At present, optical elements are almost always employed in the utilization and development of a wide variety of optical instruments. Due to improper handling during processing, scratches can appear on the surface of optical elements. Scattered light from surface scratches can reduce the beam quality, increase system noise, and reduce contrast, thereby affecting the performance and normal operation of the entire optical system. Therefore, the detection of surface scratches on optical elements is significant. As the existing light scattering methods can only detect the surface scratches of optical elements, the CCD or CMOS sensor can only receive the light field distribution formed by the scattering of surface scratches, from which the two-dimensional size of the surface scratches can be obtained. However, the depth information of the scratches cannot be detected directly. Since up to 80% of the surface information such as depths and shapes of surface scratches is characterized by phase information, we propose to apply the angular spectrum iterative algorithm and transport of intensity equation (TIE) + angular spectrum iterative algorithm to the scattering method for detecting the depth of surface scratches on optical elements. Finally, a scattered light field acquisition optical path is put forward to detect the depths of surface scratches on optical elements.

In the detection of surface scratch depths, the angular spectrum iterative algorithm and transport of intensity equation (TIE) are applied to the detection of surface scratch depths by scattering method. The scratch depths can be obtained from the reconstructed surface scratch phase distribution by the phase modulation characteristics of surface scratches. In the simulation section, the forward and reverse propagation relationship models between the optical element surface and the CMOS receiving surface are built by the angular spectrum transfer function. Based on this model, the scattered light field distributions of surface scratches with different shapes are obtained. Then, the angular spectrum iterative algorithm and TIE+angular spectrum iterative algorithm are adopted to reconstruct the scratch phases. The reconstruction process of the angular spectrum iterative algorithm is to select a random phase as the initial phase of the CMOS receiving surface and iterate repeatedly between the two surfaces. Additionally, the calculated value is replaced with the amplitude value of the initial simulated scattered light field intensity and the unit amplitude of the optical element surface until the defined error reaches the preset precision or the set maximum number of iterations. The phase distribution of scratches on the surface of optical elements can be obtained, and the depths of scratches can be calculated by the modulation characteristics of the surface scratches to the phase. TIE+angular spectrum iterative algorithm is similar to the reconstruction process of the angular spectrum iterative algorithm, which means that the initial random phase is replaced by the phase calculated by TIE. Finally, the effectiveness of the two reconstruction algorithms is evaluated from the strength error, correlation coefficient, and relative root mean square error. In the experimental section, the scattered light field acquisition device is built and the scattered light field distribution on the surface of the optical element is received by the CMOS detector. At the same time, the reconstructed scratch distribution on the surface of the optical element is reconstructed by the above two reconstruction algorithms, and then the surface scratch depth size is calculated. Finally, the reconstruction results of the two algorithms are compared with the detection results of the white light interferometry, and the relative errors of the two algorithms are calculated.

In the simulation section, scratch distribution and scattering field distribution of three different shapes, which are square scratch, triangular scratch, and oval scratch, are first simulated (Figs. 4 and 5). Then the scratch scattering field distribution is employed as the initial input of the angular spectrum iterative algorithm and TIE+angular spectrum iterative algorithm respectively to reconstruct the phase distribution of scratches on the surface of optical elements. The depth information of surface scratches is obtained based on the phase modulation characteristics of surface scratches (Figs. 6 and 7). Finally, we evaluate the effectiveness of the two algorithms from the strength error, correlation coefficient, and relative root mean square error. From the perspective of the intensity error evaluation, the number of iterations is set as 5000. The rising number of iterations leads to decreasing intensity error. Compared with the angular spectrum iterative algorithm, TIE+angular spectrum iterative algorithm has a smaller intensity error and faster convergence speed (Fig. 8). From the evaluation of the correlation coefficients, the correlation coefficients of both reconstruction algorithms are greater than 0.9 and the reconstructions are both highly correlated. However, the TIE+ angular spectrum iteration algorithm has a greater correlation coefficient and a higher degree of correlation compared to the angular spectrum iteration algorithm. From the evaluation of the relative root mean square error, the relative root mean square error of the TIE+angular spectrum iterative algorithm is 5.2%-5.3%, and that of the angular spectrum iterative algorithm is 5.8%-6.6%. The simulation results show that the scratch depth reconstructed by TIE+angular spectrum iterative algorithm is more accurate. In the experimental section, the scattered light field distribution of scratches on the surface of optical elements is collected experimentally, and the scratch depth on the surface of optical elements is reconstructed through the angular spectrum iterative algorithm and TIE+angular spectrum iterative algorithm (Fig. 12). Finally, the reconstructed results are compared with those of the white light interferometry, and the relative error range of the angular spectrum iterative algorithm is 1.35%-4.21%. The relative error range of the TIE+angular spectrum iterative algorithm is 0.90%-3.73%. The experimental results indicate that the scratch depth reconstructed by TIE+angular spectral iteration algorithm is more accurate.

In this paper, we apply the angular spectrum iteration algorithm and TIE+angular spectrum iteration algorithm to the surface scratch depth detection of optical elements by scattering method. During the experiment, only one image of the optical element surface light field distribution needs to be collected, which is employed as the initial input of two reconstruction algorithms to reconstruct the phase information of the scratch. Then the depth information of the scratch is calculated according to the modulation characteristics of the surface scratch to the phase. Compared with the angular spectrum iterative algorithm, TIE+angular spectrum iterative algorithm has a smaller scratch depth reconstruction error, faster convergence speed, higher reconstruction accuracy, and better reconstruction effect.