Location-dependent frequency tuning of vibrating micromechanical resonators via laser trimming

Location-dependent, bidirectional laser trimming of the resonance frequencies of vibrating micromechanical resonators is demonstrated in steps as small as 21 ppm over it range of 20,200 ppm, and with targeting measures that suppress unwanted variations in Q and series motional resistance. Specifically, geometrically symmetrical laser targeting is shown to be instrumental in preserving high Q on the micro-scale, much more so than on the macro-scale. A semi-empirical model is used to model the trimming process and to identify the laser trim locations that most efficiently attain a desired shift in frequency with minimal Q reduction. Different micromechanical resonator types are trimmed to demonstrate the versatility of the technique, including a clamped-clamped beam and a wine glass disk.