As an important part of optical materials, optical transmission materials are widely employed in optical display and optical communication, and their optical properties play a key role in the whole optical system. The optical properties of optical transmission materials mainly include optical uniformity, optical thickness, surface shape, fringes, and bubbles. The optical parallel plate is strictly controlled by its design parameters. If the optical uniformity, optical thickness, surface shape, and other optical parameters of the plate are inconsistent, the optical wave front will be changed when the light wave passes through, thus degrading the optical system performance. Therefore, the optical uniformity, thickness, and surface shape of optical materials are significant performance indicators for high-precision optical systems. To solve the problem of slow speed, low efficiency, and small measurement range of optical parameters of parallel plates with different thicknesses, we propose a wavelength phase-shifting interferometry method based on characteristic polynomial.

This method combines the two-step absolute measurement method to carry out theoretical research on multi-surface interference technology. Then we design a weighted multi-step wavelength shift algorithm based on the feature map and feature polynomial theory, which is employed to extract and calculate the surface shape, optical thickness changes, and optical uniformity information of the plate. The specific process is as follows. The evaluation function and Fourier expression of the phase-shifting algorithm show the immunity of the algorithm to the errors. Finally, the algorithm is compared with the OPL algorithm. Firstly, the two-step absolute measurement method is combined with the theoretical research on multi-surface interference technology. Then a weighted multi-step wavelength phase-shifting algorithm is designed based on the feature map and feature polynomial theory, which is adopted to extract and calculate the surface shape, thickness changes, and optical uniformity information of the plate surface. The evaluation function of the phase-shifting algorithm and its Fourier expression are utilized to show the immunity of the algorithm to errors. Among them, the weighted multi-step wavelength shift algorithm based on the characteristic polynomial theory is designed as follows. First, the measured plate is placed in an interference cavity, and 77 interferograms are obtained by wavelength-tuned phase-shifting interference. The superimposed interference region of each interferogram is composed of six groups of first-order interference fringes. According to these interferograms, the feature graphs are designed and the corresponding feature polynomials are written. The characteristic polynomial is polynomial expanded, and the sampling amplitude is obtained by simultaneous solution combined with the relative frequency amplitude of the target information. The phase information is obtained by taking the two into the phase calculation formula, and the corresponding wave front information is obtained after the phase information is unpacked and de-tilted. Then, the measured plate is removed for cavity measurement, and the cavity phase information obtained by cavity measurement is unpacked and de-tilted to obtain cavity wavefront information. Finally, the optical uniformity of the parallel plate can be obtained by bringing the information of the two wave fronts into the calculation formula.

The proposed method features high speed and high precision in measuring the optical uniformity of parallel plates with different thin thicknesses. The PV and RMS errors of the 77-step phase-shifting algorithm and OPL algorithm are in the order of 10-8, and the PV errors of the surface shape and optical thickness changes are within λ/100. The PV and RMS errors calculated for the optical uniformity of parallel plate are in the order of 10-7, and the PV errors of surface shape and optical thickness are within λ/100. The data show that the calculation results of the two methods are basically consistent, and the measurement accuracy of 40 mm parallel plate is slightly higher than that of 5 mm parallel plate (Table 4). However, the 77-step algorithm is much better than the OPL algorithm in computational efficiency and speed because the required number of interference samples is much smaller than that of OPL algorithm (Table 5).

We study a multi-step wavelength phase-shifting algorithm based on the characteristic polynomial theory. Based on the conventional phase-shifting algorithm and the characteristic polynomial theory, three groups of 77-step algorithms are designed according to the target requirements for measuring and calculating the surface shape, optical thickness changes, and optical uniformity of parallel flat surfaces. Additionally, the evaluation function diagram of the algorithm is drawn to show the sensitivity and immunity of the algorithm to errors. The measurement results show that the 77-step phase-shifting algorithm can suppress harmonic errors, phase-shifting errors, and other coupling errors. Meanwhile, the algorithm requires fewer interferograms with high detection efficiency, takes into account the computational efficiency with high precision, and is suitable for optical parameter measurement of parallel plates with different thicknesses. The problems are solved such as the large number of interferograms required by existing algorithms, large computational amount, partial error compensation, and sensitivity to harmonic frequency mismatch or deviation.