## 2 Optimal Design and Numerical Simulation
For the ELC diffusing unit, the retardation according to the position is:  
\[\Gamma_{ELC}(r)=[n_{eff}-n_{o}]d_{LC} \tag{4}\]  
with  
\[n_{eff}=\sqrt{\frac{n_{o}^{2}n_{e}^{2}}{n_{e}^{2}\cos^{2}\theta_{LC}+n_{o}^{2} \sin^{2}\theta_{LC}}}, \tag{5}\]  
where \(\Gamma_{ELC}(r)\) is the retardation according to the position \(r\) of the ELC diffusing unit, \(n_{eff}\), \(n_{o}\), and \(n_{e}\) are the effective, the ordinary, and the extraordinary refractive indices of liquid crystal, respectively,  
Figure 1: Diagrams of the liquid crystal display (LCD) system equipped with an electric-field-driven liquid crystal (ELC) diffusing unit: (**a**) Narrow-view mode when the ELC diffusing unit is off and (**b**) wide-view mode when the ELC diffusing unit is on. (**c**) Schematic diagram of the ELC diffusing unit. The liquid crystal (LC) molecules have different behaviors depending on the voltages applied between the driving electrode with the width of \(w\) and the common electrode of the bottom glass. d\({}_{\rm LC}\) is the thickness of the LC layer, and \(\theta_{\rm LC}\) is the angle between the optical axis of the liquid crystal and the direction of light. In the case of an isotropic diffuser, the retardation according to the position depends on the thickness z(r), as shown in Equation (3). In the case of the ELC diffusing unit, the retardation can be controlled by changing the direction of optical axis of liquid crystal, \(\theta_{\rm LC}\), which affects the effective refractive index n\({}_{\rm eff}\), as expressed by Equation (5). The direction of the optical axis of the liquid crystal can be controlled by the structure of the electrodes at both ends of the diffusing unit and the electric field distribution generated by a voltage applied to them. Accordingly, the ELC diffusing unit can be optimally designed if the electrode structure and voltage are determined so that the retardation distribution \(\Gamma_{\rm ELC}(\rm r)\) according to the position of the ELC diffusing unit in Equation (4) can be equivalent to the ideal retardation distribution \(\Gamma_{\rm ideal}(\rm r)\) of Equation (3). A wide viewing angle of the Lambertian distribution can be obtained if the optimized ELC diffusing is subject to an appropriate voltage. The LC panel of the proposed LCD system uses IPS mode. Its resolution is \(2960\times 1440\), and its pixel size is \(37\times 37\)\(\mu\)m. The diffusing unit pitch of the ELC diffuser was set to be equal to the pixel. The pitch of the LC panel minimizes the crosstalk which leads to the overlap of the images between the adjacent pixels. The diffusing unit uses a nematic liquid crystal, which has a birefringence of 0.42 at a wavelength of 550 nm, elastic constants of K\({}_{11}~{}=~{}12.5\) and K\({}_{33}~{}=~{}32.1\), and dielectric anisotropy of 7.6 [23]. A maximum thickness for the maximum retardation of 15,500 nm was determined at the center of the ideal diffusing unit by applying a diffusing unit size of 37 \(\mu\)m and a refractive index of 1.5 to Equations (1)-(3). Accordingly, the cell gap needs to be at least 36.91 \(\mu\)m for the birefringence of liquid crystal of 0.42 from Equation (4) to obtain the retardation of 15,500 nm in the ELC diffusing unit. In addition, the optical axis of the LC molecules needs to align in the vertical direction from the glass surface at both ends so that the retardation can become 0 to enable a 15,500 nm difference in the retardation at the center and both ends of the unit. Therefore, it is necessary to infinitely increase the applied voltage, which generates a strong vertical electric field at the electrodes and across the entire ELC diffusing unit. Consequently, the optical axes of the LC molecules are excessively arranged in the vertical direction from the glass surface such that the retardation distribution of the diffusing unit deviates from the ideal retardation distribution. The vertical component of the electric field across the entire diffusing unit can be reduced if the applied voltage is decreased.