Fig. 1. (a) System configuration of coupled microcavities, where r1 and r2 are radii of cavities and d0 is the inter-cavity distance. (b) and (c) are the Riemann surfaces for the real and imaginary parts of the resonant wavenumbers in the parameter space (R,D), obtained using the boundary element method. The branch-cut (interaction center) is marked by a red solid (gray dashed) curve. The insets in (b) are the spatial distributions of the two coupled modes at the interaction center.

Fig. 2. (a) FDTD results of the EDA spectra of |a1| and (b) of |a2| at D=0.37, as a function of (kin,R). The dashed curves are Re(k±r2) obtained using the BEM. In (a), solid curves represent EDA as a function of kin for fixed R=0.88 and 0.892, whereas the upper/lower triangles mark the anti-bonding/bonding modes. The inset in (b) shows the radiating pumping source (arrow) used in FDTD simulation.

Fig. 3. Couplings of WGM1 and WGM2: (a) real and (b) imaginary parts of μ12r2 (red circle) and μ21r2 (black square) obtained using Eq. (6), as a function of R at D=0.37. In (c), Im(K±r2) obtained using the BEM (gray solid) are compared to those obtained using Eq. (4) with the Hermitian (open circle) and non-Hermitian (red dashed line) couplings. The inset in (c) shows the Re(K±r2), where K±r2 are the re-expressed relative eigenvalues of the mean eigenvalues. The vertical solid lines in (c) and the inset mark the interaction center, whereas the vertical dashed line in (c) marks the branch-cut.

Fig. 4. TCMT results of the EDA spectra of |a1| with (a) true non-Hermitian and (b) artificial Hermitian couplings, respectively, at D=0.37. The insets show the values of |a2|. The dashed curves are Re(k±r2) obtained using the BEM.

Fig. 5. (a) Parameter trajectory for the branch-cut (filled symbols) and the interaction center (open circles) for five cases of WGM coupling pairs defined by angular mode numbers: [(i), (m1,m2)=(4,5)], [(ii), (7,8)], [(iii), (11,12)], [(iv), (12,13)], [(v), (13,14)]. (b) |Im(⟨μ12μ21⟩)|r22 for pairs (i), (ii), and (v) obtained at D=0.37. (c) Comparison between the mean values of Im(kr2) and μij for the same pairs at D=0.37.

Fig. 6. Couplings for pair (m1,m2)=(7,7), R≈1. (a) Real and (b) imaginary parts of μ12r2 (red circle) and μ21r2 (black square) obtained using Eq. (6), as a function of R at D=0.37. The (c) real and (d) imaginary relative eigenvalues of K±r2 obtained using the BEM (gray solid) are compared to those obtained using Eq. (4) with the Hermitian (open circle) and non-Hermitian (red dashed) couplings. The vertical solid lines in (c) and (d) indicate the interaction center.