# Unidirectional Slow Light Transmission in Heterostructure Photonic Crystal Waveguide
## 3 Model Design and Simulation Results
### Backscattering-Immune Feature  
As illustrated in Figure 4a, with an external magnetic field applied along the \(+z\) direction, the tailored four PCWs still have only positive group velocities (the slope of the waveguide dispersions). It indicates that the waveguide supports only forward transmission and the opposite direction transmission is completely prohibited. The absence of backward propagation channel avoids the possibility of backscattering loss and enables the electromagnetic wave to wrap around different types of obstacles as verified in Figure 7. We introduce four types of interfacial disorders into the waveguide to investigate the robustness of the edge modes. For comparison, Figure 7a shows the \(E_{z}\) field distributions without introducing the disorders. As shown in Figure 7b, the light bypasses the inserted metal reflector and maintain the forward transmission. This can never happen in a conventional PCW, as the metal reflector would cause strong scattering and even block the guide mode. However, in the proposed unidirectional PCW, a new interface between the metal reflector and the MOPC is formed, thus providing a way for the electromagnetic wave to flow around the metal reflector and no backscattering takes place. The second type of defect is shown in Figure 7c, when two YIG rods at the interface of the MOPC are replaced by the Copper rods with the same size, the electromagnetic wave flow around the metal rods without introducing backscattering. The third type of defect is a Z-shaped interface, as shown in Figure 7d, the light travels along the Z-shaped interface \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Current Work** & **Optimized** & **Average Group** & **Frequency** & **NDBP Values** \\  & **Parameters** & **Index** & **Bandwidth** & **NDBP Values** \\ \hline  & \(r_{1}=0.077a\), & 77 & 0.444\% & 0.342 \\  & \(r_{2}=0.113a\) & & & \\  & \(r_{1}=0.078a\), & 68 & 0.481\% & 0.327 \\  & \(r_{2}=0.115a\) & & & \\  & \(r_{1}=0.079a\), & 64 & 0.485\% & 0.311 \\  & \(r_{2}=0.117a\) & & & \\  & \(r_{1}=0.08a\), & 60 & 0.491\% & 0.295 \\  & \(r_{2}=0.118a\) & & & \\ \hline
**Previous Work** & **Modification of** & **Average Group** & **Frequency** & **NDBP Values** \\  & **the Rod Radius** & **Index** & **Bandwidth** & **NDBP Values** \\ \hline  & Ref. [31] & 22.85 & 0.813\% & 0.186 \\  & Ref. [46] & 26 & 0.896\% & 0.233 \\  & Ref. [47] & 40 & 0.65\% & 0.258 \\  & Ref. [48] & 62 & 0.43\% & 0.268 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Group index, frequency bandwidth, and NDBP values under different optimized parameters and the comparison with previous works. Figure 6: **(a)** Normalized delay-bandwidth product (NDBP) values of the four optimized PCWs as a function of the externally applied magnetic field \(H_{0}\); **(b)** The \(\beta_{2}\) as a function of the frequencies for the four optimized PCWs in Figure 5 with different \(H_{0}\). and the forward propagation is not affected by the two sharp corners. In Figure 7e, two Alumina rods and two YIG rods at the interface of the waveguide (circled in the black rectangles) are replaced by two Alumina rods with smaller rod radii \(r_{3}=0.08a\) and two YIG rods with larger rod radii \(r_{4}=0.11a\), respectively. It is found that the unidirectional transmission is not affected by the interfacial defects. This ideal robust unidirectional transmission feature is of great significance for slow light systems. The same unidirectionality and robustness of the edge modes from all four types of perturbation is observed when the external magnetic field is applied in the \(-z\) direction (bottom plots in Figure 7b-e). The propagation of the electromagnetic wave in the waveguide is switched to the opposite direction accordingly in all five cases.