We propose a novel experiment to search for axion dark matter that differentiates the phase velocities of the left- and right-handed polarized photons. Our optical cavity measures the difference of the resonant frequencies between two circular polarizations of the laser beam. The design of our cavity adopts a double-pass configuration to realize a null experiment and give a high common mode rejection of environmental disturbances. We estimate the potential sensitivity to the axion-photon coupling constant g_{aγ} for the axion mass m≲10^{-10} eV. In a low mass range m≲10^{-15} eV, we can achieve g_{aγ}≲3×10^{-16} GeV^{-1}, which is beyond the current bound by several orders of magnitude.