# Optimal Control of Passive Cascaded Liquid Crystal Polarization Gratings
## 3 Optimal Control Method of the Orthogonal Cascaded LCPG  
Assume that the total angles (\(M\Delta\theta\), \(N\Delta\theta\)), M and N are decimal deflection multiples of pitch and azimuth directions of orthogonal cascaded polarization gratings, where \(\Delta\theta\) is the angular resolution. The main control steps are as follows:  
Step 1: binarize the angle to be achieved, and calculate the layer control coefficients required for each layer of the cascaded liquid crystal polarization gratings with different deflection angles. Let \(M\) and \(N\) be converted into binary deflection multiples to obtain the weights of different bits in the meridian direction: \(n_{1}\), \(n_{3}\), \(n_{5}\), \(n_{7}\), \(n_{9}\) and the weights of different bits in the sagittal direction: \(n_{2}\), \(n_{4}\), \(n_{6}\), \(n_{8}\), \(n_{10}\). Relationship between \(n_{1}\), \(n_{2}\), \(n_{3}\), \(n_{4}\), \(n_{5}\), \(n_{6}\), \(n_{7}\), \(n_{8}\), \(n_{9}\), \(n_{10}\) and \(M\), \(N\):  
\[\left\{ \begin{array}{l}M=n_{1}+2n_{3}+4n_{5}+8n_{7}+16n_{9}\\ N=n_{2}+2n_{4}+4n_{6}+8n_{8}+16n_{10}\end{array} \right. \tag{18}\]  
Step 2: calculate the plate control coefficient of the liquid crystal adjustable half-wave plate. The plate control coefficient of the liquid crystal adjustable half-wave plate cannot be given randomly, as shown in the following figure. With the liquid crystal polarization grating layer, the incidence angle of the first liquid crystal polarization grating is \(\omega\) (\(\omega\) is  
Figure 5: (a) Schematic diagram of the internal structure of LCVR under oblique incident light. (b) Schematic diagram of the refraction phenomenon that occurs when incident light is obliquely incident LCVR. positive), and the incident angle of the second liquid crystal polarization grating can be \((\omega+\theta,\beta)\). It can also be \((\omega-\theta,\beta)\). The diffraction efficiency of the liquid crystal polarization grating is different at different incidence angles, and the diffraction efficiency is higher at small incidence angles. Therefore, when the control coefficient of the liquid crystal adjustable half-wave plate in front of the first liquid crystal polarization grating deflects the first liquid crystal polarization grating, \(-\theta\). In this way, the diffraction efficiency is better. Figure 6 is the Optimization control coefficient rendering. Two control coefficients \(n_{i1}\) and \(n_{i2}\) need to be determined for each layer of passively cascaded liquid crystal polarization gratings. \(n_{i1}\), \(n_{i2}\) as defined as follows:  
\[n_{i1}+n_{i2}=n_{i} \tag{19}\]  
For the first and second layers, since the incidence angle is \(0^{\circ}\) and the layer control is \(0^{\circ}\), the control coefficient of the first liquid crystal adjustable half-wave plate can be set to be positive, that is, \(n_{11}=0.5\), \(n_{12}=-0.5\), \(n_{21}=0.5\), and \(n_{22}=-0.5\). For layers with \(i\geq 3\) and \(n_{i}=0\), when the total angle is positive, \(n_{i1}=-0.5\), \(n_{i2}=0.5\). When the total angle is negative, \(n_{i1}=0.5\), \(n_{i2}=0.5\). Calculate the incidence angle of each liquid crystal adjustable half-wave plate \((\alpha_{ij},\beta_{ij})\), where \(i\) represents the serial number of the layer, 1-10 in the example, and j represents the serial number of two liquid crystal adjustable half-wave plates in the LCPG layer, 1 or 2:  
\[\left\{