# Expansion of Image Space in Enhanced-NA Fresnel Holographic Display
## 3 Optical Reconstruction from Enhanced-NA Fresnel Hologram without Sacrificing Image Size  
We carried out optical experiments by using a phase-only SLM (Holoeye PLUTO) with 1920 \(\times\) 1080 pixels of an 8-\(\upmu\)m pixel pitch and a blue laser with a wavelength of 473 nm. The critical distance for a pixel specification in a vertical direction was set as a reference to secure the imaging area with no aliased errors, which was calculated to be 146.1 mm. The phase holograms were synthesized using cameramen images located at distances of a  
Figure 5: Properties of image reconstruction in the extended image space from the enhanced-NA Fresnel hologram. A digital hologram is made using the letter image placed at a distance of half of \(z_{c}\). (**a**) High-order images are generated along the direction cosine surface at an angular interval, \(\theta_{m}\). (**b**) Numerically reconstructed images showing up to the second order are cropped for convenience. half, a quarter, and one eighth of \(z_{c}\), as shown in Figure 6a. It is hard to directly use the fast Fourier transform algorithm in the hologram synthesis, using the object space outside of the diffraction scope, because the pixel specifications of both planes deviate from the sampling condition in Equation (2). The digital hologram was prepared in two separate steps. First, the high-resolution holograms used to suit the extended object space in accordance with the sampling condition were calculated using the modified GS algorithm; thereafter, they were downsampled to the final resolution of the hologram. To prepare a hologram made at half of \(z_{c}\), 73.1 mm, the initial phase hologram with 3840 \(\times\) 2160 pixels of a 4-\(\upmu\)m pixel pitch was calculated using the object with the same pixel specifications. The final hologram in Figure 6b is the downsampled data by twice the pixel interval. We conveniently captured the image within the SLM size, by considering a direct-beam filtering by a polarizer. Figure 6c shows the reconstructed image with no replica noises. We observed that the other holograms reconstructed the original images well, without suffering from the high-order noises. All the images have a rectangular form with an 8.64 mm \(\times\) 8.64 mm lateral size, according to the sampling condition. Although it is not displayed here, when the viewing direction is moved to the high-order region, the  
Figure 6: Optical reconstruction from the digital hologram made using the extended object space. (**a**) Phase holograms were prepared at distances of a half, a quarter, and one eighth of \(z_{c}\). (**b**) Synthesized digital hologram at half of \(z_{c}\), (**c**) reconstructed image, and (**d**) unfocused image at 140 mm. (**e**) Reconstructed image at quarter \(z_{c}\). corresponding replica image reappears after the primary image is gone, which is consistent with the numerical analysis. The high-order images that overlapped in the image space are not viewable because of an inclined viewing direction. The image space is expanded, but the viewing angle is confined to a diffraction angle. Figure 6d is the unfocused image of the image in Figure 6c. Even though an image blurring occurs, an accommodation effect is very weak. This indicates that the diffractive wave from the image spot propagates at a narrow spreading angle, where a perspective view of the image is hardly visible to the naked eye. The energy compaction of the hologram patterns invokes this weakness, as explained in Section 2. In the second experiment, a sparse letter object was used and the down and up sampling processes were embedded within the iterative processes. Furthermore, to increase the optimization performance, the adequate background value of 37%, in comparison to the letter pixel value, was added to the object space by considering the suppression effect of the high-order noises when using the relatively dense object. A letter image is fully filled in the vacant space, whose size is equal to the hologram magnitude in vertical direction.