# Unidirectional Slow Light Transmission in Heterostructure Photonic Crystal Waveguide
## 3 Model Design and Simulation Results
### Slow Light Properties  
Since the first few rows of YIG rods have the greatest influence on the shape of the dispersion curve, we adjust the radii of the YIG rods to obtain a flat band region for slow light (a large NDBP value and a low GVD value) with the unidirectional feature. The new configuration of the waveguide is shown in Figure 2, the radii of the YIG rods in the first and second rows adjacent to the waveguide core are denoted by \(r_{1}\) and \(r_{2}\), respectively. We perform a serious of numerical simulations for different \(r_{1}\) values and find that when \(r_{1}\) goes beyond the range of \(0.075a\) to \(0.08a\), the unique unidirectionality of the edge modes vanishes. Therefore, we present the dispersion diagram for parameters \(r_{1}=0.075a\), \(r_{1}=0.077a\), \(r_{1}=0.08a\), and \(r_{1}=0.11a\), as shown in Figure 3a. The corresponding group index characteristics are illustrated in Figure 3b. The dispersion curve becomes flat at \(r_{1}=0.077a\) and a step-like region (corresponding to the slow light region) appears, which indicates a low GVD feature. In the cases of \(r_{1}=0.075a\) and \(r_{1}=0.08a\), the curve gets very steep and the step-like behavior disappears. Figure 1: The projected band diagram of the waveguide. The corresponding geometry configuration is illustrated in the inset, where the lower plane is the magneto-optical photonic crystal (MOPC) \((\varepsilon=15e_{0},\ r=0.11a)\), the upper plane is the Alumina photonic crystal (PC) \((\varepsilon=10e_{0},\ R=0.115a)\), the dashed rectangle indicates a supercell of the waveguide. The red curve is the edge mode at the interface of the MOPC. Based on the optimum \(r_{1}\), we then adjust \(r_{2}\) to further engineer the dispersion curve. Figure 4a illustrates the projected band diagram of TM modes with an external magnetic field applied in the \(+z\) direction for parameters \(r_{1}=0.077a\), \(r_{2}=0.113a\); \(r_{1}=0.078a\), \(r_{2}=0.115a\); \(r_{1}=0.079a\), \(r_{2}=0.117a\); and \(r_{1}=0.08a\), \(r_{2}=0.118a\). The slope of the curves (group velocities) have only positive values and the flat band regions for slow light are highlighted by the thick solid lines. The corresponding frequency range of the slow light region are 4.2886 GHz-4.3077 GHz, 4.2744 GHz-4.295 GHz, 4.2599 GHz-4.2806 GHz, and 4.2483 GHz-4.2692 GHz. The thick solid lines are approximate linear, which correspond to the step-like regions in Figure 5a. As illustrated in Figure 5a, the step-like behavior with a small change in group index in a relatively large frequency bandwidth is found in all cases, which indicates a large NDBP value and a low GVD characteristics. Under the constant group index criterion [16; 40], the normalized frequency bandwidths of 0.444%, 0.481%, 0.485%, and 0.491% are obtained for group indices of 77, 68, 64, and 60, respectively. Consequently, the corresponding NDBP are 0.342, 0.327, 0.311, and 0.295 respectively. This NDBP values are higher than previous works based on modifying the rod radius [31; 42; 46]. The results of our design and previous works are summarized in Table 1. The corresponding GVD characteristics are illustrated in Figure 5b, we find that the GVD parameters \(\beta_{2}\) in the slow light region keep around \(10^{4}\left(\frac{a}{2\pi c^{2}}\right)\) for all four cases. This value is two orders of magnitude smaller than the previous works [43; 44] and it indicates a more superior performance of our design. In the cases of \(r_{1}=0.077a\), \(r_{2}=0.113a\); \(r_{1}=0.079a\), \(r_{2}=0.117a\); and \(r_{1}=0.08a\), \(r_{2}=0.118a\) both negative \(\beta_{2}\) values and positive \(\beta_{2}\) values are observed, it may reveals potential applications in  
Figure 3: (a) Dispersion curves of the edge modes for different \(r_{1}\) with an externally applied magnetic field along the \(+z\) direction. The shadow region denotes the extended modes of a perfect MOPC; (b) Group indices as a function of the frequencies, the relevant step-like behavior is indicated.