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A stably acquiring and accurately demodulating interferogram is crucial for the interferometer to achieve ultra-high precision and sensitivity measurements. In this study, a robust orbital angular momentum (OAM) beams-based Fizeau interferometer is proposed and experimentally demonstrated, which is more compact and stable than the OAM interferometers with other structures due to the common optical path characteristic. In accordance with such an interferometric scheme, a phase-demodulation method operated in the domain of the OAM complex spectrum is also proposed and demonstrated in this study. In contrast to other phase-demodulation techniques, the proposed phase-demodulation technique neither requires phase shifters or phase modulators nor brings spectral leakage, which provides a robust alternative enabling to accurately and quickly extract the phase from the OAM interferogram. As a proof-of-concept of the proposed scheme, tiny displacements ranging from 0 to 800 nm were measured. The proposed OAM beams-based Fizeau interferometer and the corresponding azimuthal phase demodulation method may provide a feasible way for exploring further applications of the OAM-based interferometer in metrology.