Objective The development trend of laser heterodyne interferometry is the enhancement of measurement resolution and accuracy. The primary factor that hinders the further enhancement of laser heterodyne interferometry accuracy is the periodic nonlinear error. This research proposes a nonlinear error compensation approach for laser heterodyne interferometry based on wavelet transform. The Morlet wavelet transform is employed for the nonlinear error function, and the wavelet ridge is extracted from the wavelet coefficient matrix's information. Next, the characteristic information of wavelet ridge line is examined and the first harmonic nonlinear error is rebuilt. After compensating for the first harmonic nonlinear error based on the wavelet transform approach, the second harmonic nonlinear error is fitted iteratively by the least-squares nonlinear fitting approach. Methods First, a nonlinear error compensation approach based on a wavelet transform is proposed. Periodic nonlinear errors in laser heterodyne interference systems can be modeled as the superposition of pure sine waves. The wavelet family is created by changing the time and scale factors of complex Morlet wavelet. The pure sinusoidal model is converted using wavelet transform based on the Morlet wavelet family. By further computation, the wavelet coefficient's modulus and phase are obtained, so that the wavelet ridge is extracted. When the modulus of the wavelet system is the largest at the ridge position, the corresponding scale corresponds to the first harmonic nonlinear error frequency. The first harmonic nonlinear error phase is the corresponding phase. The second harmonic nonlinear error frequency is twice that of the first harmonic nonlinear error. Additionally, the second harmonic nonlinear error phase is achieved under the corresponding frequency's scale. Based on this, the first harmonic nonlinear error function's amplitude is computed, and the rebuilt first harmonic nonlinear error function model is achieved. After compensating for the first harmonic nonlinear error, the nonlinear fitting approach based on the least square is employed to further fit the nonlinear second harmonic. Furthermore, a laser heterodyne interferometer optical path is constructed to measure nonlinear errors. The nonlinear error compensation based on the wavelet transform is employed for the nonlinear error measurement, and the actual compensation impact is investigated. Results and Discussions Based on the principle of laser heterodyne interference, an experimental optical path of laser heterodyne interference is constructed (Fig. 3). The measurement signal's nonlinear error in the experimental device is measured. The interference signal's spectrum when the gauge block moves at 8 mm/s is examined (Fig. 4). The first harmonic's magnitude and the nonlinear error's second harmonic are 5.97 nm and 0.25 nm, respectively. The nonlinear error compensation approach for laser heterodyne interference based on the continuous wavelet transform is employed to compensate for the nonlinear error. The measurement system's nonlinear error component decreases from 5.97 nm to 1.09 nm, and the nonlinear error component is reduced to 18% of the original. Based on this approach, the impact of nonlinear error can be suppressed efficiently and the measurement accuracy of laser heterodyne interference can be enhanced. Conclusions By addressing the challenge of periodic nonlinear error compensation in laser heterodyne interferometry, a nonlinear error compensation approach based on continuous wavelet transform is proposed. The Morlet wavelet transform is employed for the nonlinear error function, and the wavelet ridge is extracted from the wavelet coefficient matrix's information. Next, the characteristic information of wavelet ridge line is examined and the first harmonic nonlinear error is rebuilt. After compensating for the first harmonic nonlinear error based on the wavelet transform approach, the second harmonic nonlinear error is fitted iteratively using the least-squares nonlinear fitting approach. Experimental findings demonstrate that the nonlinear error component decreases from 5.97 nm to 1.09 nm, and the nonlinear error component is reduced to 18% of the original when the approach is employed for laser heterodyne interferometry. Based on this approach, the impact of nonlinear errors can be suppressed efficiently and the measurement accuracy of laser heterodyne interference can be enhanced.
