# Wavefront Aberration Measurement Deflectometry for Imaging Lens Tests
## 1 Introduction  
Lenses play an important role in imaging systems. The performance of an imaging lens is affected by its aberration, which is introduced by both the manufacturing tolerance of the lenses and lens alignment. Therefore, a feasible method for measuring the wavefront aberrations of imaging lenses is important. A few methods have been proposed to test the aberrations of imaging lenses. The existing methods mainly include the Foucault knife-edge test, the Ronchi test, the star test, the interferometry test, and the Hartmann-Shack test, etc. The Foucault knife-edge test uses a knife edge to block out one part of the rays and analyzes the shadow that appears over the aberrated region. Although some algorithms have been proposed for quantitative measurement, the Foucault knife-edge test is still mainly used for qualitative analysis [1; 2; 3]. The Ronchi test measures the aberrations of imaging lenses with the interference patterns produced by a grating. However, the accuracy is limited when the light source does not have good monochromaticity, and strict alignment is needed for the grating [1; 4; 5]. The star test analyzes the aberrations by examining the images of a point source formed by the lenses under test (LUT) and compares them with the ideal image form, which depends heavily on experience [1]. The interferometry test is a widely accepted method that obtains the wavefront aberration by measuring the optical path difference (OPD) [6; 7; 8]. The interferometry test can achieve very high accuracy. However, it has limitations in its dynamic range and is easily affected by the environment. When the aberrations are too large, the dense interferometric fringe patterns cannot be analyzed correctly. In addition, the cost of an interferometer is high, which makes it difficult to be widely used. The Hartmann-Shack test is a popular method that uses lenslet arrays to sample the wavefront and obtain the slope data [9; 10]. Then, the slope data are used to reconstruct the wavefront by integration. However, the resolution of it is limited by the number of lenslet arrays. In this paper, to overcome the difficulties mentioned above, we propose wavefront aberration measurement deflectometry (WAMD), which is based on the theory of phase measuring deflectometry (PMD). As an effective method for measuring the surface shapes of specular optical elements, PMD has received wide attention [11; 12]. PMD has the advantages of high accuracy, low costs, a large dynamic range, and so on, and it can also be used to measure the transmitted wavefronts of transparent optical elements. Some methods have been proposed to obtain the transmitted wavefronts of specific optical elements, which is very helpful for further studies of transmitted metrology [13; 14; 15]. In recent years, [PERSON] et al. presented a technique by which to test the wavefront aberration of a phase object based on the transmitted fringe deflectometry [16]. In [PERSON]'s method, the propagation directions of light rays before and after deflection by a phase object under test are obtained by moving the screen. Then, the wavefront aberrations of the reference wavefront and deformed wavefront are obtained. The difference between the two wavefronts is the wavefront aberration caused by the phase object. Their method requires a complicated calibration and measurement process (a time-consuming iteration process is required, and the screen needs to be moved several times), which makes it difficult to be applied to online measurement. [PERSON] et al. proposed a computer-aided calibration method [17; 18]. They used the wavefront difference between the ideal system model and the actual test system to achieve a simultaneous multi-surface measurement of freeform refractive optics and were able to remove the alignment error, as well as the lens manufacturing tolerance, by optimization. However, what they finally obtained was only the transmitted wavefront, rather than the real wavefront aberration in geometrical optics.