# Expansion of Image Space in Enhanced-NA Fresnel Holographic Display
## 1 Introduction  
A holographic display offers a three-dimensional image, in such a way that the wavefront of the propagating wave from the digital hologram is generated in free space [1]. Various attempts to implement holographic displays have been conducted because it has the ultimate merit, in that an observer can view a complete scene without visual fatigue [2; 3; 4]. Despite technological advancements, there still remains a fundamental limitation for realizing a commercial holographic display, owing to the finite space-bandwidth product of a pixelated spatial light modulator (SLM). In particular, the issue of solving a narrow viewing angle is a crucial task. It is known that an SLM with a pixel pitch of the wavelength scale is necessary for seeing a holographic image at a wide angular view. A commercial modulator has a pixel pitch of several micrometers, inducing a viewing angle of a few degrees. Experiments to overcome this problem have been conducted by expanding the diffraction zone, using spatial or temporal multiplexing of modulators [3; 5; 6; 7; 8]. In this circumstance, massive data processing is required, and it would be inefficient, in view of energy consumption in informatics. These conventional approaches for expanding the viewing zone are based on the assumption that the diffraction angle of a pixel pitch directly relates to the angular view of the holographic image. Previously, we found that the viewing angle \(\Omega\) of the reconstructed image is fundamentally determined by the numerical aperture (NA) of the digital hologram, rather than a diffraction angle, as demonstrated by the following equation [9; 10]:  
\[\Omega=2\sin^{-1}(\text{NA}). \tag{1}\]  
The hologram numerical aperture is represented as a geometrical structure of aperture size, \(l=N\Delta x\), and a distance \(z\), \(\text{NA}=\frac{N\Delta x}{2z}\), where \(N\) and \(\Delta x\) denote the number of pixels and pixel size of the digital hologram, respectively. The viewing angle does notindependently depend on the diffraction ability of the pixel pitch or the illuminating wave. This concept provides a major technological approach in expanding the viewing angle in the holographic display system. In principle, the viewing angle varies with the reconstructed distance, even when using an SLM with a constant pixel pitch. The viewing angle of the image would be larger than a diffraction angle when the image is formed below the critical distance. We called this system the enhanced-NA Fresnel holographic display, where there is still a decrease in image size because of the Nyquist sampling criterion in hologram synthesis [9, 11, 12, 13]. However, this behavior is different from the trade-off relation, characterized by the space-bandwidth product of the digital hologram itself. For the realization of a wide viewing angle holographic display, a holographic image, without sacrificing the image size, should be reconstructed. When the image reconstruction is numerically or optically processed on the region outside of the sampling condition, the interference of high-order images is inevitable. Therefore, a detailed study on the expansion of image space and the effect of this on the changes in the viewing angle is required. A phase modulation has some advantages, such as showing no twin image and high diffraction efficiency [14, 15, 16]. A phase retrieval algorithm, such as the iterative Fresnel transform, is used to calculate the phase hologram in the Fresnel diffraction regime. The enhanced-NA digital hologram creates replica fringe patterns, which fundamentally obstruct the image reconstruction in an extended space, beyond the bounds of a diffraction scope of a hologram pixel [9]. In this study, we explore the image reconstruction in the extended image space from the phase Fresnel hologram with a high numerical aperture. Firstly, the phase hologram was successfully prepared using the iterative algorithm, based on the Fresnel propagation.