# Expansion of Image Space in Enhanced-NA Fresnel Holographic Display
## 4 Viewing Angle Enlargement of Holographic Image and Discussions  
As previously stated in Section 2, the replica fringes are formed in the enhanced-NA Fresnel hologram because of the undersampling of the point spread function, where the finite-sized object could suppress this phenomenon from the near-field diffraction property. However, it rather produces the adverse effect on the fringe concentration. To enhance the diffraction performance of a digital hologram by an incident wave, the energy compaction should be mitigated. A simple method is to diffuse an aggregated hologram pattern by adding the random phase to the real-valued object in the hologram synthesis, where the generation of the replica fringes is inevitable. In this case, it is fundamentally impossible to remove the high-order noises in the image space. We found that, as displayed in Figure 7a, the iterative algorithm including the down and up sampling processes effectively prevents the formation of the replica fringes. The hologram pattern about the reversely propagating wave seems to be created to compensate the corresponding high-order noises during the repetition processes. This is different from a spatial filtering appearing in the algorithms with the double Fourier transforms. However, as seen in Figure 8, the down and up sampling processes still weaken the viewing angle expansion of a holographic image. The Rayleigh-Sommerfeld formula using a definite integral can arbitrarily control the pixel specifications in both planes. Therefore, there is not a requirement for an unnecessary upsampling process within the iterative algorithm. As described in Section 2.2, the viewing angle dependent on the hologram numerical aperture would be conserved. In this circumstance, to obtain the optimized hologram, both the high-order images as well as the replica fringes in the digital hologram should be simultaneously removed during the repetition processes. Furthermore, there is still a drawback to consuming too much time in the iterative procedure. Figure 9 shows the experimental results using the Riemann integral during the repetition processes. The phase hologram composed of 512 \(\times\) 512 pixels was calculated using the letter object with a 4-\(\upmu\)m pixel pitch located at a half of \(z_{c}\). The fringe compaction was not completely eliminated, and there seemed to remain some replica fringes, as shown in Figure 9a. The inset in Figure 9b shows the magnified light intensity propagating from the reconstructed image. The diffracting performance of the focused image underwent no deterioration. The synthesized hologram reconstructs the image with an accommodation effect, but some high-order noises appear still, as shown in Figure 9b. The viewing angle  
Figure 8: Diffraction behavior from the focused image. Digital hologram is prepared using a circular object adding a random phase. All images are numerically reconstructed at a distance, 78.1 mm. (**a**) The diffractive wave from the circular object image spreads out well, whereas (**b**) the upsampled digital hologram by a simple duplication pixel deteriorates the diffracting behavior. is estimated to be about the first-order diffraction angle of 6.7\({}^{\circ}\) because the overlapped image is viewable. Here, the whole area of the digital hologram contributes to the optical image formation, which fulfills the viewing angle variation, dependent on the hologram numerical aperture. It is important to develop a proper algorithm to reconstruct a holographic image that maintains an angular view, and to increase image fidelity. The high angular spectrum of a diffractive wave in both hologram and image planes should survive to form a highly focused spot. Presently, it is known that optimization algorithms, such as the stochastic gradient descent, demonstrate a considerable improvement in image fidelity. There have been several studies using an image captured by a camera or a deep learning algorithm [24; 25]. In addition, a non-iterative algorithm has been studied [26]. These will become the proper methods.