# Tunable non-Hermiticity in Coupled Photonic Crystal Cavities with Asymmetric Optical Gain
## 3 Results and Discussion  
### Doubly Coupled PhC Cavity  
Figure 1 shows the coupled PhC cavity structure, referred to as the doubly coupled PhC cavity that we employed for non-Hermiticity control. The two identical vertical L3 line defect PhC cavities, referred to as component cavities, are horizontally coupled via three vertical lines of airholes. This coupled PhC cavity supports two independent eigenmodes that are slightly different from the one in the single component cavity in terms of eigenfrequency and field profile. These two eigenmodes originate from the coupling between the two fundamental modes (left and right panels of Figure 1a) of the individual component cavities. The resultant eigenmodes can be classified based on the spatial field distribution or symmetry with respect to the central line of the coupled cavity. The eigenmode with the symmetric (or in-phase) field distribution is known as the bonding mode (right, Figure 1b), whereas the eigenmode with the antisymmetric (or out-of-phase) field distribution is known as the antibonding mode (left, Figure 1b).  
The eigen-frequencies of the bonding (\(f_{+}\)) and antibonding modes (\(f_{-}\)) in the coupled cavity are determined by the eigenfrequency of the single component cavity (\(f_{0}\)) and the inter-cavity coupling constant (\(J\)). The set of coupled differential equations describing the interaction between the two component cavities is given by  
\[i\frac{d}{dt}\left[\begin{array}{cc}\phi_{1}\\ \phi_{2}\end{array}\right]=2\pi\left[\begin{array}{cc}f_{0}&J\\ J&f_{0}\end{array}\right]\left[\begin{array}{cc}\phi_{1}\\ \phi_{2}\end{array}\right] \tag{1}\]  
where \(\phi_{1}\) and \(\phi_{2}\) are the wavefunctions of component cavities 1 and 2, respectively [11, 15]. Based on the assumption of steady-state oscillations, \(\phi(t)=\exp(-2i\pi ft)\), the eigenfrequencies of the doubly coupled PhC cavity are obtained as follows:  
\[f_{\pm}=f_{0}\pm J \tag{2}\]  
Figure 1a shows the eigen-frequency relation between the single component cavity and the doubly coupled cavity: the eigenfrequency \(f_{0}\) of the single component cavity is split into two eigenfrequencies, \(f_{+}\) and \(f_{-}\), of the doubly coupled cavity with a frequency difference of \(2J\), based on Equation (2). Equation (2) reveals that the rational design of two component cavities with controllable \(J\) can allow for  
Figure 1: Doubly coupled photonic crystal (PhC) cavity. (**a**) Coupled PhC cavities that consist of two identical defect PhC cavities. The fundamental mode of each single cavity is shown to the left and right of the coupled cavities. The frequency diagram of the resonant modes in the single and coupled cavities is shown. \(f_{0}\) is the eigenfrequency of the fundamental mode of the single cavity, and \(J\) is the coupling constant in the coupled cavities. (**b**) Calculated field profiles of the two eigenmodes in the doubly coupled PhC cavity: out-of-phase anti-bonding (\(f_{-}\)) and in-phase bonding mode (\(f_{+}\)). The electric field of the cavity modes was polarized in the \(x\)-direction.  
the systematic tuning of eigenmodes and the corresponding eigenfrequencies in the coupled PhC cavity. Henceforth, we focus on controlling \(J\) while maintaining the structural parameters of the component cavities. As the L3 line defect PhC cavity supports a high-quality (high-\(Q\)) fundamental mode with strong light confinement near the cavity, alterations of the cavity structure can introduce undesirable complexities, which may lead to a considerable optical loss.