High-order nonlinear error compensation method based on error construction

In heterodyne interferometry, nonlinear error refers to a periodic error associated with phase variations caused by incomplete reflection of optical components. Errors that exhibit periodicity matching the phase variation frequency are classified as first-order nonlinear errors, while those occurring at integer multiples of this frequency are termed high-order nonlinear errors. Factors contributing to first-order nonlinear errors primarily include laser orthogonality errors, elliptical polarization errors, and incomplete light splitting by polarization beam splitters. Extensive research has been conducted on compensation methods for first-order nonlinear errors, resulting in well-established approaches. Incomplete reflection of optical components causes a portion of the measurement light to traverse the optical path multiple times, leading to ghost reflections. The high-order nonlinear errors introduced by ghost reflections represent a significant source of error that limits the accuracy of heterodyne interferometry. In heterodyne laser interferometers, as the incident angle of the measurement optical path does not influence the resulting phase difference, the phase difference between ghost reflections and the primary measurement light exhibits a fixed integer multiple relationship. Consequently, the frequency of high-order nonlinear errors maintains a proportional relationship with the phase variation frequency throughout the measurement process. Due to this inherent multiple relationship, methods like ellipse fitting and rotating polarizers, which effectively compensate for first-order nonlinear errors, are unsuitable for high-order nonlinear errors. To address this, we propose a nonlinear error compensation method based on error construction. When the interferometric signal is phase-resolved using the dual-orthogonal phase-locked amplification method, the phase is ultimately obtained by computing the inverse tangent of a pair of orthogonal signals. By applying frequency doubling and phase modulation to these signals, a constructed error analogous to the nonlinear error is generated. The frequency doubling coefficient is determined by the order of the high-order nonlinear error. The phase modulation coefficient of the measurement system can be calibrated by applying the mutual correlation coefficient method to phase data acquired during uniform motion of the test object. To validate the proposed high-order nonlinear error compensation method, we constructed a heterodyne interferometer measurement system and conducted experimental measurements. Experimental results indicate that the peak-to-peak value of the nonlinear error is reduced from 13.5 nm to 5.6 nm. This method enhances the measurement precision of the heterodyne interferometer, achieving sub-nanometer accuracy.