# Expansion of Image Space in Enhanced-NA Fresnel Holographic Display
## 2 Holographic Image Reconstructed from Enhanced-NA Phase Fresnel Hologram
### Analysis of Angular Field of View of Holographic Image Dependent on Hologram Numerical Aperture
The above-measured viewing angles are consistent with the calculated values. We have also confirmed the viewing angle variation dependent on the hologram numerical aperture in optical experiments [9]. Meanwhile, a method including a double Fourier transform, such as a convolutional algorithm or angular spectrum method, cannot enhance the angular view because spatial filtering at the Fourier plane within the formula is embedded, resizing the hologram pattern according to the distance [10]. Here, one preserves the constant NA irrespective of the distance. For clarity, we defined a hologram that maintains a high numerical aperture as the enhanced-NA Fresnel hologram [13]. Figure 1: Flow chart of the modified Gerchberg–Saxton algorithm to calculate the phase-only hologram. The letters C, A and I indicate complex field, amplitude component, and the identity value, respectively. In previous work, we found that an aliased error arises when synthesizing the enhanced-NA Fresnel hologram because of the undersampling of the point spread function \(h(x,y)\) in the front of a Fourier transform, \(FT\), in the expanded form of the Fresnel diffraction formula, which is as follows [13; 19]:  
\[h(x,y)FT\big{\{}o\big{(}x^{\prime},y^{\prime}\big{)}h\big{(}x^{\prime},y^{\prime }\big{)}\big{\}}. \tag{3}\]  
Since the sampling criterion with respect to the object plane is well satisfied, the spreading angle of the diffractive wave from the object \(o(x^{\prime},y^{\prime})\) covers the whole area of the digital hologram, which induces the suppression of aliased replica fringes when the object has a finite size. We explained that this phenomenon results from the near-field diffraction property within the diffraction zone, by object pixel resolution [9]. This behavior brings about an energy concentration in a low-frequency region in the digital hologram, which weakens the diffraction ability by an incident plane wave. Figure 3 illustrates the diffractive wave from the reconstructed image by a directly calculated digital hologram. We know that the image with original resolution should be restored digitally  
Figure 2: Change in the viewing angle of a holographic image reconstructed from the enhanced-NA Fresnel hologram. (**a**) Schematic diagram of the enhanced-NA Fresnel hologram imaging system. Intensity profiles of the diffractive wave from the focused image, reconstructed from the phase holograms synthesized at (**b**) a critical distance, \(z_{\rm c}\) and (**c**) a half of \(z_{\rm c}\). Lateral line images in the inset picture are displayed with distance. A spreading angle of a propagating wave is larger than the diffraction angle by a hologram pixel, when the focused image is placed at a distance lower than a critical distance. or optically because the high-frequency components of the hologram exist. However, the simulated light intensity from the finite-sized image does not spread out with distance, where it is difficult to observe a holographic image with a sufficient angular view. On the other hand, in a logarithmic scale, the spreading angle of a diffractive wave is clearly distinguished [10], which still ensures the validity of Equation (1). An effective method for mitigating the energy concentration is to add a random phase to the object in a hologram synthesis. In this case, as depicted in Figure 3c, the intensity profile of the diffractive wave progresses at an adequate spreading angle.