White light scanning interferometry (SWLI), as a high-precision, non-contact measurement technique, is widely used in fiber optics, automotive industry, precision measurement, and manufacturing. SWLI can employ piezoelectric ceramics (PZT) or stepper motors for vertical scanning. Although PZT features extensive application due to its high precision, it requires expensive controllers to correct the hysteresis effects brought by the piezoelectric elements. Additionally, the range of PZT is typically limited to several tens of micrometers. In contrast, high-precision stepper motors used for axial scanning provide a broader range and reduce cost. However, the nonlinear stepping vibrations of stepper motors lead to greater displacement errors than PZT, resulting in significant random errors between the spacings of adjacent frames of interferometric images. This issue can cause obvious ripple errors or height jumps in morphology calculations, resulting in lower solution accuracy and challenges for SWLI in cost reduction and measurement in vibrating environments. The elimination of this problem will enhance the applicability and accuracy of SWLI technology, further advancing its development and application across various fields.

We improve and further investigate the fitting compensation techniques proposed by previous researchers. A stepper motor is used as the scanning component to support the objective lens in vertical displacement. A high-precision grating ruler is installed on the guide rail to directly read the displacements accurately and sort the signals. The sorted interference signals are then coarsely positioned at the center of the zero-order fringe using the centroid method. Additionally, non-equidistantly sampled signals undergo trigonometric fitting using the Fourier series. By randomly sampling several coordinate points of the interference images, optimal angular frequency parameters are determined using a search algorithm, followed by matrix calculations and signal set fitting using the linear least squares method. Based on the coarse positioning, an appropriate interval is selected for high-density up-sampling of the fitting function. This interval can range between one-quarter to one-half of the wavelength to ensure coverage of the coherent peak waveforms in the interference signal, thus obtaining accurate coherent peak positions and reconstructing the object’s morphology (Fig. 3). We adopt an approach of up-sampling specific local areas to reduce computational load and enhance measurement accuracy.

We conduct both simulation experiments and actual measurement tests. In the simulation experiments, tests are carried out using simulated scanning with vibration errors. The simulated white light Gaussian source has a central wavelength of 550 nm and a spectral width of 100 nm, with a scanning step size set at 40 nm and a simulated step height of 400 nm. Certain random errors are introduced in the vertical scanning intervals, with the range of single displacement errors within 40%, and Gaussian noise is added. The centroid method and the proposed algorithm are compared, with the results of morphology restoration and errors shown in Fig. 7 and Fig. 8. Repeated measurements indicate that the error rate decreases from 3.5% to 0.11%. In the actual measurement experiments, a microscope driven by a white LED light source and a stepper motor is used to test step samples. The displacement error and test results are shown in Fig. 11. The comparison shows that our algorithm, compared to the original one, better controls the calculation errors caused by defect signals due to nonlinear sampling in the actual measurement environment. Table 2 shows the computational errors of various algorithms for the step samples, where the error rate of the proposed algorithm decreases from 1.32% to 0.36%.

We introduce an improved white light interferometry algorithm adapted for non-equidistant sampling environments. The algorithm begins with coarse positioning of the interference coherence peak to define the local sampling interval. Optimal angular frequency parameters obtained via the proposed algorithm are then used for least squares matrix fitting of the signal. Subsequent high-density up-sampling within this interval allows accurate determination of the zero optical path peak signal points, thus precisely defining the sample’s surface morphology. Both simulation and experimental results demonstrate that this algorithm effectively reduces ripple errors caused by non-equidistant signal distribution in vibrating environments and effectively reduces calculation errors due to decreased interpolation performance compared to the original algorithm. Characterized by high accuracy and repeatability, this method also significantly reduces the computational burden of up-sampling all signals by sampling selected regions, making it an efficient 3D reconstruction algorithm for white light interferometry.