## 4 Mode Properties and Broadband Characteristics
### Broadband Performances  
We further investigated the relationship between wavelength and mode properties (\(n_{\text{eff}}\), \(An_{\text{eff}}\), DMD, D, and \(A_{\text{eff}}\)) of DBT-MREC-MMF covering the whole C+L+U band. The variations of \(n_{\text{eff}}\) and \(\Delta n_{\text{eff}}\) as a function of the wavelength for each eigenmode are presented in Figure 15a,b. It was found that the DBT-MREC-MMF supports 22 eigenmodes over a range of wavelengths from 1530 to 1675 nm, and the min \(\Delta n_{\text{eff}}\) between adjacent eigenmodes is larger than 3.089 \(\times\)\(10^{-4}\) at 1530 nm. Results indicate that the DBT-MREC-MMF can work across the whole C+L+U band.  
The differential mode delay (DMD) is important to reduce the complexity and power loss of MIMO processing. The DMD between mode A and mode B can be defined as [34]:  
\[\text{DMD}\ =\ \tau_{\text{B}}-\tau_{\text{A}}\ =\ \frac{n_{\text{gB}}-n_{\text{gA}} }{\text{c}}\ =\ \frac{n_{\text{effB}}-n_{\text{effA}}}{\text{c}}-\frac{\lambda}{\text{c}} \bigg{(}\frac{\partial n_{\text{effB}}}{\partial\lambda}-\frac{\partial n_{ \text{effA}}}{\partial\lambda}\bigg{)} \tag{4}\]  
where \(\tau_{\text{B}}\) and \(\tau_{\text{A}}\) are the group time delays of the two modes, \(n_{\text{gB}}\) and \(n_{\text{gA}}\) are the group refractive indices, \(n_{\text{effB}}\) and \(n_{\text{effA}}\) are the effective refractive indices, and c and \(\lambda\) are the speed of light and the operating wavelength in vacuum, respectively. Fiber dispersion is a  
Figure 14: Bending loss as a function of bending radius \(R\) along the (**a**) \(x\)-axis and (**b**) \(y\)-axis.  
key factor in determining transmission capacity, and the dispersion of each eigenmode is calculated by [16]:  
\[\text{D}\ =\ -\frac{\lambda}{\text{C}}\frac{\partial^{2}n_{\text{eff}}}{ \partial\lambda^{2}} \tag{5}\]  
As shown in Figure 16a,b, the DMD and dispersion of all 22 eigenmodes as a function of wavelength were calculated. All of the eigenmodes exhibit a relatively small DMD (\(-15.41\)-\(12.4\) ns/km), which can induce negligible power penalties in short-haul optical communication links. Moreover, the dispersion values for lower-order eigenmodes covering the whole C+L+U band vary from 17.39 to 63.75 ps/nm/km, indicating that dispersion can be neglected when applied in short-haul optical interconnects. The dispersion value for higher-order eigenmode HG\({}_{31}^{Y}\) is larger, and ranges from \(-77.57\) to \(-70.85\) ps/nm/km over the whole C+L+U band. To compensate for the dispersion problem in fibers, specialized dispersion-compensating fibers can be used. This method helps to ensure that the signal transmitted through the fibers remains intact and undistorted, enabling high-quality data transmission [35; 36; 37]. Hence, this fiber can be considered to be a promising candidate for MIMO-free MDM systems.  
The mode effective area (\(A_{\text{eff}}\)) of the fiber reflects the lateral distribution of the optical field on the fiber cross-section. The nonlinear coefficient and \(A_{\text{eff}}\) are reciprocal to each other. \(A_{\text{eff}}\) is calculated by Equation (6) [38]:  
\[A_{\text{eff}}\ =\ \frac{\left(\iint\abs{E}^{2}dxdy\right)^{2}}{\iint\abs{E}^{ 4}dxdy} \tag{6}\]  
where \(E\) represents the mode field power distribution across the fiber cross-section wavelength. The \(A_{\text{eff}}\) of the proposed DRT-MREC-MMF within the wavelength range of 1530 to 1675 nm is shown in Figure 17. The effective area gradually increases with increasing wavelength. The \(A_{\text{eff}}\) values range from 66.8 to 96.39 \(\upmu\)m\({}^{2}\) over the whole C+L+U band.  
Figure 15: Variations of (**a**) \(n_{\text{eff}}\) and (**b**) min \(\Delta n_{\text{eff}}\) as a function of wavelength.