Fig. 1. System configuration comprised of two microdisks and one scatterer. The scatterer is magnified in the dotted box.

Fig. 2. Re(kR) of supermodes as a function of r1/R and their basis modes in coupled microdisks without a scatterer. The solid and the dashed curves are the even and odd parity modes, respectively. In the region marked by a blue dashed circle, the four modes originating from (l,m)=(1,6) and (1, 5) are highlighted by the thick solid and dotted curves. The insets I, II, III, and IV are the four basis modes defined as ϕ1, ϕ2, ϕ3, and ϕ4, respectively.

Fig. 3. (a) and (b) are the θ-dependent Re(kR) and Im(kR) of the four modes originating from the modes in the blue dashed circle in Fig. 2, respectively; (c) shows the absolute values of wavenumber differences for all combinations of the four modes. The black-solid curve in (c) is for the black-solid and black-dashed curves in (a) and (b). The orange-dashed curve in (c) is for the black-solid and orange-solid curves in (a) and (b). The blue-dotted curve in (c) is for the orange-solid and black-dashed curves in (a) and (b). Vertical lines at A, B, and C mark the local minima of these three curves in (c).

Fig. 4. (a) and (b) Riemann surfaces of Re(kR) and Im(kR), respectively, around the supermode EP kEPR≈4.219−i0.0430. The dashed curves are the branch cuts, and the large dots are the EPs. Here Δr2=r2−rEP and Δθ=θ−θEP. (c) Coalescent mode at the EP; (d) and (e) the two remaining modes at this EP parameter.

Fig. 5. (a) and (b) Riemann surfaces of Re(kR) and Im(kR), respectively, around the chiral EP kEPR≈4.232−i0.0438. The dashed curves, the large dots, Δr2, and Δθ are the same as in Fig. 4. (c) Coalescent mode at the EP; (d) and (e) the two remaining modes at this EP parameter.

Fig. 6. Riemann surfaces and modes of the EP3 by hybridizing a supermode and a chiral EP; (a) and (b) are Re(kR) and Im(kR), respectively. The dashed curves are the branch cuts and the large dots are the EPs. Here Δr2=r2−rEP and Δθ=θ−θEP. (c) and (d) The coalescent and the remaining modes at this EP parameter.

Fig. 7. (a) Accumulated Husimi function of the coalescent mode as a function of p. The black-solid and the red-dashed curves are for the left and right microdisk, respectively. (b) Overlaps of all combinations of the four modes at EP3. The subscripts 1, 2, and 3 are the modes forming EP3, and the subscript 4 is the remaining mode.

Fig. 8. Wavenumber splittings depending on the outside refractive index of the microdisk in double logarithmic scale. Thick magenta, blue, and green curves correspond to an EP3, a supermode EP, and a chiral EP, respectively. Thin black-solid and black-dashed curves are the fittings of |ΔkR|∼(Δn)1/3 and ∼(Δn)1/2, respectively.