# Investigation of High-Q Lithium Niobate-Based Double Ring Resonator Used in RF Signal Modulation
## 3 Results and Discussion
### Analysis of Double Ring Structure  
Assuming no waveguide loss under the analysis and comparison of the coupling coefficients \(k_{1},k_{2},k_{3}\) on the transmission characteristics of the double ring resonator, as shown in Figure 6. Figure 6a shows that with \(k_{1}=0.1\), the Q-factor and extinction ratio of the inner ring resonance peak gradually decrease as \(k_{2}\) and \(k_{3}\) gradually increase, and the inner ring resonance peak drifts toward the longer wavelength. Figure 6b shows that with \(k_{2}=k_{3}=0.1\), \(k_{1}\) gradually increases, and the extinction ratio of the inner ring resonance peak gradually decreases, but the resonant peak does not drift apparently. Figure 6c shows that with \(k_{2}=0.1\), \(k_{1}\) and \(k_{3}\) increase gradually at the same time, and the position and extinction ratio of the inner ring resonance peak do not change significantly, but the Q-factor of the resonance peak decreases accordingly. By comparing Figure 6a-c, it can be concluded that in the preparation process, if the two gaps of the straight and curved waveguides are different, the coupling coefficients will be different, which destroys the resonance enhancement of the double ring and thus affects the critical coupling state. Figure 6d shows that in the case of \(k_{1}=k_{2}=0.1\), with the gradual increase of \(k_{2}\), the extinction rate of the resonant peak of the inner ring does not change significantly, but the position of the resonant peak drifts toward the longer wavelength and the Q-factor decreases. Based on the above discussion, it is concluded that since a smaller coupling coefficient leads to a smaller bandwidth of light being coupled into the microring, making the resonant enhancement of light at specific wavelengths more intense, resulting in a high Q-factor. Therefore, it is necessary to ensure a smaller coupling coefficient in the structural design and a higher precision in the coupling region during the chip fabrication to obtain a high Q-factor of the device. When the waveguide loss of the microring is considered, the greater the loss, the worse the transmission characteristics of the microring. The field attenuation of outer ring and inner ring for one round trip are characterized by \(A_{1}\) and \(A_{2}\), respectively, and the relation between the transmission loss \(\rho\) and the field attenuation is \(A=e^{-\rho L}\). Figure 7a shows that in the case of no loss in the inner ring, the Q-factor and extinction ratio of the double ring resonant peak gradually decrease with the reduction of \(A_{1}\). Figure 7b shows that with no loss in the outer ring, the resonance peak of the inner ring deteriorates significantly with the continuous reduction of \(A_{2}\). This result reveals that the transmission loss of the microring has a great impact on the performance of the modulator. As LN is recognized as a difficult material to be etched, large losses will inevitably be introduced in the actual preparation process of the waveguide, it is important to determine the critical coupling conditions to remedy this problem. Figure 5: The Q-factor of resonator with the ratio (\(k\)) of outer ring length to inner ring length. Figure 6: (**a**) The transmissivity of the double ring resonator at different wavelengths and coupling coefficients (\(k_{2}\), \(k_{3}\)); (**b**) the transmissivity of the double ring resonator at different wavelengths and coupling coefficients (\(k_{1}\)); (**c**) the transmissivity of the double ring resonator at different wavelengths and coupling coefficients (\(k_{1}\), \(k_{3}\)) and (**d**) the transmissivity of the double ring resonator at different wavelengths and coupling coefficients (\(k_{2}\)). All other unchanged coupling coefficients are fixed at 0.1. Figure 7: (**a**) The transmissivity of the double ring resonator at different wavelengths and the field attenuation of outer ring (\(A_{1}\)) and (**b**) the transmissivity of the double ring resonator at different wavelengths and the field attenuation of inner ring (\(A_{2}\)).