Fig. 2: Frequency matching, dispersion, and coupling.

a Frequency mismatch Δν = − 2ν_p + ν_s + ν_i for the hOPO in orange. The solid circles are experimental measurements using a wavemeter (see Methods) to determine the cold-cavity resonance frequencies (uncertainties are within the size of the data points), while the solid curve is from finite-element method (FEM) simulations where the idler, pump, and signal modes are from the TE₀, TE₀, and TM₀ mode families, respectively. The frequency mismatch when all modes are chosen from the TE₀ family is shown in the green solid circles and green solid curve for experiment and simulations, respectively. The top images are the simulated transverse electric field profiles for the hOPO modes, where the cross-sectional parameters of ring width (RW), thickness (H), and sidewall angle (θ) are indicated. The lower inset magnifies the right zero crossing of the hOPO Δν curve, showing an ≈ 9 GHz offset when μ = 76. b Dispersion parameter (D) for the TM₀ and TE₀ mode families, with experimental data shown as solid points and fits shown as solid curves. The gray boxes highlight the relevant spectral bands (signal for TM₀ and idler and pump for TE₀). D < 0 for the relevant mode family for each of the idler, pump, and signal bands. c Coupling parameter (K), defined as the ratio of the resonator-waveguide coupling rate (κ_ext) to the resonator intrinsic loss rate (κ_int), for the idler, pump, and signal modes (blue, green, and orange) as a function of resonator-waveguide gap. The top inset shows an optical micrograph of one device, where the scale bar is 5 μm, along with the refractive index (thin black) profile along a cross-section through the ring and waveguide (wg), and the radial component of the evanescent tail of the idler and pump modes (black and green curves). d Transmission spectra for the idler, pump, and signal modes at a gap of 300 nm, along with fitting results for the intrinsic and coupling quality factors.