# Tunable non-Hermiticity in Coupled Photonic Crystal Cavities with Asymmetric Optical Gain
## 3 Results and Discussion
### Asymmetric Gain in the Doubly Coupled PhC Cavity
For example, at optical gains of \(\gamma_{1}=\gamma_{\text{max}}\) and \(\gamma_{2}=0\), the fields are confined exclusively in cavity 1 for the amplifying mode and in cavity 2 for the decaying mode (Figure 4c). This unique non-Hermitian behavior can be exploited in experiments to help develop a new active tuning mechanism, without altering the cavity structure, that enables dramatic changes in terms of resonant frequency and field profile. In laser applications, for example, one can readily excite two eigenmodes (e.g., bonding and antibonding modes) with slightly asymmetric pumping and judiciously induce an unbalanced modal competition that favorably acts on one of the eigenmodes, resulting in the final laser mode [8; 17; 18]. In addition, one can exclusively excite a single amplifying laser mode by applying for the further increased levels of the asymmetry. Next, we performed the vertical airhole shifting of cavity 2 to independently control \(J\) in Equation (4) and observed the corresponding changes in non-Hermiticity (Figure 5). The calculated complex eigenfrequencies of the coupled cavity with one and two vertical airhole shifts are shown in Figure 5a,d, respectively. As discussed earlier, the vertical shift results in the reduction of \(J\), which suggests that the system is substantially affected by the gain contrast \(|\Delta\gamma|\). For example, the eigenmode coalescence in Re(\(f\)) with bifurcation in Im(\(f\)) appears at EPs with lower values of \(|\Delta\gamma|\) (e.g., 0.271 THz in (a) and 0.100 THz in (d)). In addition, regarding the decrease in \(\gamma_{2}\) from \(\gamma_{\text{max}}\) with fixed \(\gamma_{1}=\gamma_{\text{max}}\), one branch of the bifurcated Im(\(f\)), which contributes to the amplifying mode, varies slowly, whereas the other branch undergoes an abrupt change until \(|\Delta\gamma|\) equals \(|\Delta\gamma|_{\text{max}}\). On the contrary, regarding the increase in \(\gamma_{1}\) for  
Figure 4: Evolution of eigenfrequencies in doubly coupled PhC cavity. (**a**) The calculated eigenfrequencies of the coupled cavity modes in the doubly coupled PhC cavity without vertical airhole shifting. The optical gain in component cavities 1 and 2 followed the trajectory shown in Figure 3b. The real (black squares) and imaginary parts (red circles) of the eigenfrequencies obtained from the numerical full-wave simulation were co-plotted with the analytical solutions (black and red lines) obtained from Equation (4). (**b**) The cavity mode profiles of the antibonding and bonding modes at \(\gamma_{1}=\gamma_{2}=\gamma_{\text{max}}\). The field profiles clearly show the in-phase and out-of-phase characteristics in the coupled modes. (**c**) Calculated intensity profiles of the amplifying and decaying modes at \(\gamma_{1}=\gamma_{\text{max}}\) and \(\gamma_{2}=0\). The intensity distributions clearly show the strong and spatially exclusive confinement of the fields of the amplifying and decaying modes. two passive cavities (\(\gamma_{1}=\gamma_{2}=0\)), the same branches exhibit an opposite behavior. These controlled non-Hermitian properties can be advantageously exploited in several applications. For example, the readily obtainable phase transition with low asymmetric gain enables sensitive and rapid switching operations between optical states. Moreover, the stable development of mode amplification guarantees steady and robust single-mode operation at varied pumping powers, which is hardly obtainable in nano and microcavity-based optical devices. Figure 5b,c,e,f show the field (Figure 5b,e) and intensity profiles (Figure 5c,f) of the non-degenerate (\(\gamma_{1}=\gamma_{2}=\gamma_{\text{max}}\)) and degenerate eigen-modes (\(\gamma_{1}=\gamma_{\text{max}}\), \(\gamma_{2}=0\)) in the coupled cavity with one and two vertical airhole shifts. Despite the applied structural modifications, which slightly break the structural symmetry, the mode profiles are similar to those shown in Figure 4b,c.