This work reports a novel structure for a Fabry-Pérot micro cavity that combines the highest reported quality factor for an on-chip Fabry-Pérot resonator that exceeds 9800, and a very high sensitivity for an on-chip volume refractometer based on a Fabry-Pérot cavity that is about 1000 nm/refractive index unit (RIU). The structure consists of two cylindrical Bragg micromirrors that achieve confinement of the Gaussian beam in the plan parallel to the chip substrate, while for the perpendicular plan, external fiber rod lenses (FRLs) are placed in the optical path of the input and the output of the cavity. This novel structure overcomes number of the drawbacks presented in previous designs. The analyte is passed between the mirrors, enabling its detection from the resonance peak wavelengths of the transmission spectra. Mixtures of ethanol and deionized (DI)-water with different ratios are used as analytes with different refractive indices to exploit the device as a micro-opto-fluidic refractometer. The design criteria are detailed and the modeling is based on Gaussian-optics equations, which depicts a scenario closer to reality than the usually used ray-optics modeling.