# Tunable non-Hermiticity in Coupled Photonic Crystal Cavities with Asymmetric Optical Gain
## 3 Results and Discussion
### Coupling Control Schemes in the Doubly Coupled PhC Cavity  
We propose three different coupling control schemes in the doubly coupled PhC cavity. First, we controlled the size of the shared airholes (blue circles, radius \(R\)) in the coupling region between the two component cavities while fixing the size of regular airholes (radius \(R_{0}\)), as shown in Figure 2a. Figure 2b shows the calculated eigenfrequencies \(f_{0}\) (black) and coupling constants \(J\) (red) plotted as a function of the ratio of the shared airhole size to the regular airhole size (\(R/R_{0}\)). As expected, the coupling constant varied in response to the change of \(R/R_{0}\): \(J\) slowly increased with \(R/R_{0}\). Moreover, the eigenfrequency sensitively varied in response to the change of \(R/R_{0}\) as well because the L3 line defect cavity is readily affected by the cavity size and the airholes around the cavity. Consequently, this control scheme can allow for the substantial tuning of eigenfrequencies of bonding and antibonding modes with small structural variations.  
Second, we vertically shifted one of the component cavities (cavity 2) using a designated number of airholes with respect to the other component cavity (cavity 1), as shown in Figure 2c. Although the vertical shift only allows for discrete levels of control regarding \(J\), unlike the shared airhole-based control, this scheme preserves the fundamental modes of the individual component  
Figure 2: Coupling control schemes in doubly coupled PhC cavity. (**a**,**c**,**e**) Three different coupling control methods: (**a**) changing the size of the shared airholes (blue dots) between two component cavities, (**c**) vertical shifting of cavity 2, and (**e**) lateral shifting of cavity 2. (**b**,**d**,**f**) The estimated fundamental mode frequency of a single cavity, \(f_{0}\), and the intercavity coupling constant \(J\) as a function of (**b**) the ratio of the shared airhole size to the regular airhole size (\(R/R_{0}\)), (**d**) the number of vertical airhole shifts of cavity 2, and (**f**) the number of lateral airhole shifts of cavity 2.  
cavities well. Figure 2d shows the calculated eigenfrequencies, \(f_{0}\) (black), and coupling constants, \(J\) (red), plotted as a function of the number of airhole shifts. Interestingly, while the eigenfrequency remained almost constant, the coupling constant exhibited more structured behaviors as the number of shifts increased. For example, at first, the maximum \(J\) at zero shift decreased as the number of shifts increased; subsequently, it became minimum at three airhole shifts and recovered at five airhole shifts. We attribute this characteristic behavior to the unique field distribution of the fundamental mode, which shows directional field leakage at 60 degrees to the cavity axis.  
Third, we increased the level of control of \(J\) by laterally shifting cavity 2 using the designated number of airholes with respect to cavity 1, as shown in Figure 2e. This lateral shift entailed a pairwise correspondence between airholes owing to the triangular symmetry of the PhC structure, such that the centers of the component cavities could be aligned horizontally. As expected, the fundamental eigenfrequency \(f_{0}\) (black) showed a trend similar to the one observed in the vertical shift scheme, but the coupling constant \(J\) (red) exhibited a drastic decrease as the number of airhole shifts increased (Figure 2f). Taken together, the vertical and lateral shift schemes are more desirable than the shared airhole control scheme because they guarantee the independent control of \(J\) while keeping\(f_{0}\) unaffected. In addition, the vertical shift scheme allows for finer levels of control of \(J\) than the lateral one, which is beneficial for the detailed study of non-Hermiticity when an asymmetric gain is introduced. Therefore, we exclusively employ the vertical shift scheme for the systematic tuning of non-Hermiticity (explained in the following section).