# Expansion of Image Space in Enhanced-NA Fresnel Holographic Display
## 2 Holographic Image Reconstructed from Enhanced-NA Phase Fresnel Hologram
### Expansion of Image Space and Its Reconstruction Property Depending on Viewing Angle  
For the realization of a viable holographic display based on the enhanced-NA Fresnel hologram, a sufficient amount of image space should be secured. Figure 4 shows the simulation results for the generation of high-order images in the image space. The digital hologram with an 8-\(\upmu\)m pixel pitch was conveniently calculated using a rectangular object consisting of 256 \(\times\) 256 pixels, and then, the pixels were upsampled tenfold. The pixelated structure plays a role in planar gratings, and the modulated replica images were generated at an interval of \(\lambda z/p\) in the Fresnel regime, where \(p\) indicates a pixel interval [10]. In this specification, it is expected that the Fraunhofer pattern starts to appear at a submillimeter distance. The digital hologram made at a distance, \(2z_{c}\), of 69.3 mm reveals the enveloped images by a sinc function. However, there is no intensity modulation in the reconstructed image from the enhanced-NA Fresnel hologram, although the image is placed at a millimeter distance. Here, the bright areas of all the images are comparable to hologram size on the same scale. The image intensity has been known to be uniform, irrespective of its lateral location [20,21]. This makes it possible to reconstruct a uniform image in the extended image space larger than the diffraction scope by a hologram pixel. Figure 3: Propagating waves of the focused image reconstructed from the directly calculated complex holograms. Intensity profiles of the diffractive wave in (**a**) a linear scale and (**b**) a logarithmic scale, which are simulated using the hologram synthesized at a half of \(z_{c}\). (**c**) Diffractive behavior of the image reconstructed from the hologram with no energy compaction. Figure 5 displays the schematics of image reconstruction in the extended image space. We consider the hologram made using the letter object placed at a distance of half of \(z_{c}\), 17.3 mm. The object has 512 \(\times\) 512 pixels with a 4-\(\mu\)m pixel pitch, which is 2 times larger than the diffraction scope by a hologram pixel pitch. The angular separation of diffracted orders from the planar gratings is well described in the direction cosine space [22; 23]. Each diffraction vector carries the high-order image in the new direction. As depicted in Figure 5a, the first-order images are created in the corresponding areas outside of the diffraction scope. The viewing direction of the high-order images is tilted at a diffraction angle \(\theta_{m}\), which is shown in the following equation:  
\[\sin\theta_{m}=\frac{m\lambda}{p}. \tag{4}\]  
Figure 5b is the numerically reconstructed image showing up to the second-order term. The Rayleigh-Sommerfeld formula was used to preserve non-paraxial diffraction. The high-order images are unfocused because the image plane vertical to the optical axis is slightly apart from the direction cosine surface. The image blur is larger at the higher order terms. Figure 4: Simulation results for the generation of high-order images in the enhanced-NA Fresnel diffraction regime. A rectangular object consisting of 256 \(\times\) 256 pixels has a boundary filled with zeros to distinguish high-order terms. The bright areas of all the images are of the same size as a digital hologram. We characterize the reconstruction behavior in two separate cases. The first case is that the viewing angle depends on the hologram numerical aperture, whose value is double a diffraction angle in this configuration. Here, the overlapped image with the high-order terms is observable. This type of high-order image would be removed through the iterative optimization algorithm, which will be discussed later. Secondly, the viewing angle could be restricted to a diffraction angle, due to the nature of algorithm. In this case, the complete image without the interference of high-order noises can be seen in the viewing direction of the zeroth order.