# Coupled Dual-Loop Optoelectronic Oscillator Based on Stimulated Brillouin Scattering
## 2 Principle of Operation
Then, one optical output of the DOMZM is directly sent into the MZM while the other optical output of the DOMZM is converted into microwave signal by the PD2 after passing through the SMF. The output \(V_{2}(t)\) of the PD2 is proportional to \(\left|E_{DOMZM}(t+\tau_{2})\right|^{2}\). \(\tau_{2}\) is the time delay introduced by Loop2. According to Equation (3), ignore the higher order terms, \(V_{2}(t)\) can be described as:  
\[V_{2}(t)\propto 2R_{2}G_{EA2}{E_{0}}^{2}J_{0}(\beta_{1})J_{1}(\beta_{1})\exp \bigg{[}j(\omega_{m}t+\omega_{m}\tau_{2}+\frac{\pi}{4})\bigg{]} \tag{4}\]  
where \(R_{2}\) is the responsivity of the PD2, \(G_{EA2}\) is the gain of the EA2. Then the microwave signal \(V_{2}(t)\) of the PD2 can be applied to the MZM. The output optical field of the MZM can be expressed as:  
\[E_{MZM}(t)=\frac{\sqrt{2}}{2}E_{DOMZM}(t)+\frac{\sqrt{2}}{2}E_{DOMZM}(t)\exp \Bigg{\{} \begin{array}{l}j\frac{\pi}{\
u_{\pi 2}}V_{DC2}\\ +j\frac{\pi}{\
u_{\pi 2}}R_{2}G_{EA2}{E_{0}}^{2}J_{0}(\beta_{1})J_{1}(\beta_{1})\exp \Big{[}j(\omega_{m}t+\omega_{m}\tau_{2}+\frac{\pi}{4})\Big{]}\end{array} \Bigg{\}} \tag{5}\]  
where \(V_{DC2}\) is the bias voltage applied to the MZM, \(V_{\pi 2}\) is the half-wave voltage of the MZM. The bias voltage of MZM is set to \(V_{\pi}/2\). The reversed pump light stimulated the output optical signal of the MZM after passing through the HNLF. After processing by the SBS gain spectrum, the optical signals are sent into the PD1. Thus, ignoring the frequency-doubling signal and DC term, the output of the PD1 can be simply described as:  
\[V_{1}(t) \propto\frac{R_{1}{\rm{E_{0}}}^{2}}{2}\Bigg{\{} \begin{array}{l} \left[J_{0}(\beta_{1})J_{0}(\beta_{2})+J_{0}(\beta_{1})J_{0}(\beta_{3})\right] \exp(j\omega_{m}t)\\ +[J_{0}(\beta_{1})J_{1}(\beta_{2})+J_{0}(\beta_{1})J_{1}(\beta_{3})]\exp[j \omega_{m}(t+\tau_{2})]\end{array} \Bigg{\}}\] (6) \[=\frac{R_{1}{\rm{E_{0}}}^{2}}{2}\Bigg{\{\[V_{1}(t) \propto\exp[j[\omega_{m}(t+n\tau_{1})]]\sum\limits_{n=0}^{\infty} \left\{ \begin{array}{c}\frac{G_{EA1}R_{1}E_{0}2}{2}\\ +J_{0}(\beta_{1})I_{1}(\beta_{2})\exp(j\omega_{m}\tau_{2})\\ +J_{0}(\beta_{1})I_{1}(\beta_{3})\exp(j\omega_{m}\tau_{2})\end{array} \right] \right\}^{n}\] \[=\exp(j\omega_{m}t)\sum\limits_{n=0}^{\infty}\left\{\exp(j\omega_ {m}n\tau_{1})[G_{e}(\omega_{m}\tau_{2})]^{n}\right\}\]  
where \(G_{EA1}\) is the gain of EA1, \(n\) is the number of oscillating times around the OEO loop and \(G_{e}(\omega_{m}\tau_{2})\) is called effective open-loop gain, written as:  
\[G_{e}(\omega_{m}\tau_{2})=\left\{ \begin{array}{c}\frac{G_{EA1}R_{1}E_{0}2}{2 }\\ +J_{0}(\beta_{1})I_{1}(\beta_{2})\exp(j\omega_{m}\tau_{2})\\ +J_{0}(\beta_{1})I_{1}(\beta_{3})\exp(j\omega_{m}\tau_{2})\end{array} \right\} \tag{8}\]  
From Equation (9) we can find that the effective open-loop gain \(G_{e}(\omega_{m}\tau_{2})\) is not a constant but a sinusoidal function of \(\omega_{m}\tau_{2}\), it means that the \(G_{e}(\omega_{m}\tau_{2})\) is a sinusoidal comb filter and the microwave signal is selective amplified. Thus, the frequency transfer function will be the product of the sinusoidal comb-filtering spectrum and SBS gain spectrum. When the TOEO stably oscillate, the effective open-loop gain \(G_{e}(\omega_{m}\tau_{2})\) is a little less than at oscillating threshold, and Equation (8) can be described as:  
\[V_{1}(t)\propto\frac{\exp(j\omega_{m}t)}{1-G_{e}(\omega_{m}\tau_{2})\cdot\exp( j\omega_{m}\tau_{1})} \tag{9}\]  
The corresponding microwave power \(P(\omega_{m},t)\) with stable oscillation is then given:  
\[P(\omega_{m},t)\propto\frac{\left|V_{1}(t)\right|^{2}}{2R}=\frac{1}{2R\left[1 +G_{e}^{2}(\omega_{m}\tau_{2})-2G_{e}(\omega_{m}\tau_{2})\cdot\cos(\omega_{m} \tau_{1})\right]} \tag{10}\]  
where \(R\) is the load impedance of PD1. The oscillation frequency must satisfy the loop phase matching condition and SBS frequency matching condition as:  
\[\left\{ \begin{array}{c}\omega_{m}=\frac{2\kappa n}{\tau_{1}}\quad k=0,1,2,3,\ldots\\ \omega_{m}=\omega_{p}-\omega_{B}-\omega_{s}\end{array} \right.