Axions have attracted attention promising candidates for dark matter (DM).Although axions have been intensively searched for, they have not been observedyet. Recently, novel experiments to search for axion DM have been proposed thatuse optical cavities to amplify polarization rotation of laser light induced bythe axion-photon interaction. One such experiment employs a ring cavitycomposed of four mirrors. However, its sensitivity to the axion-photon coupling\gag in the low axion mass region is limited due to the reflection phasedifference between s- and p-polarizations. In this paper, we propose a newmethod to improve the sensitivity using zero-phase shift mirrors and awavelength tunable laser. Moreover, the laser makes it easier to scan the highaxion mass region by tuning the reflection phase difference between s- andp-polarizations. We experimentally confirmed that the phase differencegenerated upon reflection on a zero phase shift mirror satisfies therequirement of 8.6 \times 10^{-3}~\si{deg}, which corresponds to the halfwidth at half maximum (HWHM) of the p-polarization with the mirror fixed on afolded cavity and a wavelength tunable laser.<br
