# Research on Nanometer Precision Measurement Method of High Order Even Aspheres
## 1 Introduction  
Optical aspheres have the incomparable superiority of flats and spheres in improving image quality, correcting system aberration, and reducing system size and weight [1]. They have been widely used in large optical systems under extreme usage conditions, such as space telescopes [2], extreme ultraviolet lithography (EUVL) [3], and inertial confinement fusion (ICF) [4], to meet the functional requirements of the devices. Due to the mature development of ultra-precision machining technologies such as Magnetorheological Finishing (MRF) [5, 6] and Ion Beam Figuring (IBF) [7, 8, 9], the surface shape accuracy of optical aspheres has been able to reach the nanometer scale. For example, the surface shape accuracy of projection optics for extreme ultraviolet lithography has been better than 0.1 nm [10]. Measurement results guide the deterministic optical processing, which means that machining accuracy is directly limited by that of measurement. Therefore, a precondition for realizing a breakthrough in the accuracy of surface shape is to improve the accuracy of measurement. A number of methods have been used to achieve the high-accuracy measurements of optical aspheres, such as Sub-aperture Stitching Interferometry (SSI) and interferometry with compensators [11, 12, 13, 14, 15, 16, 17]. Sub-aperture Stitching Interferometry has enabled the high-precision measurements of non-rotationally symmetric optical freeforms and guided processing, resulting in the convergence of the error of freeform surface shape from 76 to 11 nm RMS [11]. Annular Sub-aperture Stitching Interferometry (ASSI) has been validated on steep aspheric and the measurement accuracy can reach the nanometer scale [12]. Usinga computer-generated hologram (CGH) as a null compensator can realize a null test of optical off-axis aspheres with high accuracy [13; 14]. The mid-spatial frequency error of large aperture long-focal-length lenses can also be accomplished with high accuracy with the help of CGH and the RMS of the measurement results has reached the sub-nanometer level [15]. A novel SSI method based on monoscopic deflectometry has achieved the measurement of large aperture optical aspheres by rotating the measured mirror, and the system error has been calibrated [16]. The combination of SSI and the null compensator has completed the high-precision measurement of 320 mm large aperture optical asphere, realizing the null test of SSI [17]. Therefore, the null test is the preferred method to achieve the high-precision measurement of optical aspheres. It is necessary to use absolute testing to separate the system error of interferometry when the accuracy of the test surface is comparable to that of the transmission flat (TF) or transmission sphere (TS). The two-sphere method, the three-flat test, the shift-rotation method, the N-position method, etc. are the commonly used absolute testing methods that are able to separate the system error to break through the measurement limits [18; 19; 20; 21; 22]. For example, eight rotation positions can be measured to separate the non-rotationally symmetric system errors of the lateral shearing interferometer [18]. The combination of the three-flat test and the N-position method has made it possible to measure large aperture optical flats up to 800 mm with nano-precision [20]. A new absolute testing method combining the even and odd function method and the three-flat test method is proposed, and with the help of a dedicated rotating device, in situ absolute testing can be realized on an interferometer with an aperture of 600 mm [21]. The combination of the N-position method and the random sphere method can separate the interferometer system error and the CGH error one by one, realizing the absolute testing of an optical asphere with an aperture of 300 mm, and the deformation due to fixture mounting and gravity is also taken into account [22]. Consequently, absolute testing is the key to pushing the limits of measurement accuracy.