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This paper discusses relative intensity noise (RIN) characterization of triangular ring lasers, fabricated through a self-aligned dry etching process in InGaAsP-InP. Unusual low frequency noise enhancement was observed in each of the output beams, that is strongly dependent on the mirror backscattering strength. We explain the physical origin of the unusual RIN results as a partition noise effect between the two output beams of the ring laser, which can be best interpreted as superposition of two incoherent spatially bidirectional modes, generated by incoherent spontaneous emission photons traveling in clockwise and counterclockwise directions. This effect is unique to the ring laser geometry, and reported here for the first time. General noise predictions of our analysis are consistent with experimental measurements. This partition noise enhancement is expected to be present in the entire class of bidirectional semiconductor ring lasers, affecting the noise performance of not only the triangular ring laser, but also circular and square geometries, and possibly microdisk type lasers, in integrated photonic circuit applications.