# Wavefront Aberration Measurement Deflectometry for Imaging Lens Tests
## 4 Experiment  
Next, to verify the feasibility of WAMD, an experimental setup was built, as shown in Figure 6a. The lenses to be tested and a CCD camera (TXG12, Baumer, Switzerland) with an external pinhole (working as a projection center of the camera) were aligned on the optical axis. An LCD screen (E-2M21 GM, MTIPH, China)was installed perpendicular to the optical axis of the system. The resolution of the CCD camera was 1296 \(\times\) 966 pixels with a pixel size of 3.75 \(\upmu\)m, and the resolution of the screen was 1600 \(\times\) 1200 pixels with a pixel size of 0.2705 mm. In our experiment, the distance between the camera and the screen was 1005.8 mm. In the experiment, we calibrated the directions of the emergent rays (represented by \(IS_{0}\) in Figure 7) at first by recording the phase shifting patterns displayed on the screen without the LUT in the system. The phase was extracted with the phase shifting algorithm, [27] and then the intersection coordinates \((x_{S_{0}},y_{S_{0}})\) of the rays on the screen were obtained, as shown in Figure 7. Combining the pinhole position with the coordinates \((x_{S_{0}},y_{S_{0}})\), the ray directions \(\overrightarrow{p}\) was calibrated. Then, the intersection coordinates \((x_{F},y_{F})\) of the rays and the first surfaces of the LUT, shown in Figure 1b, were obtained by ray-tracing with the ray directions \(\overrightarrow{p}\) and the nominal structure parameters of LUT. After this, the lens to be tested was aligned into the system, and the phase shifting patterns captured by the camera were distorted due to the aberration of the LUT. Then, the coordinates \((x_{S_{2}},y_{S_{2}})\) on the screen were obtained by the phase shifting algorithm. The angu  
Figure 5: Comparisons between the actual wavefront aberrations and wavefront aberrations measured by WAMD of: (**a**) lenses with different aperture diameters (radius of curvature: 500 mm; thickness: 8 mm; refractive index: 1.5168); (**b**) lenses with different radius of curvatures (aperture diameter: 50 mm; thickness: 15 mm; refractive index: 1.5168). Figure 6: (**a**) WAMD system in the experiment; (**b**) interferometry test system in the experiment. can be calculated by Equation (2) with the two sets of coordinates \((x_{F},y_{F})\), \((x_{5_{2}},y_{5_{2}})\) and the distance \(D\). Finally, the wavefront aberration can be determined by the modal method. The single plano-convex lens and the telephoto lens mentioned in the simulation were tested in the experiment under the same circumstances, and the results measured by WAMD were compared with those measured by an interferometer (DynaFiz UHR1200, Zygo, Middlefield, CT, USA). Since the defocus aberration is greatly affected by the distance between the interferometer and the focal point of the LUT, the first four terms of the Zernike polynomials were removed in the results. For the single plano-convex lens (Edmund Optics #63-483; the parameters are shown in Figure 2a), the result is shown in Figure 8. The results measured by WAMD and the interferometer appear close to each other, and spherical aberration was measured with reasonable accuracy. The RMS of the difference reached 58.9 nm (4.74 %). The difference between them is mainly constituted by coma and astigmatism, which is caused by the inaccuracy of the alignment tools used, resulting in a tilt and off-axis of the LUT. The telephoto lens tested in the experiment consisted of two single lenses (#63-561, Edmund Optics, US; GL13-040-080, Golden Way Scientific, Beijing, China) and had the structural parameters shown in Figure 2b without aberration optimization. The result for the telephoto lens is shown in Figure 9 in which we can observe that the wavefront aberration measured by WAMD was nearly the same as that measured by the interferometer. The RMS of the difference was 166.8 nm (5.71 %). The spherical aberration was measured accurately, and the fifth, seventh, and eighth terms were measured effectively. The result verified the feasibility of this method.