# Wavefront Aberration Measurement Deflectometry for Imaging Lens Tests
## 3 Numerical Simulations  
To verify the correctness and accuracy of our proposed method, numerical simulations were conducted. Based on ray-tracing in computers, the WAMD model for wavefront aberration measurement was established using numerical software, and a plano-convex lens and a telephoto lens were tested (the parameters of the lenses under test are shown in Figure 2). The length between the pinhole camera and the screen was set as 1100 mm, and the pinhole was located at the second focal point of the lenses under test. It was assumed that the emergent rays from the pinhole of the camera passed through the LUT. The coordinates \(\left(x_{F},y_{F},z_{F}\right)\) and \(\left(x_{S_{2}},y_{S_{2}},z_{S_{2}}\right)\) obtained by ray-tracing were taken into Equation (2). Then, the obtained angular aberrations were used to reconstruct the wavefront aberrations by the modal method [26], and the results are given below.  
For the plano-convex lens, the actual wavefront aberration with parallel light beams entering from the flat surface and leaving the convex surface is shown in Figure 3a. The wavefront aberration reconstructed by WAMD is shown in Figure 3b. The difference between Figure 3a,b is shown in Figure 3c, which shows that the difference was smaller than 1.23 % in RMS and 0.89 % in PV. Additionally, Figure 3d compares the Zernike coefficients of Figure 3a,b. The result shows that the error of WAMD is small. The simulation result for the telephoto lens is given in Figure 4, in which the actual aberration, the reconstructed aberration, and their difference are shown, respectively. It was observed that the difference was smaller than 6.32 % in RMS and 7.59 % in PV. The main error can be attributed to the wrong power. However, defocus was not considered as one of the monochromatic aberrations because it is closely related to the position of the image plane. Therefore, the correctness and accuracy of WAMD were verified.  
Figure 2: The parameters of imaging lenses in the simulation: (**a**) the parameters of the plano-convex lens; (**b**) the parameters of the telephoto lens.  
Without the loss of generality, lenses with different parameters were simulated to determine the feasibility of the proposed method. Symmetrical biconvex lenses were adopted, and their aperture diameters and radius of curvatures were changed, respectively, in the simulation. The aperture diameter increased from 10 mm to 100 mm, as shown in Figure 5a. The radius of curvature increased from 150 mm to 600 mm, as shown in Figure 5b. For all of these simulated lenses, WAMD performed well and the errors were all smaller than 1.93 % in RMS. The correctness and accuracy of WAMD were further verified.  
Figure 4: Wavefront aberration of the telephoto lens in the simulation: (**a**) actual wavefront aberration; (**b**) wavefront aberration measured by WAMD; (**c**) difference between (**a,b**); (**d**) comparison of Zernike coefficients.  
Figure 3: Wavefront aberration of the plano-convex lens in the simulation: (**a**) actual wavefront aberration; (**b**) wavefront aberration measured by WAMD; (**c**) difference between (**a,b**); (**d**) comparison of Zernike coefficients.