric vectors (white arrows) for 22 eigenmodes of the proposed DBT-MREC-MMF at the wavelength of 1550 nm.  
related to many factors, such as the size of the stress-applying area, temperature, core ellipticity, core-cladding refractive index difference, thermal expansion coefficient, and wavelength of light. When a fiber has an asymmetric structure and stress-applying areas, it is necessary to analyze its modal birefringence (\(B_{\rm m}\)). \(B_{\rm m}\) can be classified into two types: geometric birefringence (\(B_{\rm g}\)) and stress birefringence (\(B_{\rm s}\)). \(B_{\rm g}\) is caused by the anisotropy of the refractive index of the material due to the asymmetry of the fiber structure. \(B_{\rm s}\) is caused by the photoelastic effect in the fiber after the addition of stress-applying areas. \(B_{\rm m}\) is defined as [31]:  
\[B_{\rm m}=N_{\rm x}-N_{\rm y}=B_{g}+B_{\rm s}=N_{\rm x0}-N_{\rm y0}+(\rm C_{1}- \rm C_{2})\,(\sigma_{\rm x}-\sigma_{\rm y}), \tag{1}\]  
where \(N_{\rm x}\) and \(N_{\rm y}\) are the refractive indices of the material along the x and y directions. \(N_{\rm x0}\) and \(N_{\rm y0}\) are the refractive indices of the stress-free material. \(\sigma_{\rm x}\) and \(\sigma_{\rm y}\) are the normal stress along the x and y directions of the material, respectively. \(\rm C_{1}\) and \(\rm C_{2}\) are the stress-optic coefficients. \(B_{\rm s}\) is defined as:  
\[B_{\rm s} =(\rm C_{1}-\rm C_{2})(\sigma_{1}-\sigma_{2}). \tag{2}\]  
Figure 13a,b shows the distribution of the normal stress \(\sigma_{\rm x}\) and \(\sigma_{\rm y}\) in the transverse cross-section of the proposed MMF. Stress along the \(x\)-axis (\(\sigma_{\rm x}\)) is much larger than that along the \(y\)-axis (\(\sigma_{\rm y}\)). The normal stress along the \(x\)-axis (\(\sigma_{\rm x}\)) and \(y\)-axis (\(\sigma_{\rm y}\)) are -5.38 \(\times\) 10\({}^{7}\) N/m\({}^{2}\) and \(\sim\)\(-\)5.19 \(\times\) 10\({}^{7}\) N/m\({}^{2}\), respectively. Stress-induced birefringence, \(B_{\rm s}\) = (\(C_{1}-C_{2}\)) \(\times\) (\(\sigma_{\rm x}-\sigma_{\rm y}\)), is -3.626 \(\times\) 10\({}^{-4}\). According to Equation (1), we obtain the \(B_{\rm m}\) of each point in the cross-section of the MMF. Figure 13c,d shows the distributions of the von Mises stress distribution and geometric and stress-induced birefringence (\(N_{\rm x}-N_{\rm y}\)) in the transverse cross-section of the DBT-MREC-MMF. From Figure 13d, the maximum birefringence is 9.55 \(\times\) 10\({}^{-4}\) N/m\({}^{2}\), and birefringence at the core is \(\sim\)3.627 \(\times\) 10\({}^{-4}\) N/m\({}^{2}\). As for the \(B_{\rm g}\), which can be derived from Equations (1) and (2), the value is -0.001 \(\times\) 10\({}^{-4}\). Since the \(B_{\rm s}\) is much larger than the \(B_{\rm g}\), the main part of the \(B_{\rm m}\) is \(B_{\rm s}\).  
Figure 13: Distributions of (**a**) normal stress \(\sigma_{\rm x}\) (**b**) normal stress \(\sigma_{\rm y}\), (**c**) von Mises stress, and (**d**) modal birefringence Bm (\(\sim-N_{\rm y}\)) in the transverse cross-section of the proposed DBT-MREC-MMF.