The depth resolution of a wavelength-scanning interferometric measurement system is limited by the wavenumber scanning bandwidth. When the laser discontinuous output restricts the available bandwidth, the depth resolution of the measurement system significantly decreases. Currently, the mainstream solution internationally is to synthesize the wavenumber sequences before and after the discontinuous wavelength scanning. Under discontinuous wavelength output, the initial phase of the scanning interferometric signal experiences jumps, leading to abnormal sidelobe amplitudes, which can cause interference between adjacent sidelobe peaks and affect subsequent depth-resolved phase reconstruction. This paper proposes a multi-surface interferometric measurement method suitable for discontinuous wavelength scanning. By constructing a least-squares spectrum-based phase reconstruction method for scanning interferometry, the impact of phase jumps in discontinuous scanning interferometric signals is eliminated, thereby suppressing the abnormal sidelobes caused by the discontinuous sweep synthesis and improving the applicability of wavelength-scanning interferometry.

The algorithm proposed in this paper, based on least-squares spectrum correction of discontinuous interferometric signals, aims to suppress abnormal sidelobes caused by discontinuous laser wavelength scanning, thereby fully utilizing the spectral information from the discontinuous bands to achieve high-resolution and high-precision phase reconstruction. First, a constant drive current, temperature tuning range, and acquisition frame rate are set. Using the principle of scanning interferometric imaging, interference patterns from multiple surfaces are obtained on a self-built multi-surface scanning interferometric measurement hardware system. After removing the direct current component and synthesizing the wavenumber sequences before and after the discontinuous wavelength scanning, the necessary preprocessing is carried out to obtain the discontinuous scanning interferometric signals required for the experiment. The discontinuous interferometric signals are decomposed into a linear combination of sine and cosine basis functions using the least-squares error minimization, and the amplitude parameters of the sine and cosine basis functions, which contain the frequency and phase information of the scanning interferometric signal, are estimated using matrix least-squares estimation to demodulate the interference phase. Two experimental sets are designed in this paper. In the numerical simulation comparison experiments, the mean difference and standard deviation between the interferometric phase and the ideal phase are used as evaluation metrics for quantitative analysis. In the actual experimental results, high-precision parameter estimation for multiple surfaces is achieved, validating the effectiveness of the proposed method.

In the numerical simulation experiment, this paper compares the interference phase maps before and after the synthesis of discontinuous scanning wavenumber sequences (Fig. 6). Due to the limited bandwidth of continuous scanning wavenumber sequences, Fourier transforms struggle to effectively separate the spectral components of the interferometric signal, leading to inadequate phase reconstruction accuracy. To improve depth resolution, this paper synthesizes discontinuous scanning wavenumber sequences. However, the initial phase jumps introduced by these sequences cause abnormal sidelobes during Fourier transform demodulation, leading to errors in peak frequency extraction and ultimately resulting in phase aliasing. To suppress the impact of these sidelobe anomalies, an algorithm based on the least-squares spectrum is used to correct the discontinuous interferometric signals, reducing the sidelobe amplitudes, minimizing their influence on the main peak, and ensuring accurate peak frequency extraction for correct phase reconstruction. To assess phase quality, error analysis is conducted on the results of the discontinuous interferometric phase (Table 1 and Table 2), calculating the average error and standard deviation between each interferometric phase and the ideal phase. The interference phase errors calculated by the least-squares spectrum are all less than 0.036 rad, significantly smaller than those obtained from Fourier transform calculations. In the multi-surface scanning interferometric measurement experiment, the interferometric phase results for each surface are consistent with those from the numerical simulation experiment (Fig. 10 and Fig. 11), demonstrating the robustness of the algorithm. High-precision parameter estimation for each surface are as follows (Fig. 12): the light wedge tilt angle is 6.34′, with an error of 0.34′, corresponding to a percentage error of 5.67%; the standard error for the plano-convex lens surface is 21.66 nm; and the radius of curvature of the convex surface of the plano-convex lens is 516.8130 mm, with an error of 13 μm, corresponding to a percentage error of 0.0025%.

The least-squares spectrum method proposed in this paper significantly enhances the applicability of wavelength-scanning interferometry under discontinuous laser output. The method effectively suppresses abnormal sidelobes caused by discontinuous wavelength scanning, improves depth resolution, and accurately reconstructs the phase. Both numerical simulation and multi-surface interferometric experimental results demonstrate that the least-squares spectrum has good robustness in correcting discontinuous interferometric signals to achieve high-precision multi-surface measurements, providing an effective solution for high-precision optical measurements.