## 4 Bandwidth Study  
First, we carried out the theoretical study on the bandwidth of the polymer waveguide using a beam propagation method (BPM) and the refractive index distribution of the calculation model is the same as that of the fabricated ones. In order to simulate the overfilled launch condition to excite as many modes as possible in the polymer waveguide, we chose an optical field of 50 \(\upmu\)m-core MMF with a random distribution of modes as the input optical field, as shown in Figure 11.  
At a wavelength of 850 nm, altogether, 170 modes of the waveguide with a normalized power percentage higher than 1% of that of the mode with the highest power can be excited, and their power distribution is shown in Figure 12.  
The group refractive indices of these modes can be calculated by  
\[n_{g}=n_{eff}-\lambda\frac{\partial n_{eff}}{\partial\lambda} \tag{1}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Length** & **Underfilled** & **Overfilled** \\ \hline
1 m & \(\leq\)0.03 dB & \(\leq\)0.05 dB \\
1.7 m & \(\leq\)0.05 dB & \(\leq\)0.05 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measurement results of the polarization-dependent loss of the polymer waveguides. Figure 11: Optical field input by 50-\(\upmu\)m MMF. Figure 12: Normalized mode power distribution of the waveguide under the simulated overfilled launch condition. where \(n_{eff}\) is the effective refractive index of the mode, and \(n_{g}\) is its group refractive index [37]. The differential mode delay (DMD) of the waveguide under the overfilled launch condition is defined by the difference between the maximum and minimum values of the group refractive indices [38], which can be calculated by  
\[DMD=\text{max}\big{(}n_{g}\big{)}-\text{min}\big{(}n_{g}\big{)} \tag{2}\]  
The calculated BLP of the polymer waveguide under the strictest overfilled launch condition is about 52.4 GHz\(\cdot\)m, which is the reciprocal of DMD. Under the same simulation conditions, the BLP of the waveguide with a horizontal refractive index gradient is larger than that of the waveguide with a step refractive index (48.4 GHz\(\cdot\)m) of the same size. Although we successfully measured the BLP of the polymer waveguides by connecting 20 cm long polymer waveguides into a meter-scale long waveguide link using a unique optical sampling method, the implementation of many polymer waveguide connectors will inevitably result in bandwidth degradation due to the misalignment at the interface [31]. However, it is challenging to characterize the BLP of the waveguide using well-established measurement technology of S\({}_{21}\) parameters due to the short length of the waveguide and the limited bandwidth of the vector network analyzer. In order to directly evaluate the BLP of fabricated waveguides, a vector network analyzer (VNA), was used to measure the S\({}_{21}\) response curves of waveguide links. A schematic of the experimental setup is shown in Figure 13. We obtained waveguide links with different lengths by connecting waveguides with a length of 0.2 m, 1 m, and 1.7 m, respectively. The light from a distributed Bragg reflection (DBR) laser operating at 850 nm passed through a polarization controller (PC) and entered an intensity modulator (IM) to be modulated. The modulated light was then injected into a multimode variable optical attenuator (VOA) and launched into the waveguide through a 50 \(\upmu\)m-core OM3 multimode fiber. The normalized S\({}_{21}\) response curves measured under underfilled launch conditions are shown in Figure 14a. The attenuation of the S\({}_{21}\) response curve at 32.2 GHz is less than 3 dB for a waveguide link with a length of 2.1 m and an insertion loss of 10 dB. As a result, the BLP of the waveguide under the underfilled launch condition is greater than 67.6 GHz\(\cdot\)m. The normalized S\({}_{21}\) response curves measured under the overfilled launch condition are shown in Figure 14b.