## 2 Theoretical Analysis
\begin{array}{rcl}\tau\delta\dot{\alpha}_{s}+(\gamma+i\varphi)\delta \alpha_{s}=2\chi\overline{\alpha}_{p}\delta\alpha_{i}^{*}+2\chi\overline{ \alpha}_{i}^{*}\delta\alpha_{p}+\sqrt{2\gamma_{s}}\delta\alpha_{s,in}+\sqrt{2 \mu_{s}}\delta\beta_{s,in}\\ \delta\alpha_{i}=\chi\overline{\alpha_{s}}\delta\alpha_{p,in}-\chi^{2} \overline{\alpha_{s}}\overline{\alpha_{s}}\delta\alpha_{i}-\chi^{2} \overline{\alpha_{s}}\overline{\alpha_{i}}\delta\alpha_{s}+\chi\overline{ \alpha}_{p}\delta\alpha_{s}^{*}\end{array} \right.. \tag{5}\]  
Figure 1: Schematic of the singly resonant optical parametric oscillator. Hence, combining Equations (4) and (5) leads to  
\[\left\{ \begin{array}{l}\tau\delta\dot{a}_{s}+(\gamma+i\varphi)\delta\alpha_{s}= \frac{\sqrt{2}(\sigma-1)^{1/2}\left(\gamma^{2}+\varphi^{2}\right)^{1/4}}{\sigma} (\delta\alpha_{p,in}^{*}+\delta\alpha_{p,in})+\sqrt{2\gamma_{s}}\delta\alpha_{s,in}+\sqrt{2\mu_{s}}\delta\beta_{s,in}\\ \sigma\delta\alpha_{i}=(\sigma-1)^{1/2}\delta\alpha_{p,in}-\frac{(\sigma-1) \left(\gamma^{2}+\varphi^{2}\right)^{1/4}}{\sqrt{2}}\delta\alpha_{s}+\frac{ \left(\gamma^{2}+\varphi^{2}\right)^{1/4}}{\sqrt{2}}\delta\alpha_{s}^{*}\end{array}  \right.. \tag{6}\]  
The fluctuations of amplitude and phase quadratures can be defined as \(\delta X_{j}=\delta a_{j}+\delta a_{j}^{*}\) and \(\delta Y_{j}=-i(\delta a_{j}-\delta a_{j}^{*})\). Taking the Fourier transform of Equation (6), the fluctuation equations of signal and idler in the cavity can be expressed by  
\[\left\{ \begin{array}{l}-(\omega\tau+\varphi)\delta Y_{s}(\omega)+\gamma \delta X_{s}(\omega)=\frac{2\sqrt{2}(\sigma-1)^{1/2}\left(\gamma^{2}+\varphi^{ 2}\right)^{1/4}}{\sigma}\delta X_{p,in}(\omega)\\ +\sqrt{2\gamma_{s}}\delta X_{\alpha_{s,in}}(\omega)+\sqrt{2\mu_{s}}\delta X_{ \beta_{s,in}}(\omega)\\ (\omega\tau+\varphi)\delta X_{s}(\omega)+\gamma\delta Y_{s}(\omega)=\sqrt{2 \gamma_{s}}\delta Y_{\alpha_{s,in}}(\omega)+\sqrt{2\mu_{s}}\delta Y_{\beta_{s,in }}(\omega)\\ \sigma\delta X_{i}(\omega)=(\sigma-1)^{1/2}\delta X_{p,in}(\omega)+\frac{\left( \gamma^{2}+\varphi^{2}\right)^{1/4}}{\sqrt{2}}(2-\sigma)\delta X_{s}(\omega) \\ \sigma\delta Y_{i}(\omega)=(\sigma-1)^{1/2}\delta Y_{p,in}(\omega)+\frac{\left( \gamma^{2}+\varphi^{2}\right)^{1/4}}{\sqrt{2}}(2-\sigma)\delta Y_{s}(\omega) \end{array} \right., \tag{7}\]  
where \(\omega\) is the analysis frequency with the unit of hertz. The fluctuations for the amplitude quadrature of signal and idler in the cavity can be expressed by  
\[\left\{ \begin{array}{l}\delta X_{s}(\omega) = \frac{\frac{2\sqrt{2}(\sigma-1)^{1/2}\left(\gamma^{2}+\varphi^{2} \right)^{1/4}}{\sigma}\delta X_{p,in}(\omega)+\gamma\sqrt{2\gamma_{s}}\delta X _{\alpha_{s,in}}(\omega)+\gamma\sqrt{2\mu_{s}}\delta X_{\beta_{s,in}}(\omega)} {(\omega\tau+\varphi)^{2}+\gamma^{2}}\\ + \frac{(\omega\tau+\varphi)\sqrt{2\gamma_{s}}\delta Y_{\alpha_{s,in}}(\omega)+( \omega\tau+\varphi)\sqrt{2\mu_{s}}\delta Y_{\beta_{s,in}}(\omega)}{(\omega \tau+\varphi)^{2}+\gamma^{2}}\\ \delta X_{i}(\omega) = \frac{(\sigma-1)^{1/2}}{\sigma}\delta X_{p,in}(\omega)+\frac{(2- \sigma)\left(\gamma^{2}+\varphi^{2}\right)^{1/4}}{\sqrt{2}}\delta X_{s}(\omega) \end{array} \right., \tag{8}\]  
Using the input-output relation of Equation (2), the output variance for the amplitude quadrature of the output signal and idler can be deduced as  
\[\left\{ \begin{array}{l}Var(\delta X_{s,out})=1+\frac{16\gamma^{2}\gamma_{s }AC^{2}}{\delta^{2}D^{2}}Var(\delta X_{p})\\ Var(\delta X_{i,out})=\frac{2\gamma\gamma_{s}B^{2}C}{\sigma^{2}D^{2}}+\left( \frac{2\gamma_{i}A}{\sigma^{2}}+\frac{8\gamma\gamma_{i}ABC^{2}}{\sigma^{3}D}+ \frac{8\gamma^{2}\gamma_{i}AB^{2}C^{4}}{\sigma^{4}D^{2}}\right)Var(\delta X_{p })\end{array}