Objective The level of optical metrology determines the level of optical manufacturing technology, and the phase-shifting interferometry (PSI) as an easy, high-speed and accurate optical testing tool is usually used during or after optical fabrication. Both accuracy and efficiency are important to PSI. Outstanding phase-shifting algorithms (PSAs) can reduce the requirements for the interferometer hardware and environment, and further improve the accuracy and speed of PSI. Traditional PSAs with known phase shifts are easily affected by the miscalibration of piezo-transducer and environmental errors. In order to save time, many single-step PSAs were developed. Nevertheless, the sign of phase is difficult to judge by only one interferogram. In some high-precision events, accurate phase reconstruction is of interest. Hence, the multi-step PSAs with more than three interferograms were developed. However, it's difficult to reconstruct the phase with high accuracy and efficiency simultaneously. Comparatively, two-step random PSAs can avoid the effect of phase shift error, solve the sign ambiguity problem of the single-step PSAs, and balance the accuracy and speed. However, general two-step random PSAs need pre-filtering or use some complex methods to calculate background, these methods will cost more time. To balance the computational time and accuracy, a fast and high-precision two-step random phase-shifting algorithm based on principal component analysis and VU decomposition method is proposed in this paper.
Methods A two-step random phase-shifting algorithm based on principal component analysis and VU decomposition method is proposed in this paper. Firstly, two-step principal component analysis method is used to calculate the initial phase of iteration by two filtered phase-shifting interferograms, and then VU decomposition and iteration of two unfiltered phase-shifting interferograms are used to calculate the final phase. Finally, the proposed method is compared with four good two-step random phase-shifting algorithms for different fringe types, noise, phase shift values and fringe numbers to verify its superior performance in the computational time and accuracy.
Results and Discussions Compared with four good two-step random phase-shifting algorithms, the proposed method has the best comprehensive performance for different fringe types, noise, phase shift values and fringe numbers. The proposed method has the highest accuracy. Meanwhile, its effective phase shift range and fringe number range are the largest. When the size of interferograms is 401 pixel×401 pixel, the proposed method takes only 0.035 s more than Gram-Schmidt orthonormalization algorithm and two-step principal component analysis method. Under ideal conditions, the proposed method can get exactly correct result. If high precision is required, it is best to suppress the noise in advance, while setting the phase shift value away from 0 and π, and the fringe number greater than 2.
Conclusions In order to balance the accuracy and speed of phase calculation, a fast and high-precision two-step random phase-shifting algorithm based on principal component analysis and VU decomposition method is proposed in this paper. The method is characterized by high accuracy, high speed and no filtering. It takes approximately the time of non-iterative algorithm to obtain the accuracy of iterative algorithm, and breaks the limit that iterative algorithm costs more time. It is suitable for high-precision optical in-situ measurement and has wide development future.
DownLoad: