“…The high density of grating inscriptions and the finer the grating pitch, combined with the process limitations of the instrument itself, result in three main sources of error in the ultra-precision grating interferometric signal in displacement measurement: (1) at the same noise level, the single-period measurement step of the ultraprecision grating interferometric signal is relatively small, and the impact of the error on the signal becomes larger; (2) at the same acquisition frequency and within the same measurement time, the ultra-precision grating interferometric signal acquires more outliers. (3) Fluctuations in the wavelength of the laser source, environmental changes, inaccurate optical circuit mounting, unsatisfactory performance of the optics, and processing of the raw signal by the photodetector and subsequent processing circuitry can all bias the measurement results. For problem (3), Heydemann proposed a phase error correction model for orthogonal interferometric signals in 1981 and used a least squares non-linear fit to eliminate the nonlinear error [22]; Wu and Eom of Physikalisch-Technische Bundesanstalt (PTB) carried out a detailed analysis of the nonlinear error, confirming that it is caused by the performance of the light source and optical components as well as interferometer adjustment errors, and claiming that its response to signal as the phase calculation error of the interferometric signal with unequal amplitude, non-orthogonal phase, and DC offset [23]; further, Jin et al studied the improvement based on the above conclusion [24][25][26][27][28].…”