## 5 Fabrication Method and Tolerance  
Table 1 clearly lists the tolerance parameters of the DBT-MREC-MMF. The min \(\Delta n_{\text{eff}}\) of the target value is 3.515 \(\times\) 10\({}^{-4}\). It is noteworthy that, within all the tolerance ranges listed in Table 1, the value of min \(\Delta n_{\text{eff}}\) is greater than 2 \(\times\) 10\({}^{-4}\), and the fiber supports 22 eigenmodes. These results demonstrate that the fiber exhibits a considerably high tolerance and exceptional performance.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \begin{tabular}{c} **Tolerance** \\ **Range** \\ \end{tabular} & \begin{tabular}{c} **Target** \\ **Value** \\ \end{tabular} & \begin{tabular}{c} **Tolerance** \\ **Range** \\ \end{tabular} & \begin{tabular}{c} **Target** \\ **Value** \\ \end{tabular} & \
\begin{tabular}{c} **Tolerance** \\ **Range** \\ \end{tabular} &
\begin{tabular}{c} **Target** \\ **Value** \\ \end{tabular} \\ \hline \(a_{2}/\mu\)m & 5.7–6.3 & 6.1 & \(\varepsilon\) & 1.6–1.71 & 1.65 & \(\Delta n_{2}\) & 0.016–0.02 & 1.78 \\ \(a_{1}/\mu\)m & 2.5–3.6 & 3.3 & \(a_{3}/\mu\)m & 8.4–9.2 & 9 & \(\Delta n_{1}\) & 0.01–0.0148 & 0.013 \\ \(a_{4}/\mu\)m & 10.25–13 & 11 & \(\Delta n_{3}\) & 0.0055–0.009 & 0.007 & \(h/\mu\)m & 7–15 & 11 \\ \(r_{1}/\mu\)m & 16–26 & 20 & 9 & 70–100 & 90 & \(\varepsilon_{0}\) & 0.9–1.9 & 1 \\ \(a_{8}/\mu\)m & 17.5–20.5 & 18.5 & \(\theta_{0}\) & 30–60 & 45 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Optimal parameters with fabrication tolerance of the DBT-MREC-MMF.  
Figure 16: Variations of (**a**) DMD and (**b**) dispersion as a function of wavelength.  
Figure 17: Variations of \(A_{\text{eff}}\) as a function of wavelength.  
The fabrication process of the proposed weakly-coupled DBT-MREC-MMF typically involves the following steps: (1) Preform fabrication: The first step is to fabricate a preform of the proposed fiber structure. This is typically done using the modified chemical vapor deposition (MCVD) method, where a glass tube is heated and rotated while reactive gases are introduced to deposit the desired dopants and form the core and cladding. (2) Drawing the fiber: After the preform is fabricated, it is placed in a fiber drawing tower where it is heated and drawn into a thin fiber. The fiber diameter is controlled by the drawing speed and the temperature gradient along the preform. (3) Etching: Once the fiber is drawn, it is etched to create the air-hole structure. This can be done using hydrofluoric acid or a similar etchant, which selectively removes the cladding material around the air holes. (4) Cleaving and polishing: The final step is to cleave the fiber to the desired length and polish the end faces to ensure good optical coupling.  
Several novel ring core fibers have been successfully developed [39; 40], exhibiting excellent performance in data transmission. The fabrication process of stress-applying regions, particularly the widely experimented bow-tie or panda-type in polarization-maintaining fibers, has reached a mature stage [41; 42]. Additionally, air-hole-assisted fibers have been extensively studied and manufactured [43; 44]. Drawing on these well-established fabrication techniques and experiences, it is reasonable to suggest that the proposed fully degeneracy-lifted DBT-MREC-MMF for MIMO-free direct fiber MDM transmission could be manufactured. We are confident that the fabrication of the proposed fiber will be successfully realized.