  The highly efficient coupling of light from conventional optical components
to optical mode volumes lies in the heart of chip-based micro-devices, which is
determined by the phase-matching between propagation constants of fiber taper
and the whispering-gallery-mode (WGM) of the resonator. Optical gyroscopes,
typically realized as fiber-optic gyroscopes and ring-laser gyroscopes, have
been the mainstay in diverse applications such as positioning and inertial
sensing. Here, the phase-matching is theoretically analyzed and experimentally
verified. We observe Sagnac effect in a millimeter-scale wedged resonator
gyroscope which has attracted considerable attention and been rapidly promoted
in recent years. We demonstrate a bidirectional pump and probe scheme, which
directly measures the frequency beat caused by the Sagnac effect. We establish
the linear response between the detected beat frequency and the rotation
velocity. The clockwise and counterclockwise rotation can also be distinguished
according to the value of the frequency beat. The experimental results verify
the feasibility of developing gyroscope in WGM resonator system and pave the
way for future development.
