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 .