Fig. 1. The MS and its modes. (a) Diagram of transverse phase matching. (b) MS design and the coordinate axes: θ is the polar and ϕ the azimuthal angle. (c) BIC wavelengths versus the gap between the two holes. (d) The dispersion for two quasi-BICs: (α) lower and (β) upper surfaces. Dotted white lines: two-BIC phase matching for λp=774.22nm. (e), (f) Fields |E| of the two BICs.

Fig. 2. SPDC at a single BIC. (a) Solid line: the generation rate for a horizontally polarized, normally incident pump, |HH⟩ signal and idler state, ϕ=79deg and θs=0.2deg for the signal, opposite angle for the idler. Dashed line: the product of the maximum intensity inside the resonator of the signal and idler fields. (b) Solid line: generation rate versus the pump wavelength. The red dot marks the wavelength used in (d) and Fig. 4. Dashed: ky at phase matching when kx=0. (c), (d) Generation rate in k-space for (c) phase matching at the Γ point and (d) for λp=774.43nm. White dashed line: phase matching condition; black dashed line: the path of constant θs in Fig. 4(a).

Fig. 3. Nondegenerate signal and idler at different BICs. (a) Transverse phase matching condition according to Eq. (2). The dotted white line corresponds to λp=774.22nm. (b) Solid line: generation rate versus the pump wavelength. The red circle is the integration of (d). Dashed line: ky at transverse phase matching when kx=0. (c), (d) Generation rate in k-space at (c) phase matching at the Γ point (λp=774.2165nm) and (d) λp=774.22nm. White dashed line: phase matching condition; black dashed line: the path of constant θs in Fig. 4(c).

Fig. 4. Entanglement of the signal and idler photons generated from: (a), (b) a single BIC with λp=774.43nm; (c), (d) two different BICs with λp=774.2165nm. (a), (c) Schmidt number versus the photon angle. (b) Schmidt decomposition of the wavefunction plotted on the Poincaré sphere for the red dot in (a). (d) Schmidt number versus the pump polarization and photon angle.