## 2 Optimal Design and Numerical Simulation
However, in this case, the optical axes of the liquid crystals at the driving electrodes are not sufficiently arranged in the vertical direction, which increases the retardation to more than 0. Accordingly, the cell gap needs to be increased as much as the increase of the retardation at the electrodes to make the overall retardation distribution equivalent to the ideal retardation distribution. As a result, the retardation at the center of the unit is increased. However, the driving voltage and the response time of ELC diffusing unit increase if the cell gap becomes excessively large. To solve this problem, the optimal design of the ELC diffuser should consider both the driving voltage and the cell gap of the ELC diffuser. [PERSON]'s design of experiments method was used for the optimal design of factors that are influential on the optical characteristics of the ELC diffusing unit. This allows the number of simulations to be reduced effectively, and the correlation between the design factors of the ELC diffuser and retardation distribution can be obtained. Before applying the Taguchi method, the performance characteristics of the target diffuser need to be defined. The performance characteristics can be obtained by quantifying the proximity between the retardation distribution and the ideal retardation distribution. The retardation distribution can be expressed based on retardation values at each position and their rates of change. The performance characteristics are determined by applying appropriate weights to the differences of retardation and rates of change between the ideal diffuser and the designed diffuser at each position. As the performance characteristic becomes closer to 0, the design becomes more ideal,which is a smaller-the-better (STB) situation. [PERSON]'s signal-to-noise ratio (SNR) for the performance characteristics can be calculated as follows:  
\[\text{SNR}=-10\log\left(\frac{1}{n}\sum_{i=1}^{n}y_{i}^{2}\right), \tag{6}\]  
where  
\[y_{i}=w_{A}\;\Delta\Gamma_{i}\;+\;w_{B}\;\Delta\frac{d\Gamma_{i}}{dr}, \tag{7}\]  
The ELC diffusing unit, respectively and \(w_{A}\) and \(w_{B}\) are weighting factors, which have values of 1 because they are equally important. Moreover, the selected control factors were the width of the electrode, the voltage applied to the electrode, and the cell gap. These were selected because they influence the behavior and light transmittance of liquid crystal molecules. As presented in Table 1, three levels were set for each factor by considering the physical properties of materials, the response time of liquid crystal, and the convenience of the production process. Figure 2 shows the modeling results for analyzing the light transmittance of the ELC diffuser. The LC layer of the ELC diffusing unit was divided into grids. Each grid was 1 \(\upmu\)m in the r direction and 0.1 \(\upmu\)m in the z direction. The directions of each incident ray in the grid and the corresponding retardation can be obtained by using [PERSON]'s law and the effective refractive indices in the grids. The direction of light transmitted through the ELC diffusing unit was identified by using the wavefront obtained from the directions of light in each grid and the corresponding retardation. The effective refractive index of liquid crystal in each grid was calculated using Equation (5). The angle (\(\theta_{LC}\)) was numerically obtained by using a finite difference method (FDM), which is suitable for highly nonlinear equations with de Gennes' order tensor and the Ericksen-Leslie equation [19].