## 3 Simulations and Experiments
### Numerical Simulations
#### 3.1.1 Verification of PC Method  
The purpose of the first set of simulations is to prove the correctness of the proposed PC method by generating the SCGHs by PC or PS method and reconstructing them by PS or PC, respectively. The object planes are square images with \(128\times 128\) pixels. The flare angle of WRP is \(7.5^{\circ}\), and the distance \(d\) between WRP and SCGH is 345.5 mm. The results including PSNR(dB) are shown in Figure 4.  
The relatively high values of PSNR verify the bidirectionality of the proposed PC method, which means it can be used to calculate both diffracted and reconstructed optical field between a plane and a sphere. There is almost no difference visually between the results reconstructed by PC and PS methods, and the actual difference between them is extremely small, according to the little disparity of PSNR between Figure 4b,d,f and Figure 4h,j,l. However, there is no denying that the difference does exist when the PS and PC methods are used separately in synthesis and reconstruction. The PS method is regarded as accurate in order to further discuss this issue. When taking it in the generation process as shown in Figure 4g-l, the diffracted optical fields from the object plane are recorded precisely, while the images are approximate reconstructed when the other method is used during reconstruction; whereas, the images in Figure 4a-f are reconstructed using the opposite method. Due to the asymmetry of these two processes, the error of the PC methods introduced in generation and reconstruction cannot be eliminated. In the second set of simulation, object planes of different resolution are compared by PS and PC methods. The reconstructed results with different resolution are shown in Figure 5 and Table 1. In addition to PSNR, the SSIM is used simultaneously as a quality measurement in this simulation. With the resolution of the object plane increasing, the PSNR and SSIM of the results reconstructed by both methods decline because each pixel introduces error in the process of applying the PC method. However, the clearly reconstructed images in Figure 5 prove that the proposed method works well with various of resolution. Figure 4: Verification of the proposed method. Panels (**a**,**c**,**e**) are diffracted field by PC method; panels (**b**,**d**,**f**) are reconstructed field by point source (PS) method; panels (**g**,**i**,**k**) are diffracted field by PS method; panels (**h**,**j**,**l**) are reconstructed field by PC method. Figure 3: Original object images of the simulations (**a**–**c**) and experiments (**d**–**f**). According to the analysis in Section 2.4, the error introduced by the PC method will be increasing as the curvature of SCGH increases, and the PSNR of the reconstruction results should decrease dramatically. In order to verify this point of view, we conducted the third set of simulation comparing the quality of reconstructed images of SCGHs with different curvatures. In this experiment, we tested the flare angles of WRP ranged from 180 to 7.5\({}^{\circ}\) as they determine different curvatures of the SCGH segments. For simplicity, the graphic results present only four main angles, and the rest refer to the PSNR curve, as shown in Figure 6 (considering the computation time, the resolution of the object plane is set to \(64\times 64\)). According to the results of graphs and PSNR, the quality of the reconstructed image is better with the decreasing on the flare angle of WRP, which means that the error of PC method is greater with wider range of curvature of SCGH. Furthermore, the error reaches the maximum when the flare angle is 180\({}^{\circ}\). This result coincides with our analysis above. However, for SCGHs with relatively large curvatures, the images are still of good quality despite degradation. It proves the feasibility of the proposed PC method used in SCGHs with wide range of curvatures.