## 2 Optimal Design and Numerical Simulation  
Figure 1 shows diagrams of the wide- and narrow-view modes of the proposed LCD system and structure of ELC diffusing unit. Each diffusing unit of the ELC diffuser is an ELC lens that can electrically control the direction of light. The LC molecules of the ELC diffusing unit are homogeneously aligned parallel to the polarizer direction of the LCD panel so that maximum retardation occurs in the field-off state. If no voltage is applied to the diffusing unit, the diffusing unit transmits collimated light from the QCBL without a wavefront change, as illustrated in Figure 1a. Thus, a light distribution appears with a narrow viewing angle. However, if an appropriate voltage is applied to the diffusing unit, the polarized light that passes through the LC panel undergoes different retardations because of the different spatial refractive indexes of the liquid crystals, as shown in Figure 1b. Accordingly, the wavefront of the light becomes spherical. This spherical wave forms a focus and then generates a wide light distribution in the viewing zone. To form a spherical wave with a wide light distribution, the retardation needs to monotonically decrease from the center of the diffusing unit to both ends. In the case of liquid crystal with positive birefringence, the effective refractive index decreases if the tilt angle of the optic axis increases, which decreases the retardation at the corresponding position. Therefore, driving electrodes are installed at both ends of the diffusing unit as shown in Figure 1c, and an appropriate voltage is applied. As a result, LC molecules can be arranged in the vertical direction from the glass surface or in the direction where the tilt angle increases. The retardation distribution can be precisely controlled when the retardation distribution of the ELC diffusing unit is controlled and the unit has multiple driving electrodes. However, in this study, a single driving electrode was arranged at each end to reduce the thickness of the diffusing unit, as well as to ensure structural simplicity and convenience of the production process. Since the length of the lenticular-like ELC diffusing unit coincides with the vertical length of the LC panel, it does not affect the vertical viewing angle of the display and only changes the horizontal viewing angle. Before the ELC diffusing unit was designed, the ideal retardation distribution of the diffusing unit (the Lambertian distribution) was calculated. This distribution is defined as a criterion for the retardation distribution of the ELC diffusing unit. The ideal shape of a diffusing unit with a Lambertian distribution was designed using the following equation, which is based on ray tracing and energy conservative mapping [21; 22]:  
\[\iint T(r,\ z)E(r,\ z)\ ds=\iint I(\theta,\ \varphi)\ d\Omega, \tag{1}\]  
where \(r\) is the distance from the center of the diffusing unit (the position in the horizontal direction), and \(z\) is the height from the bottom of the unit (the position in the vertical direction). \(T(r,z)\) and \(E(r,\ z)\) are the transmittance and the illuminance of the incident light at \((r,z)\) of the diffusing unit, respectively, and \(I(\theta,\ \varphi)\) is the intensity of the emitted light at tilt angle \(\theta\) and azimuth angle \(\varphi\). In the case of Lambertian emission, \(I(\theta,\ \varphi)\) can be expressed as follows:  
\[I(\theta,\varphi)=I_{0}cos\theta, \tag{2}\]  
where \(I_{0}\) is the peak intensity. The height \(z\) for any position \(r\) in the ideal diffusing unit can be obtained from Equation (1). Accordingly, the retardation distribution of the ideal unit is obtained by:  
\[\Gamma_{ideal}(r)=(n_{diffuser}-n_{i})z(r), \tag{3}\]  
where \(\Gamma_{ideal}(r)\) is the retardation according to the position \(r\) of the ideal diffusing unit, \(n_{diffuser}\) and \(n_{i}\) are the refractive indexes of the diffusing unit material and the incident area, respectively, and \(z(r)\) is the thickness of the diffusing unit according to the position \(r\).