## 2 Principle and Setup
For a single resonance, \(R(\omega)\) can be expressed as [33]:  
\[R(\omega)=1-\frac{2\omega_{0}/q_{\mathrm{e}}}{\omega_{0}/q_{0}+\omega_{0}/q_{ \mathrm{e}}+2i(\omega-\omega_{0})}, \tag{8}\]  
where \(\omega\) is the laser angular frequency, \(\omega_{0}\) is the resonant angular frequency, \(q_{0}\) is the intracavity Q, and \(q_{\mathrm{e}}\) is the external cavity Q. When two resonances overlap or weakly couple, \(R(\omega)\) can be described by [34]:  
\[R(\omega)=1+2\frac{\kappa_{01}X_{2}+\kappa_{02}X_{1}+2\mathrm{Re}(g)\sqrt{ \kappa_{\mathrm{e1}}\kappa_{\mathrm{e2}}}}{X_{1}X_{2}-g^{2}}, \tag{9}\]  
where \(\kappa_{0n}=\omega_{n}/(2q_{0n})\), \(\kappa_{\mathrm{en}}=\omega_{n}/(2q_{\mathrm{en}})\), \(q_{n}=(q_{0n}{}^{-1}+q_{\mathrm{en}}{}^{-1})^{-1}\), \(X_{n}=-\kappa_{0n}-\kappa_{\mathrm{en}}+\mathrm{i}(\omega-\omega_{n})\), \(\omega_{n}=2\pi\!f_{n}\) (\(n=1\), \(2\)), and \(\Delta\lambda_{\mathrm{d}}=\mathrm{c}/\!f_{1}-\mathrm{c}/\!f_{2}\); \(\kappa_{0n}\) is the intracavity decay rate of resonances; \(\kappa_{\mathrm{en}}\) is the external cavity decay rate of the two resonances; \(f_{n}\) is the resonant frequency of the two resonances; \(q_{0n}\) is the intracavity Q of the two resonances; \(q_{\mathrm{en}}\) is the external cavity Q of the two resonances; \(q_{n}\) is the resonance total Q of the two resonances; \(g\) is the coupling coefficient; \(\omega_{n}\) is the angular resonant frequency of the two resonances; \(\omega\) is laser angular frequency; \(c\) is the light velocity in a vacuum; and \(\Delta\lambda_{\mathrm{d}}\) represents the two resonances' distances as wavelengths. The parameters \(q_{0n}\), \(q_{\mathrm{en}}\),\(f_{n}\) (\(n=1\), \(2\)), and \(g\) can be obtained by fitting the IQ curve. The other parameters can be derived by the fitting parameters. When two resonances strongly couple, according to the coupled mode theory, the coupled mode equations can be written as [14]:  
\[X_{3}E_{3}-g_{1}E_{4}+2i\sqrt{\kappa_{\mathrm{e3}}}E_{\mathrm{in}}=0, \tag{10}\]  
\[Q_{0}(\omega) =kP_{0}J_{0}(\beta)I_{1}(\beta)\mathrm{Im}[R(\omega)R^{*}(\omega+ \omega_{\mathrm{m}})-R^{*}(\omega)R(\omega-\omega_{\mathrm{m}})] \tag{2}\]0  
where \(\kappa_{0p}=\omega_{p}/(2q_{0p})\), \(\kappa_{\mathrm{ep}}=\omega_{p}/(2q_{\mathrm{ep}})\), \(q_{p}=(q_{0p}{}^{-1}+q_{\mathrm{ep}}{}^{-1})^{-1}\), \(X_{p}=i(\kappa_{0p}+\kappa_{\mathrm{ep}})+(\omega-\omega_{p})\), \(\omega_{\mathrm{p}}=2\pi\!f_{\mathrm{p}}\) (\(p=3\), \(4\)), and \(\Delta\lambda_{\mathrm{d2}}=\mathrm{c}/\!f_{3}-\mathrm{c}/\!f_{4}\); \(E_{\mathrm{in}}\) is the optical field amplitude of the input light; \(E_{p}\) (\(p=3\), \(4\)) is the optical field amplitude of the two resonances; \(\kappa_{0p}\) is the intracavity decay rate of the two resonances; \(\kappa_{\mathrm{ep}}\) is the external cavity decay rate of the two resonances; \(f_{p}\) is the resonant frequency of the two resonances; \(q_{0p}\) is the intracavity Q of the two resonances; \(q_{\mathrm{ep}}\) is the external cavity Q of the two resonances; \(q_{p}\) is the resonance total Q of the two resonances; \(g_{1}\) is the coupling coefficient; and \(\omega_{p}\) is the angular resonant frequency of the two resonances. The optical field amplitude of the output light (\(E_{\mathrm{out}}\)) can be described by:  
\[Q_{0}(\omega) =kP_{0}J_{0}(\beta)I_{1}(\beta)\mathrm{Im}[R(\omega)R^{*}(\omega+ \omega_{\mathrm{m}})-R^{*}(\omega)R(\omega-\omega_{\mathrm{m}})] \tag{2}\]1The \(R(\omega)\) can be described by:  
\[Q_{0}(\omega) =kP_{0}J_{0}(\beta)I_{1}(\beta)\mathrm{Im}[R(\omega)R^{*}(\omega+ \omega_{\mathrm{m}})-R^{*}(\omega)R(\omega-\omega_{\mathrm{m}})] \tag{2}\]2  
The parameters \(q_{0p}\), \(q_{ep}\), \(f_{p}\) (\(p\) = 3, 4), and \(g_{1}\) can be obtained by fitting the IQ curve. The other parameters can be derived by the fitting parameters.