## 3 Results and Analysis on a Meta-Fiber Platform
### Analysis Using FDTD on a Single-Fiber Platform  
To further verify the results, we conduct a full-wave simulation using Lumerical FDTD. The results are shown in Figure 5. The metasurface collimating splitter with a total size of 80 \(\upmu\)m \(\times\) 80 \(\upmu\)m is simulated and a Gaussian beam with a waist diameter of 10.4 \(\upmu\)m and single wavelength of 1.55 \(\upmu\)m is incident from the substrate, about 500 \(\upmu\)m below the metasurface. The conditions of all the boundaries are set to perfect matched layers (PMLs). Due to the symmetry of the phase profiles and the unit cell, the device is polarization-independent, as shown in Figure 5a-d. The incident Gaussian beam is well  
Figure 4: Far-field results of the metafiber-based collimating splitter using angular spectrum propagation theory. (**a,b**) Two-dimensional and one-dimensional (when \(u_{y}=0\)) far-field intensity of the incident Gaussian beam without collimating. (**c,d**) Two-dimensional and one-dimensional far-field intensity of the collimated Gaussian beam using a metalens with a phase profile of \(\phi_{L}\). (**e,f**) Two-dimensional and one-dimensional (when \(u_{y}=0\)) far-field intensity of the collimated beam array using the designed metasurface collimating splitter with a phase profile of \(\phi\). \(u_{x}\) and \(u_{y}\) denote the direction cosine and are equal to \(\sin\theta_{x}\) and \(\sin\theta_{y}\), respectively. (**g**) Diffraction efficiency of the Dammann grating extracted from different diffraction orders for comparison. The dashed line is 0.107 (to guide the eye). (**h**) Diffraction efficiency of the metasurface collimating splitter extracted from different diffraction orders corresponding to (**e**). The dashed line is 0.11 (to guide the eye).  
collimated and splits into nine different orders for both polarizations, matching well with the results simulated using angular spectrum propagation theory. The same diffraction efficiency and uniformity are derived for both polarizations, i.e., about 57.87% and 1.29%, respectively. The diffraction efficiency and uniformity results are both worse than the results derived using angular spectrum propagation theory. For further investigation, we extract the diffraction efficiency for each order, shown in Figure 5e,f. The deterioration of the uniformity comes from the decrease of the diffraction efficiency at the four corner orders, i.e., \((-1,-1)\), \((-1,1)\), \((1,-1)\) and \((1,1)\), which correspond to much larger diffraction angles (46.94\({}^{\circ}\)) than the other orders (31.1\({}^{\circ}\)). This will bring strong coupling between the adjacent nano-cuboids, making the practical phase deviate from the designed value. Moreover, the coupling also leads to a drop in the diffraction efficiency by introducing a reflection of about 31.86%, as shown in the FDTD simulation.  
The deviation between the results of the scalar simulation and the vectorial simulation is also deeply discussed in Ref. [35]. To eliminate the deviation requires a decrease in strong coupling between the adjacent nano-cuboids. First, the diffraction angle can be designed to be much smaller by increasing the period of the supercell (even though more diffraction orders may be involved) to avoid the strong coupling. This is suitable to most of the design methods. Second, the Dammann grating should be optimized using a vectorial method, like RCWA or FDTD. The coupling effect between the unit cells is considered and optimized in these methods. The drawback is that the phase synthesis can be no longer suitable, since it works assuming that there are small interactions between the unit cells and that the phase profile can be discretized by the unit cells.