## 2 Principle and Setup  
In classical PDH technique theory, we can obtain the bipolar voltage wavelength discriminating signal (I-term and Q-term signals), and \(I_{0}(\omega)\) and \(Q_{0}(\omega)\) can be expressed as [27]:  
\[I_{0}(\omega) =kP_{0}J_{0}(\beta)I_{1}(\beta)\mathrm{Re}[R(\omega)R^{*}(\omega+ \omega_{\mathrm{m}})-R^{*}(\omega)R(\omega-\omega_{\mathrm{m}})] \tag{1}\] \[Q_{0}(\omega) =kP_{0}J_{0}(\beta)I_{1}(\beta)\mathrm{Im}[R(\omega)R^{*}(\omega+ \omega_{\mathrm{m}})-R^{*}(\omega)R(\omega-\omega_{\mathrm{m}})] \tag{2}\]  
where the phase difference (\(\varphi\)) between the detection signal and the local oscillation signal is 0 and \(\pi\)/2, respectively. \(\omega\) is the laser angular frequency; \(\omega_{\mathrm{m}}\) is the angular frequency of the signal input into the phase electro-optical modulator (PM) (\(\omega_{\mathrm{m}}=2\pi f_{\mathrm{m}},f_{\mathrm{m}}=50\) MHz); \(I_{0}(\omega)\) is the in-phase term; \(Q_{0}(\omega)\) is the quadrature term; \(k\) represents the scale factor containing the light power-voltage conversion coefficient, the insertion loss in every electronic element, and the gain of the amplifier; \(P_{0}\) represents the laser power; \(J_{0}\) and \(J_{1}\) are the first kind of 0- and 1-order Bessel function; \(\beta\) is the modulation degree (\(\beta=\pi V/V_{\pi}\)); \(V\) is the signal of the peak-to-peak value voltage input into the PM; \(V_{\pi}\) is the half-wave voltage of the PM; and \(R(\omega)\) is the complex spectral response of the resonance. When the bandwidth of the resonance is much larger than the PM modulation frequency (i.e., low quality factor, Q), the sidebands can be modulated by the resonance, and \(\omega_{\mathrm{m}}\) is negligible, thus \(R^{*}(\omega+\omega_{\mathrm{m}})-R^{*}(\omega)\approx[\mathrm{d}R^{*}(\omega )/\mathrm{d}\omega]\omega_{\mathrm{m}}\), \(R(\omega)-R(\omega-\omega_{\mathrm{m}})=[\mathrm{d}R(\omega)/\mathrm{d} \omega]\omega_{\mathrm{m}}\); the I-term and Q-term signals can be expressed as [27]:  
\[I_{0}(\omega) \approx kP_{0}J_{0}(\beta)I_{1}(\beta)[\mathrm{d}\big{|}R( \omega)\big{|}^{2}/\mathrm{d}\omega\big{]}\omega_{m}, \tag{3}\] \[Q_{0}(\omega) \approx 0. \tag{4}\]  
By comparison, when the bandwidth of resonance is much less than the PM modulation frequency (i.e., high Q), the sideband light is totally transmitted, thus \(R(\omega\pm\omega_{\mathrm{m}})\approx 1\) and \(R^{*}(\omega\pm\omega_{\mathrm{m}})\approx 1\), and the I-term and Q-term signals can be expressed as [27]:  
\[I_{0}(\omega) \approx 0, \tag{5}\] \[Q_{0}(\omega) \approx 2 kP_{0}J_{0}(\beta)I_{1}(\beta)\mathrm{Im}[R(\omega)]. \tag{6}\]When the PM modulation frequency is close to the bandwidth of resonance, the I-term and Q-term signals exist simultaneously. The \([\mathrm{d}|R(\omega)|^{2}]/\mathrm{d}\omega]\omega_{\mathrm{m}}\) (low Q condition) and \(2\mathrm{Im}[R(\omega)]\) (high Q condition) are both sharp dispersion curves like the PDH discriminating frequency curve [27]. The PDH system transforms the sharp imaginary information of the resonance and the slope information of the resonance into the I-term and Q-term signals. Any tiny change of resonance will significantly influence the I-term and Q-term signals. Therefore, the I-term and Q-term signals can be used to analyze the complicated resonances that the transmitted spectrum cannot be used to analyze. However, the \(\varphi\) is an unknown value in the experiment, and the phase difference will influence the I-term and Q-term signals [31]. Here, we use the orthogonal demodulation technique [31,32] to eliminate the phase difference and make the OD-PDH spectrum (IQ curve) stable. In sensing applications, this technique can help obtain the highest sensing sensitivity. In this experiment, we use Equation (7):  
\[IQ(\omega)=I_{0}^{2}(\omega)+Q_{0}^{2}(\omega) \tag{7}\]  
to obtain the \(IQ\) curve, which does not relate to \(\varphi\).