# _electronics_
## 2 Waveguide Design
To present two pseudospin states, we plot the distributions of the \(x\)-component of time-averaged power flow (i.e., along the propagation direction) with Poynting vectors (shown as magenta arrows) in Figure 2c where the forward and backward modes exhibit clockwise and anticlockwise spins, respectively. In a nutshell, the two unidirectional topological states have opposite spins; thus, we believe these two modes propagated in a single waveguide path would have negligible crosstalk between each other. Here, we simulate the topological waveguide supercells with different \(Nc\) and \(h\) values to optimize the transmission performances, including loss, group velocity dispersion (GVD), and corresponding bandwidth. Figure 3 shows the transmission losses and GVDs of both forward and backward modes for \(Nc=7\), 11, and 15 while maintaining \(h=\Lambda\). The results illustrate that the cluster numbers (\(Nc\) above seven) in cladding have minor influences on transmissions except for the loss of backward mode--the low-loss transmission window can be broadened particularly at low frequencies (see Figure 3b). The reason is that the  
Figure 1: (**a**) Schematic structures of honeycomb-latticed unit cells, upper: expanded cluster, lower: compressed cluster. (**b**) Topological bandgap of unit cell where blue solid and red dashed curves represent the modes of expanded and compressed clusters, respectively. (**c**) Relative bandwidth of topological bandgap as a function of \(d/\Lambda\) for different \(r/\Lambda\) values. backward mode propagates at lower frequencies of the topological bandgap, corresponding to a larger mode size than that of the forward mode, which can be better confined in the lateral direction with wider cladding. Moreover, we should note that the GVD curves of both modes have high slopes, because of a huge index contrast between the air region and the high-index pillars--the mode power is partially distributed in pillars, as shown in Figure 2c. Nevertheless, we can still find zero GVD frequencies for both modes, which would achieve high data rate transmission [35]. Figure 3: Losses as a function of frequency for (**a**) forward and (**b**) backward modes, and GVDs as a function of frequency for (**c**) forward and (**d**) backward modes for different \(Nc\) values. Figure 2: (**a**) Schematic of the topological waveguide in \(xy\)-plane where the shaded region represents the supercell for numerical modeling, the red zigzag path shows topological propagation path, and the inset presents schematic of a honeycomb lattice in \(xz\)-plane. (**b**) Band diagram of the forward (blue curve) and backward (red curve) modes, and the dashed line represents the light line. (**c**) \(x\)-component of time-averaged power flow distributions of the forward and backward modes, and the magenta arrows indicate Poynting vectors where the size of the arrow indicates the strength of vector field. We investigate the effect of \(h\) values on the supercell simulations of topological waveguides. Here, \(Nc\) was fixed at 11, and \(h\) values were 0.8 \(\Lambda\) and 1.2 \(\Lambda\). The results are shown in Figure 4. Thanks to a large plate separation, a smaller portion of propagating wave power was suffered from the Ohmic loss of the metal surface when \(h=1.2\)\(\Lambda\) for both forward and backward modes, as shown in Figure 4a,b. Nevertheless, it is worth noting that the silicon pillar height was restricted from the silicon-based microfabrication technique because the high aspect ratio pillars would significantly improve the fabrication complexity. From the GVD curves presented in Figure 3c,d, we observe that the waveguide GVDs are independent of \(Nc\) and \(h\) since the GVD extracted from the band diagram is determined by the photonic crystal unit cell. To achieve the maximum possible bandwidth with low losses, we finalize the waveguide parameters as follows: \(\Lambda=530\)\(\upmu\)m, \(r=0.111\)\(\Lambda\), \(d=0.375\) (0.278) \(\Lambda\), \(Nc=11\), and \(h=1.2\)\(\Lambda\).