Fig. 1. Schematic of the SOHPMR. Cross-sectional views along the (a) X−Y and (b) X−Z planes.

Fig. 2. Δn=nSH−nFF as functions of the radius and width of the bended waveguides. The black line represents the zero value, i.e., the phase-matching line.

Fig. 3. (a) neffkFFR and (b) loss of a 90° bend at the phase-matching line as functions of the radius. The inset in (a) is an enlarged view near neffkFFR=9.

Fig. 4. Ez distributions for the phase-matched modes at (a) FF and (b) SHF when R=2.325  μm and w=351  nm. (c) is the normalized Ez distribution on the center line of the slot along the r direction.

Fig. 5. SHG efficiency η2 as a function of the transmission coefficients for FF tFF and SHF tSH.

Fig. 6. (a) Transmission coefficients for two wavelengths, (b) final SHG conversion efficiency, and (c) enhancement factor in the ring as a function of the gap.

Fig. 7. Transmission spectra of the 3D MRR structure around (a) SHF of 1.55 μm and (c) FF of 3.1 μm when R=2.325  μm, wa=358.3  nm, w=351  nm, and g=175  nm. The two resonant wavelengths are 1548.4 and 3099 nm. (b) and (d) are the Ez distributions at the SHF and the FF, respectively.

Fig. 8. SHG efficiency η2 as a function of the attenuation coefficients for FF (αFF) and SHF (αSH).

Fig. 9. Schematic of SHG in a single-pass MRR.

Fig. 10. Analytic solution and numerical solution of the SHG efficiency as functions of the propagation length in (a) straight waveguide and (b) bended waveguide.