  We propose a novel experiment to search for axion dark matter which
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 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 $g_{a\gamma}$ for the axion
mass $m \lesssim 10^{-10}$ eV. In a low mass range $m \lesssim 10^{-15}$ eV, we
can achieve $g_{a\gamma} \lesssim 3\times 10^{-16} ~\text{GeV}^{-1}$ which is
beyond the current bound by several orders of magnitude.
