# Optimal Control of Passive Cascaded Liquid Crystal Polarization Gratings
## 2 Theory
### Diffraction Characteristics of Quadrature Cascaded Liquid Crystal Polarization Grating  
A multi-layer LCPG combination can accomplish a wide range of deflection by cascading several LCPG layers together in a sequence. Each LCPG layer can achieve a fixed angle of deflection. By varying the phase of the adjustable liquid crystal half-wave, it is possible to deflect a \"LCPG layer\" made up of two identical passive LCPGs and two adjustable liquid crystal half-wave elements in series, as shown in Figure 3.  
Cascaded LCPGs combine multiple LCPG layers in series. Each LCPG layer can achieve a fixed angle of deflection. Multilayer LCPGs can achieve deflection in a large angle range. The combination forms include binary, quasi-binary, and ternary. The most common combination form is binary. The binary combination method for large angle deflection determines the resolution for the first layer, and the diffraction angle of each layer is twice that of the previous layer (for example: 1.25\({}^{\circ}\), 2.5\({}^{\circ}\), 5.0\({}^{\circ}\), 10.0\({}^{\circ}\), 20.0\({}^{\circ}\)), The total angle is an approximate integer multiple of the resolution. For an N-layer cascaded LCPG, 2N+1 angles can be controlled theoretically. In order to achieve 1.25\({}^{\circ}\) resolution and \(\pm\)40\({}^{\circ}\) angle deflection control, a total of five layers of LCPG (2 \(\times\) 40\({}^{\circ}\)/1.25\({}^{\circ}\) = 64, 2N+1 = 64, N = 5) are required. In order to achieve two-dimensional beam deflection, the same five-layer LCPGs are required in the orthogonal direction. The best series connection mode is: pitch 1.25\({}^{\circ}\), azimuth 1.25\({}^{\circ}\), pitch 2.5\({}^{\circ}\), azimuth 2.5\({}^{\circ}\)  
The control parameters include the layer control coefficient and slice control coefficient. The layer control coefficient mainly indicates that the deflection angle of a layer is 0\({}^{\circ}\), positive or negative, and the values are 0, \(-\)1, or +1; The plate control coefficient mainly indicates that the deflection angle of a liquid crystal adjustable half-wave plate in the layer is positive and negative, and the values are \(-\)0.5 and +0.5. The middle layer control coefficient of an LCPG layer is the sum of two slice control coefficients. For example, to achieve beam deflection with an azimuth angle of 22.5\({}^{\circ}\) and an elevation angle of \(-\)16.25\({}^{\circ}\), the layer control coefficient and plate control coefficient of the adjustable liquid crystal half-wave plate in each layer of LCPG in the azimuth direction and elevation are shown in Table 1, Table 2 is the Orthogonal cascaded liquid crystal polarization gratings structural cloth. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Direction**} & \multirow{2}{*}{**Controlparameters**} & \multirow{2}{*}{**Floor 1**} & \multirow{2}{*}{**Floor 2**} & \multirow{2}{*}{**Floor 3**} & \multirow{2}{*}{**Floor 4**} & \multirow{2}{*}{**Floor 5**} \\  & & & (1.25\({}^{\circ}\)) & & (2.5\({}^{\circ}\)) & & (5.0\({}^{\circ}\)) & & (10.0\({}^{\circ}\)) & & (20.0\({}^{\circ}\)) \\ \hline \multirow{2}{*}{Totalpitch 22.5\({}^{\circ}\)} & Floorcontrolcoefficient & 0 & \multicolumn{2}{c}{\(1\)} & \multicolumn{2}{c}{\(0\)} & \multicolumn{2}{c}{\(0\)} & \multicolumn{2}{c}{\(1\)} \\ \cline{2-10}  & \multicolumn{2}{c}{Slicecontrol coefficient} & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & \(-\)0.5 & 0.5 & \(-\)0.5 & 0.5 \\ \hline Azimuthsummation & Floorcontrol coefficient & \multicolumn{2}{c}{\(-1\)} & \multicolumn{2}{c}{\(0\)} & \multicolumn{2}{c}{\(-1\)} & \multicolumn{2}{c}{\(-1\)} & \multicolumn{2}{c}{\(0\)} \\ \cline{2-10} \(-\)16.25\({}^{\circ}\) & Slicecontrol coefficient & \multicolumn{2}{c}{\(-0.5\)} & \multicolumn{2}{c}{\(-0.5\)} & \multicolumn{2}{c}{\(0.5\)} & \multicolumn{2}{c}{\(-0.5\)} & \multicolumn{2}{c}{\(-0.5\)} & \multicolumn{2}{c}{\(-0.5\)} & \multicolumn{2}{c}{\(-0.5\)} & \multicolumn{2}{c}{\(0.5\)} \\ \hline \hline \end{tabular}
\end{table}