# 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 Fresnel field, which propagates from the digital hologram to the image plane, is described by the Fresnel diffraction formula. The specifications of both planes are determined by the Nyquist-Shannon sampling theorem [10, 11, 12]. The relationship between the pixel pitches \((\Delta x,\Delta y)\) and \((\Delta x^{\prime},\Delta y^{\prime})\) in the hologram \((x,y)\) and image \((x^{\prime},y^{\prime})\) coordinates is represented as follows:  
\[\Delta x=\frac{\lambda z}{N\Delta x^{\prime}},\ \Delta y=\frac{\lambda z}{N \Delta y^{\prime}}. \tag{2}\]  
For convenience, a one-dimensional description is considered hereafter. The sampling condition of both planes is characterized by the critical distance having a value, \(z_{c}=N\Delta x^{2}/\lambda\), when the pixel pitches of both planes are the same. The image size, \(l^{\prime}=N\Delta x^{\prime}\), is smaller than the hologram size, \(l=N\Delta x\), when the image is placed at a distance less than \(z_{c}\). We know that the aliased errors appear if they are digitally or optically processed in the regions that deviate from this specification. A practicable phase hologram \(\phi\) for the real-valued object is extracted from the modified Gerchberg-Saxton (GS) iterative algorithm [14, 17, 18]. Figure 1 is a flow chart of the modified GS algorithm to obtain the phase-only hologram. The Fresnel transform, consisting of a single Fourier transform, is used in iterative procedures. Only the phase term is captured after the Fresnel transform, where the constraint in the spectral domain is a uniform modulus. The inverse Fresnel transform of the exponential term \(e^{i\phi}\) is carried out to modulate the phase hologram. To repeat this process, the amplitude of the reconstructed image is replaced by the object value. The algorithm rapidly converges to an optimized solution. The phase holograms, composed of 1920 \(\times\) 1920 pixels with an 8-\(\upmu\)m pixel pitch, are synthesized using a letter image, as depicted in Figure 2a. Here, the critical distance, considering a wavelength of \(\lambda=473\) nm, is calculated to be 259.8 mm. To measure the viewing angle of a holographic image, the propagating wave far away from the reconstructed image was simulated. Figure 2b shows the propagating waves in the simulation, using the digital hologram made at a critical distance. The intensity profiles of the diffractive wave, indicated by a lateral line in the inset picture, are displayed with distance. An observer can view the image within the spreading angle of the diffractive wave. From this, the viewing angle of the reconstructed image is estimated to be 4.2\({}^{\circ}\). Figure 2c is the simulation result for a digital hologram made at a half of the critical distance. The propagating wave from the image largely spreads, where the viewing angle appears to be about 7.7\({}^{\circ}\). As illustrated in Figure 2a, a holographic image is reconstructed by illuminating the digital hologram with a plane wave. Considering a point object, it appears that the point image is focused on a particular place through the digital hologram acting as a lens. From Equation (2), we can deduce that the lateral size and viewing angle \(\Omega\) of the reconstructed image have a trade-off relation, which is unlike the interpretation found by controlling the size of the hologram itself. The viewing angle is larger than the diffraction angle by a hologram pixel, \(\theta=2\sin^{-1}\left(\frac{\lambda}{2\Delta x}\right)\), when the image is formed at a distance less than the critical distance. The high-definition image diffracts to a larger spreading angle. We know that the angle value \(\Omega\) is pertinent to the hologram numerical aperture, \(\text{NA}=n\sin(\Omega/2)\), where the refractive index \(n\) is equal to one in free space. Hence, the viewing angle of the reconstructed image obeys the relation in Equation (1).