Fig. 1. (a) The cavity resonance of MRRs can be linearly tuned by varying the chip temperature δT or nonlinearly tuned by controlling the intra-cavity pump energy I_p through nonlinear TO effects. (b) In the wavelength domain, when δT increases, the cold-cavity resonance wavelengths of the TH mode (in red dot line) and the pump mode (in blue dot line) will linearly redshift ∆λ_TL and ∆λ_TLp, respectively. When up sweeping the CW input pump wavelength λ_p and δT are fixed, the corresponding I_p will induce nonlinear redshift ∆λ_TNL and ∆λ_TNLp to the resonance wavelengths of the TH and pump modes, respectively. For the pump mode, when λ_p is fixed, the trace of blue open circles determines τ_p which is the ratio between δT and I_p. Similarly, the trace of red open circles in the upper part of (b) gives the ratio τ_p. The overall effects of ∆λ_TL and ∆λ_TNL determine the effective TDWS of the TH mode which becomes a δT-λ_p relationship by mapping the trace of red open squares to the lower part of (b). For the pump mode, its TDWS is only related the linear TO redshift ∆λ_TLp whose trace was shown as blue open squares plotted in the δT-λ_p diagram. (c – e) The thermal mismatch between τ_p and τ_t will determine the effective TDWS of the TH mode, which leads to (c) a positive TDWS when τ_p < τ_t and (d) a zero TDWS when τ_p ≈ τ_t and (e) a negative TDWS when τ_p > τ_t in the δT-λ_p diagram.

Fig. 2. (a) The relationship between the normalized pump detuning α and the intra-cavity pump power ρ when the input pump power F² is fixed. The existence of stable athermal TH modes were shown in blue (F² = 1.6) and green (F² = 1.9) solid lines when |α| > \begin{document}$\sqrt 3 $\end{document}. (b) Four-port MRR add-drop filter schematic showing a variable bus-ring gap separation design. (c) The captured green side-emissions via THG in MRRs at different chip temperatures. (d) Schematic experimental setup of THG in MRR. TLS: tunable laser source; EDFA: erbium-doped fiber amplifier; OBF: optical bandpass filter; PC: polarization controller; TEC: thermoelectric cooler; OSA: optical spectrum analyzer.

Fig. 3. (a) Measured pump wavelength corresponding to the peak resonances of three types of TM TH modes. The calibrated cold-cavity resonance wavelengths were obtained by subtracted Ω_TNL which are shown in plus center symbols. The calibrated TDWS of Type Ⅰ (green lines) and Type Ⅱ (purple lines) TH modes are also shown. (b) Measured third harmonic photon counts (THPC) on the resonance peaks of the TH modes as a function of I_p showing the cubic TH-pump relationship. The solid lines show the cubic TH-pump relationship for comparison. (c – d) The measured I_p (upper) and filter response (lower) of corresponding type Ⅰ TH modes as functions of Ω_t in the devices of R-1 (c) and R-2 (d) respectively. The Q-factor of TH modes Q_TH can be indirectly determined by using Θ_t = 2.55×10⁹ rad/pJ into Eq. (4). The corresponding fitted curves using estimated Q_TH are also shown in (c) – (d). Here, the symbols of the measured data in (b) – (d) are the same as those in (a).

Fig. 4. (a – b) Measured spectra of all the TE athermal TH modes in MRRs of R-1 (a) and R-2 (b) that are thermal matched, i.e. ∆τ ≈ 0. In (a) and (b), the measured wavelengths of the resonance peak of TH modes at different T are shown in open diamonds. The corresponding fitted TDWS of TH modes are shown in grey dashed lines. (c) The measured photon count on the peak of different athermal TH modes as a function of I_p. The solid lines show the cubic TH-pump relationship for (a) – (b) as well as Figs. 5(c) and 5(e) for comparison. The data and fitting curves for R-1, R-2, R-3 are shown in blue, red, and green, respectively.

Fig. 5. (a) Thermal mismatch ∆τ < 0, with a TM pump generated a TM TH mode with a TDWS of 3d_TH = 7.05 pm/℃. (b) Thermal mismatch ∆τ > 0, with a TE pump generated a TE TH mode with 3d_TH = −8.53 pm/℃. (c, f) When thermal mismatch ∆τ ≈ 0, with a TM pump generated TE TH mode with 3d_TH = 0.14 pm/℃ (c) as well as 3d_TH = −0.27 pm/℃ (f). (d, g) The extracted Q-factor of TE TH modes by using the data in (c) at 26 ℃ and in (f) at 46 ℃, with Eq. (4) is used to generate the fitted lines. (e, h) The temperature dependence of the athermal TH mode resonance fluctuation with fixed ∆τ and F² for the TH modes shown in (c) and (f), with the measured values in gray squares. In (a – c) and (f), the measured wavelengths of the resonance peak of TH modes at different T are shown in open diamonds. The corresponding fitted TDWS of TH modes are shown in grey dashed lines.

Table 1. The corresponding parameters ofFigs. 3(a)and3(b). Details of corresponding pump mode parameter measurement are discussed in Supplementary information.

Table 2. Comparison of athermal TDWS schemes using different micro resonators.