A method for measuring the relative phases (RPs) among the three beams in the case of homodyne three-beam interference (TBI) is proposed and verified by the ZEMAX simulation in this paper. The method requires that the interference beams are not on the same plane, that is, any two of the three beams interfere at an angle in different planes. Based on the phase delay of the beam in space, the inclined beams have different phases at different positions within the beams' range. By arranging the photodetector array within the interference area, the RPs can be calculated using the intensity of the interference light received by specific photodetector units. The application of the algorithm for the displacement measurement of the homodyne three-beam interferometer (TBIR) has been verified by simulation. The beam number of three-beam interference is one less than that of two two-beam interference when measuring two relative phases. Compared with the two-beam interferometer, the TBIR applied to two-degree-of-freedom displacement measurement is more compact in structure due to the more compact phase measurement components.
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