Significance Asphere is a general term for surfaces deviating from a sphere. Different from spherical surfaces with similar curvature, the curvature of aspheric surfaces varies everywhere. Aspheric surfaces have higher degrees of freedom than spherical ones, allowing them to achieve more functions than spherical surfaces. In addition to correcting high-order aberrations and improving imaging quality, aspheric surfaces can reduce the sizes of optical systems by yielding effects that are only possible with multiple spherical mirrors. Employing aspheric surfaces can simultaneously improve image quality and reduce the volume of optical systems. As a result, optical designers are increasingly adopting aspheric surfaces in modern optical systems, such as biomedical, lithography, astronomical optics, and high-power laser systems. The measurement technique is vital in manufacturing aspheric surfaces. The measurement technique of the aspheric surface is mainly for the surface form and parameters, and both techniques can assess the aspheric surface quality. Interferometry is an efficient method widely applied in measuring aspheric surfaces in optical shops. Our paper reviews the interferometric measurement of optical aspheric surface form and parameter error, with attention to the research on partial compensation and digital Moire interferometry. Additionally, the future trend of interferometric measurement technology for optical aspheric surfaces is discussed. Progress According to whether the interferometer can obtain the null interferogram, the surface form measurement technique can be divided into two categories of null interferometry and non-null interferometry. Null interferometry includes the no-aberration point method and compensation method. The no-aberration point method is widely adopted to measure quadratic surfaces but cannot be utilized to measure high-order aspheric surfaces. The compensation method is implemented by an interferometric system with a compensator or computer-generated hologram (CGH) and has been widely applied to measure various aspheric surfaces. The non- null interferometry mainly includes the sub-aperture stitching interferometry, sub- Nyquist interferometry, two- wavelength phase shifting interferometry, shearing interferometry, tilted-wave interferometry, point diffraction interferometry, and partial compensation interferometry. The research group from the University of Arizona proposed the sub-aperture stitching interferometry, two-wavelength phase shifting interferometry, and sub-Nyquist interferometry in 1981, 1985, and 1987, respectively. These three methods can be employed to measure aspheric surfaces with small asphericity. The research group from the University of Stuttgart put forward the tilted-wave interferometry in 2007, which was applied to measure aspheric surface forms with large gradient variations ( Fig. 2). The tilted- wave interferometry introduces multiple off-axis point sources by a microlens array to generate multiple spherical waves with different inclinations to compensate the gradients in various local areas of the tested surface. Then the interferogram corresponding to each local region can satisfy the Nyquist sampling theorem, and the complete surface form is obtained by a phase retrieval algorithm. Point diffraction interferometry can achieve high accuracy and is applied in the measurement of the extreme ultraviolet lithography aspheric mirror. The research group from the Beijing Institute of Technology proposed partial compensation and digital Moire interferometry in 2003 ( Fig. 3). The partial compensator only compensates part of normal aberrations, and the surface form can be obtained when the interferogram satisfies the Nyquist sampling theorem. A partial compensator can measure multiple aspheric surfaces with various parameters, and it has good versatility and can be adopted to test the aspheric surfaces with a large aperture and asphericity. The interferometric method for aspheric surface parameter error establishes the relationship between the compensation distance and residual wavefront. The research group from Changchun Institute of Optics utilizes the null compensator to measure parameter error (Fig. 6). The research group from Zhejiang University obtains the vertex radius of curvature by axially moving the test surface (Fig. 7). The research group from Beijing Institute of Technology obtains the parameter error by analyzing the aberration at the best compensation position (Fig. 8). Conclusions and Prospects The measurement of aspheric surface form error and parameter error is crucial for ensuring the performance of advanced optical systems. In recent decades, intensive efforts have been made to the measurement technique of aspheric surfaces. We are delighted to see the booming development of this field, and many efficient approaches have been proposed for various test scenarios. Nevertheless, the development of advanced optics always poses new challenges. Complex boundary conditions would impose harsh requirements on optical shop testing. For example, the primary mirror of the European Extremely Large Telescope (E-ELT) is composed of 798 regular hexagonal aspheric mirrors with a diameter of 1. 4 m, and the manufacturing should be finished in seven years. Such a task poses extremely high requirements for the accuracy, cost, and efficiency of measurement techniques. The optical shop handles the testing tasks of various aspheric surfaces, and the requirement of testing cost means to ensure the versatility of the measurement techniques. Further progress will undoubtedly be associated with a test method that comprehensively considers high accuracy, high efficiency, low cost, and good versatility.
