Objective Topography measurement is an indispensable part of mechanical processing, which has important guiding significance for processing quality evaluation and compensation. Topography measurement can be divided into on-machine measurement and offline measurement. Among them, offline measurement requires secondary clamping and repeated positioning, which will introduce machining errors and reduce the overall production efficiency and processing quality. However, on-machine measurement can avoid these problems. According to the measurement principle, on-machine measurement can be further divided into contact measurement and non-contact measurement. Contact measurement is not suitable for measuring optical precision structures due to the formation of scratches on the measurement surface and the destruction of microstructures. Therefore, how to realize accurate and efficient non- contact on-machine measurement has become an important research direction in the evaluation of optical components. Meanwhile, in optical design, the Fresnel structure has been widely used due to its smaller volume compared with traditional lenses. However, there is still a lack of effective on-machine non-contact evaluation methods for optical devices with Fresnel microstructures. The conventional chromatic confocal probe often used in on- machine measurement cannot effectively evaluate the Fresnel structure due to its small measurement angle and large focus spot. In this paper, an on-machine non-contact measurement system based on the point autofocus principle is established. With the developed temperature compensation and modified coordinate calibration algorithm, the accurate evaluation of Fresnel microstructures can be realized. Methods In traditional measurement, due to the inconsistency and uncertainty of the transformation matrix between the measurement coordinate and the workpiece coordinate, methods such as the iterative closest point (ICP) algorithm are often used to alternate the measured point cloud to coincide with the model point cloud. However, the ICP algorithm suffers from problems such as high computation complexity and sensitivity to initial values. Especially for a twodimensional contour measurement problem, the transformed point cloud cannot be guaranteed to pass through the generatrix, which will cause undesired errors during evaluation. To solve this problem, this paper analyzes the relative position between coordinate systems based on the proposed on-machine measurement equipment first (Fig. 5). Then, an optimization model is established based on the measurement data and the sphere constraint by scanning a calibrated sphere through the on- machine measurement system. By solving the optimization problem, the coincidence of the measurement coordinate system and the workpiece coordinate system can be realized. In addition, in order to obtain more accurate measurement values, a temperature-based compensation algorithm is developed in this paper. First, according to the frequency analysis of the measurement results for an optical plane (Fig. 4), the correlation between the measurement error and the temperature of the sensor is validated. Then, the Gaussian process is applied to establish the implicit mapping relationship among the temperature, temperature variation rate, and measurement error. Finally, the effectiveness of the system and algorithm proposed in this paper is verified by the measurement results for the optical plane (Fig. 6) and the spherical Fresnel structure (Fig. 9). Results and Discussions Through the measurement of an optical plane (Fig. 6), the temperature compensation algorithm proposed in this paper successfully reduces the measurement error by approximately 60%. Moreover, the spectrum analysis also verifies that the main peak of the spectrum due to the temperature variation no longer exists after compensation. The optimization method for pose calibration proposed in this paper reduces the deviation of calculation results under different optimization initial values to about 10% (Table 3 and Table 4), thus improving the accuracy of probe coordinate calibration. Finally, the on- machine measurement system proposed in this paper is comprehensively evaluated through the measurement of the spherical Fresnel structure. The deviation between the measurement result after compensation and optical design [Fig. 9(d)] is consistent with the offline measurement result [ Fig. 9( b)] in terms of both value and morphology, with a maximum detection error of 210 nm. Additionally, the result provided by the conventional chromic confocal sensor is presented (Fig. 8), and there are significant defects. The possible causes of the defects are explained from the perspectives of local topography and reflected spectral intensity. Through comparison, the point autofocus on-machine measurement system manifests significant advantages while measuring the complex Fresnel microstructure. Conclusions In this paper, a highly precise on- machine measurement system is established based on a point autofocus instrument and an ultra- precision machine tool. Additionally, a temperature compensation method based on the Gaussian process and an optimized coordinate calibration method are developed. Through spectrum analysis, a positive correlation between the probe reading and its temperature is validated. Furthermore, the Gaussian process model is conducted to reduce the error to 39% before compensation. Meanwhile, the improved optimization method in this paper further improves the accuracy of the probe pose calibration. When measuring the spherical Fresnel structure, the system developed in this paper reveals an excellent consistency with the results of the offline point autofocus measurement system, and the maximum deviation is about 210 nm, which is significantly better than those of the offline white light interferometer and the traditional online confocal sensor. In summary, the on- machine measurement system constructed in this paper provides a feasible solution for the ultra-precise on-machine non- contact measurement of complex and high-steep optical microstructures.
