# Investigation of High-Q Lithium Niobate-Based Double Ring Resonator Used in RF Signal Modulation
## 2 Materials and Methods  
The schematic structure of the waveguide and the substrate is shown in Figure 1a. The underlying substrate is 0.5 mm thick silicon with a coating layer of 2 \(\upmu\)m thick silicon dioxide. The core layer is composed of 600 nm thick x-cut LN film. Lumped gold electrodes deposited on the left and right sides of the inner ring waveguide are used to apply RF signals. In order to prevent external dust, water vapor, and other pollutants from affecting the performance of the structure, a layer of SiO\({}_{2}\) is deposited on the top to enhance the stability of the device. The light of 1550 nm is used as the carrier wavelength and the refractive indices of the waveguide are \(n_{e}\) = 2.138, \(n_{o}\) = 2.211, and \(n_{SiO_{2}}\) = 1.444, which possess a large core-envelope refractive index difference of the LNOI structure (\(\Delta n\) > 0.7), providing strong light field restriction of the proposed waveguide. The outer ring adopts the add-drop structure. By optimizing the dimensions of the waveguide cross section, bending radius, upper coating thickness, electrode spacing, and other parameters, the insertion loss of the transmitted light can be minimized. For the light induced into the inner ring resonator, it will meet its resonant conditions and start to oscillate, which will help to enhance the interaction with the electric field and achieve higher modulation efficiency. The transmission parameters of the double ring modulator are shown in Figure 1b, which consists of waveguide length (\(L\)), intensity of light field (\(E\)), transmission coefficient (\(t\)), and coupling coefficient (\(k\)). Assuming that the amplitude of incident light is \(E_{i1}\) in the case of no coupling loss, the transmission characteristics of the three coupling regions can be obtained by using the transmission matrix method, and the matrix relationship among the coupling regions can be expressed as in Equations (1)-(3);  
\[\left[ \begin{array}{c}E_{i1}\\ E_{1}\end{array} \right]=\left[ \begin{array}{cc}t_{1}&jk_{1}\\ jk_{1}&t_{1}\end{array} \right]\left[ \begin{array}{c}E_{i1}\\ E_{8}\end{array} \right] \tag{1}\]  
\[\left[ \begin{array}{c}E_{i2}\\ E_{7}\end{array} \right]=\left[ \begin{array}{cc}t_{3}&jk_{3}\\ jk_{3}&t_{3}\end{array} \right]\left[ \begin{array}{c}E_{i2}\\ E_{6}\end{array} \right] \tag{2}\]  
\[\left[ \begin{array}{c}E_{3}\\ E_{4}\end{array} \right]=\left[ \begin{array}{cc}t_{2}&jk_{2}\\ jk_{2}&t_{2}\end{array} \right]\left[ \begin{array}{c}E_{2}\\ E_{5}\end{array} \right] \tag{3}\]  
where \(t\) and \(k\) represent the transmission and coupling coefficients of the coupling regions, and the relationship between these two parameters is: \(\left|t\right|^{2}+\left|k\right|^{2}=1\). When add port has no add input (\(E_{i2}=0\)), the transmission relationship between different coupling regions can be expressed by Equations (4)-(7);  
\[E_{2}=e^{-\rho L_{1}}e^{i\theta_{1}}E_{1} \tag{4}\]  
\[E_{6}=e^{-\rho L_{2}}e^{i\theta_{2}}E_{3} \tag{5}\]  
\[E_{8}=e^{-\rho L_{3}}e^{i\theta_{3}}E_{7} \tag{6}\]  
\[E_{5}=e^{-\rho L_{4}}e^{i\theta_{4}}E_{4} \tag{7}\]  
where \(\rho\) is the loss coefficient of the microring, \(L_{i}\) (\(i\) = 1,2,3,4) is the length of each part of the microring, and \(\theta_{i}\) (\(i\) = 1,2,3,4) is the phase change over the corresponding length \(L_{i}\). The final transmission equation of the through port can be obtained through the normalization of the light field amplitude at each port, as shown in Equation (8):  
Figure 1: (**a**) Schematic diagram of the structure of the double ring modulator, where the enlarged area is a crossing-section view of the devices. (**b**) Parameter distribution in the schematic diagram of double ring structure, including transmission coefficient (\(t\)), coupling coefficient (\(k\)), waveguide length (\(L\)), and the intensity of light field (\(E\)).