# Resolution Enhancement of Spherical Wave-Based Holographic Stereogram with Large Depth Range +
## 4 Reducing Pixel Size \(\Delta x_{1}\) with Aperture Filtering  
According to Equation (5), reducing pixel size \(\Delta x_{1}\) will reduce the value of the first term, or enhance the resolution at the CDP. However, the increase of the second term may decrease the depth range, similar to reducing \(L\) in the above analysis. Instead of choosing  
Figure 5: (**a**) A bee model and a school badge built in the 3 ds Max modeling software. The numerical reconstruction with capturing depth (**b**) \(L=80\) mm, (**c**) \(L=45\) mm, and (**d**) \(L=30\) mm. Figure 6: The optical reconstruction results with capturing depth (**a**) \(L=80\) mm, (**b**) \(L=45\) mm, and (**c**) \(L=30\) mm. a proper value of \(\Delta x_{1}\), we propose to use an aperture filtering technique to enhance the resolution more effectively. Figure 7a shows a typical case in which the pixel size \(\Delta x_{1}\) of EI equals the pixel size \(\Delta x_{2}\) of hogel. In Figure 7b, \(\Delta x_{1}\) is reduced and according to Equation (2), the hogel size \(p=N\Delta x_{2}\) is increased accordingly. Since the hogel size is larger than EI, the adjacent hogels will have overlap. It can be seen that the focused image spot at the CDP is decreased, making a better reconstruction quality near the CDP. However, the increased hogel size will cause wider light rays, making larger defocused spots away from the CDP. As a result, the depth range is decreased. To address this issue, an aperture filter is introduced to limit the width of the light ray in Figure 7c. The calculated complex amplitude distribution of each hogel is multiplied with a rectangle function \(\text{Rect}(u/p_{0})\cdot\text{Rect}(v/p_{0})\), where \(p_{0}\) is the pitch of the EI. Then each filtered hogel can be spliced seamlessly. The aperture filter reduces the width of light rays, making better reconstruction qualities within the total depth range. Figure 8a-c show three groups of numerical simulation results corresponding to the three cases in Figure 7. In Figure 8a, the parameters are set as \(p=0.3\) mm, \(f=3\) mm, \(\Delta x_{1}=\Delta x_{2}=5\) um, \(L=80\) mm. The captured EIA consists of 60 \(\times\) 60 EIs, each with 60 \(\times\) 60 pixels. The CDP is set at the center of the bee model, and the distance between the school badge and CDP is set as \(\Delta z=40\) mm. In Figure 8b, \(\Delta x_{1}\) is reduced to 2.5 um and each EI contains 120 \(\times\) 120 pixels. The hogel size \(p\) is increased to 0.6 mm, so that adjacent hogels will have overlap. As a result, Figure 8b shows better reconstruction quality at the CDP (see the foreleg of the bee), but with degraded reconstruction quality of the school badge. It just verifies the above analysis that the increased hogel size will cause larger defocused spots away from the CDP. In Figure 8c, each transformed hogel is first multiplied with an aperture filter to reduce its size to 0.3 mm; then, all the filtered hogels are spliced seamlessly to form the final hologram. It shows the best reconstruction quality at the marginal depth (the school badge), and the reconstruction quality at the CDP is better than that in Figure 8a. A slight degradation at the CDP can be observed compared with Figure 8b. That is because the numerical filtering causes limited aperture diffraction. Since a spherical wave phase is multiplied with each hogel, the diffraction pattern at the CDP can be thought of as the Fraunhofer diffraction of the rectangular hogel. The zero-order diffraction contains the most energy, with width of \(2\lambda L/p\). The zero-order diffraction width determines the upper limit of the resolution. The filtering process reduces the hogel  
Figure 7: (a) A typical case in which the pixel size \(\Delta x_{1}\) of EI equals to the pixel size \(\Delta x_{2}\) of hogel. (b) Reduced \(\Delta x_{1}\) causes enlarged hogel size. (c) The proposed aperture filtering technique. size \(p\), resulting in a degraded diffraction spot.