Precision optical components are widely employed in various optical systems, and the surface shape quality of optical components directly affects the performance of optical devices. Therefore, surface shape detection of optical components is of great significance. Interferometry is widely recognized as the most effective method for surface shape detection, among which phase-shifting interferometry has higher detection accuracy. However, during the continuous collection of multiple interferograms with phase differences, it is constrained by the performance of the phase-shifting components and easily affected by such objective factors as mechanical vibration and air disturbance in the environment, which decreases the detection accuracy. Therefore, it is not suitable for on-site production testing. In recent years, researchers have proposed a method that combines carrier interferometry with Fourier analysis technology to achieve phase extraction of a single interferogram. However, generally, there are still shortcomings such as large tilt direction edge errors, stripe stacking phenomenon, and low recovery accuracy. To solve the problem of low accuracy in phase extraction of single interferograms in the above phase-solving methods, we propose a new single interferogram phase extraction method based on light intensity iteration. Meanwhile, simulations and experimental research are conducted, with the stability of the algorithm analyzed.
We adopt a combination of simulation and experiment methods, analyze and explore the principle of the light intensity iteration method, and employ MATLAB to write algorithm programs while conducting simulation verification. The feasibility, stability, and noise resistance of the algorithm are explored via simulations to ensure the algorithm performance. By conducting 100 sets of simulation simulations, the final phase residuals are compared, and the convergence conditions suitable for solving single interference fringes and the solution interval with the best measurement performance are obtained. To ensure the innovation and optimization ability of the algorithm, we conduct a comparison with the Fourier transform method. Finally, multiple experiments are carried out using the ZYGO-Verifire PE phase-shifting interferometer to measure optical components. Multiple sets of experiments are conducted in an experimental environment with temperature of 23 ℃ and air humidity of 75.3%. Meanwhile, a single interference fringe pattern is collected and the phase is solved using the proposed algorithm. The results are compared, and the effectiveness of the algorithm is evaluated by residual PV and RMS values to achieve phase extraction of the single interference fringe.
Our algorithm can ensure the algorithm stability while improving detection accuracy. By adopting the Bernsen algorithm to binarize the original interferogram and further obtain a stepped predicted phase (Fig. 3), initial information is provided for subsequent light intensity iterations. The use of binarization to predict phases provides a new approach for iterative methods. The feasibility and anti-noise ability of this method are demonstrated by comparing it with the Fourier transform method (Fig. 4). Compared with the Fourier transform method, the proposed method has higher solving accuracy and faster solving speed. Meanwhile, its anti-noise ability is not significantly different from that of the Fourier method, both of which have sound anti-noise ability. By conducting hundreds of simulation experiments, convergence conditions that do not affect computational efficiency and avoid excessive iterations are obtained. The study on the effect of the fringe number on the accuracy of the algorithm solution shows that generally the size of the algorithm residual presents a trend of first decreasing and then increasing with the rising number of fringes (Fig. 8). Data comparison shows that the algorithm has the highest solution accuracy when processing a single interference fringe pattern with 4 to 5 fringes.
We propose a phase solution method based on the light intensity iteration method. Firstly, the original interferogram is binarized and the initial phase is obtained by phase unwrapping. Then, the background light and modulated light are preliminarily estimated by adopting the least square method using the interference intensity expression. The measured phase is calculated using a variation of the interference intensity expression. The measured phase is compared with the initial phase as a convergence judgment. The initial phase is replaced with a surface shape that does not meet the accuracy requirements. The background light and modulated light are updated, and the phase solution process is repeated. By light intensity iteration, the phase is extracted from a single interferogram. Meanwhile, solution accuracy, noise resistance, and algorithm stability are simulated and analyzed. Experimental measurements are conducted on a 100 mm planar element, and the results show that the obtained phase distribution of the proposed method is consistent with the phase obtained by the four-step phase-shifting algorithm of the ZYGO-Verifire PE phase-shifting interferometer. Compared to those obtained by the interferometer, the residual PV and RMS values obtained by the light intensity iteration method are 2.49 nm and 0.35 nm respectively. This indicates that the proposed method featuring high stability and efficiency can extract phase distribution from single fringes and can meet the testing needs of the production site environment.
