Compensation of frequency split by directional lapping in fused quartz micro wineglass resonators

We present, for the first time, a permanent structural asymmetry compensation method for fused quartz micro wineglass resonators. Using the technique, we demonstrated a near six times reduction of structural asymmetry (n = 2 wineglass mode), culminating in reduction of the frequency split from 41 to 7 Hz. This is an iterative process. In each iteration, the structural asymmetry was first identified by measuring the mode shape of the resonators. Then, directional lapping was performed with specially designed lapping fixtures to accurately control the lapping angle down to 1 degrees. Analytical predictions and numerical simulations were conducted to study the structural asymmetry phenomenon and the effects of compensation on the quality factor of the structure, showing the ability of this process to reduce the structural asymmetry, while not affecting the overall quality factor of the resonators.