# Coupled Dual-Loop Optoelectronic Oscillator Based on Stimulated Brillouin Scattering
## 2 Principle of Operation  
The schematic diagram of the coupled dual-loop SBS-based TOEO is shown in Figure 1a. The TOEO consists of a signal laser (SL), a dual-output Mach-Zehnder intensity modulator (DOMZM), a single-mode fiber (SMF), two photodetectors (PD), a single output Mach-Zehnder intensity modulator (MZM), an optical isolator (ISO), a polarization controller (PC), an optical circulator (OC), a high non-linear fiber (HNLF), two electrical amplifier (EA), an electrical splitter (ES). A light wave from the SL is sent to the DOMZM, which has two optical output terminals. One optical output of the DOMZM is directly sent to the MZM, and the other optical output of the DOMZM passes through the SMF and is transformed into microwave signal by PD2. Then the microwave signal modulates the MZM. The optical output of the MZM passes through the HNLF and then the reverse pump light stimulates the optical output of the MZM. After being filtered and amplified by the SBS gain spectrum, the optical output of the OC is injected into the PD1. Then, the microwave output of PD1 modulates the DOMZM to realize a closed loop after amplification by the EA1. The ES is placed between the EA1 and the DOMZM to separate part of the oscillating signal for measurement. In order to clearly understand the principle of the coupled dual-loop SBS-based TOEO, we firstly explain the spectrums at different points of the TOEO. The schematic diagram of spectrums is shown in Figure 1b. The point A shows the double sideband optical signal of the DOMZM. The point B shows the microwave signal of the PD2 to modulate the MZM. The point C shows the double sideband optical signal of the MZM after modulating by the microwave signal of PD2. The point D shows the filtered and amplified process of SBS. The point E is the result after SBS processing. Then the optical signal is sent to the PD1 after SBS processing. When the loop gains over the oscillating threshold of the TOEO, the oscillation mode will be stably established. Assuming that the DOMZM is modulated by:  
\[V_{1}(t)=V_{\text{DC1}}+V_{RF1}\cos(\omega_{m}t) \tag{1}\]  
Figure 1: (**a**) Schematic diagram of the the coupled dual-loop stimulated Brillouin scattering (SBS)-based tunable optoelectronic oscillator (TOEO). SL: signal laser; DOMZM: dual-output Mach–Zehnder intensity modulator; SMF: single mode fiber; PD, photodetector; MZM: Mach–Zehnder intensity modulator; ISO: optical isolator; HNLF, high non-linear fiber; OC: optical circulator; PL, pumped laser; PC, polarization controller; EA: electrical amplifier; ES, electrical splitter. (**b**) The spectrums at different point of the TOEO. where \(V_{DC1}\) is the bias voltage applied to the DOMZM, \(V_{RF1}\) is the amplitude of the oscillating signal and \(\omega_{m}\) is the angular frequency of the oscillating signal. The DOMZM is set in the quadrature point so the \(V_{DC1}\) is set to 0. The output optical field of the DOMZM, under a condition of small signal modulation, can be written as:  
\[E_{DOMZM}(t)=\frac{1}{2}E_{0}\exp(j\omega_{0}t)\Bigg{[}\exp\Bigg{(}j\frac{\pi V _{1}(t)}{2V_{\pi 1}}\Bigg{)}+j\exp\Bigg{(}-j\frac{\pi V_{1}(t)}{2V_{\pi 1}} \Bigg{)}\Bigg{]} \tag{2}\]  
where \(E_{0}\) is the amplitude of the input optical signal of the DOMZM and \(\omega_{0}\) is the angular frequency of the input optical signal of the DOMZM. \(V_{\pi 1}\) is the half-wave voltage of the DOMZM. Then the Equation (2) can be expanded with Bessel functions:  
\[E_{DOMZM}(t)\approx\frac{\sqrt{2}}{2}E_{0}\Bigg{\{} \begin{array}{l}J_{0}( \beta_{1})\exp j(\omega_{0}t+\frac{\pi}{4})+J_{1}(\beta_{1})\exp j\Big{[}( \omega_{0}+\omega_{m})t+\frac{\pi}{4}\Big{]}+\\ J_{1}(\beta_{1})\exp j\Big{[}(\omega_{0}-\omega_{m})t+\frac{\pi}{4}\Big{]} \end{array} \Bigg{\}} \tag{3}\]  
where \(\beta_{1}=\pi V_{RF}/2V_{\pi 1}\). \(J_{0}(\beta_{1})\) is the zero-order Bessel functions of the first kind. \(J_{1}(\beta_{1})\) is the first-order Bessel functions of the first kind.