Fig. 1. Schematics of the (a) direct GVM,²³ (b) QGVM realized by TDL,³² and (c) anomalous GVMM in the phase-matched SHG process, where $\delta =V_{\mathrm{gSH}}^{-1}-V_{\mathrm{gFW}}^{-1}$.

Fig. 2. (a) Schematic of the birefringent racetrack resonator on X-cut TFLN, where SH-band light experiences a mode-hybridization in the half-circle waveguide; (b) principle of SQPM [inset: varying SQPM SHG intensity with the periodically inverted efficient nonlinear coefficient ($m=5$), and a comparison among the SHG processes under the perfect phase-matching, QPM, SQPM, and phase mismatching]; (c) effective refractive indices of the hybrid mode in SH-band and TE0 mode in FW-band in the half-circle waveguide, and the vector mismatch dispersion between them; (d) average vector mismatch dispersion versus different FW wavelengths, which is positive in the straight waveguide and negative in the half-circle waveguide.

Fig. 3. (a) Calculated phase mismatch $\mathrm{\Delta }{\varphi }_1$ in the straight waveguide and phase mismatch $\mathrm{\Delta }{\varphi }_2$ in the half-circle waveguide of a perfect 111th-order SQPM racetrack resonator, and (b) their summation, presented by $\mathrm{\Delta }{\varphi }_1+\mathrm{\Delta }{\varphi }_2-293\pi$. A detailed comparison between (c1), the perfect narrowband SQPM, and (c2), a phase-compensated broadband SQPM. The solid and dashed lines denote the absolute phase-mismatch summation $|\mathrm{\Delta }{\varphi }_1+\mathrm{\Delta }{\varphi }_2-293\pi |$ and its dispersion $|\text{d}(\mathrm{\Delta }{\varphi }_1+\mathrm{\Delta }{\varphi }_2)/\text{d}\lambda |$, respectively.

Fig. 4. (a) Experimental setup. EDFA, erbium-doped optical fiber amplifier; PC, polarization controller; TEC, thermal electronic cooler; WDM, wavelength division multiplexer; OSA, optical spectrum analyzer; PD, photodetector; OSC, oscilloscope; (b) transmission spectrum of the SQPM racetrack resonator in C-band and (c) Lorentzian fitting of the marked resonance dip; (d) SHG intensity obtained at each FW resonance mode.

Fig. 5. (a) Schematic of the SQPM bent-waveguide on the X-cut TFLN; (b) scanning electron microscopic image of the first-order SQPM waveguide; (c) pump light spectrum and measured nonlinear upconversion spectrum, corresponding to a 3-dB bandwidth of about 16 nm.