Frequency splitting of hemispherical resonators trimmed with focused ion beams

This paper studies a high-precision etching process based on focused ion beam static trimming of the defective mass of fused silica hemispherical resonators to eliminate frequency splitting. This trimming method effectively overcomes the shortcomings of inefficiency and poor robustness, and the accuracy is seriously dependent on the stability of the trimming equipment in the traditional approach based on the harmonic form of trimming. The vibration model of an imperfect hemispherical resonator is firstly established by equating structural defects into point mass defective units, revealing that the frequency splitting is a function of the size of the equivalent defect mass and that the angular position of the stiffness axis is determined by the spatial distribution of the defect mass, which further elucidates that the frequency splitting of the resonator is the equivalent form of the defect mass on its low-frequency axis. To validate the accuracy of the present method, a simulation model of the imperfect resonator is established, and the results show that the theoretical model is in good agreement with the finite element method. Then, a high-precision nonlinear parameter identification model containing frequency splitting and low-frequency axis azimuth parameters is established based on the standing wave oscillation effect and spectral analysis, and the frequency splitting and low-frequency axis azimuth are identified by analyzing the vibration function of the hemispherical resonator. Finally, based on the parameter identification results and the ion beam removal function analysis, an ion beam trimming process is designed to eliminate frequency splitting by trimming the defective mass on the low-frequency axis. The results show that the frequency splitting after mass trimming is better than 0.001 Hz, significantly improving the hemispherical resonator's performance.