# Optimal Control of Passive Cascaded Liquid Crystal Polarization Gratings
## 2 Theory  
### Diffraction Characteristics of Monolithic Liquid Crystal Polarization Grating  
To achieve beam deflection, a liquid crystal polarization grating introduces a continuous gradient tilt phase distribution based on the geometric phase. A planar uniaxial birefringent liquid crystal molecule, which is the same as a grating, makes up the structure of a monolithic LCPG. The liquid crystal molecule vector varies periodically with its position [8], and its structure is depicted in Figure 1.  
The LC director of its two polarization states is as follows:  
\[n(y)=[\sin(y/\Lambda),\cos(y/\Lambda),0] \tag{1}\]  
wherein \(\Lambda\) is the grating period, and the azimuth of the liquid crystal molecular vector \(\varphi=\pi y/\Lambda\) linearly changes periodically along the y axis [9]. The ideal diffraction efficiency at positive incidence can be derived from the Jones matrix [10]:  
\[ \begin{array}{c}\eta_{0}=\cos^{2}(\frac{\pi\Lambda md}{\Lambda})\\ \eta_{\pm 1}=\frac{1\mp S_{3}^{\prime}}{2}\sin(\frac{\pi\Lambda md}{\Lambda}) \end{array}  \tag{2}\]  
where \(\eta_{m}\) is the m-class diffraction efficiency and \(S_{3}^{\prime}=S_{3}/S_{0}\) is the normalized Stokes parameter. From Equation (2) it can be seen that the liquid crystal polarization grating has only three diffraction stages: 0 and \(\pm 1\), with the intensity distribution between the diffraction stages depending on the phase delay \(\delta\) and incident polarization. When the birefringence phase retarder of the liquid crystal retarder is an odd multiple of \(\pi\), the zero-level diffraction will be zero and all light will be biased to \(\pm 1\) level. When the incident light is a right-handed circular polarized light, \(S_{3}^{\prime}=-1\), then the diffraction efficiency will be \(\eta_{+1}=1\) and \(\eta_{-1}=1\), all light passing through the LPG is diffracted to +1 level, as shown in Figure 2a. When the incident light is an L-circle-polarized light, \(S_{3}^{\prime}=+1\), all light is diffracted to \(-1\) level, as shown in Figure 2b. When the LCPG is at a high voltage, the incident light is transmitted along the axis (zero-level diffraction direction), as shown in Figure 2c [11]. It is theoretically possible to achieve a single stage with a diffraction efficiency of 100%, in fact; however, due to the influence of error, experimentally it is not. The diffraction effectiveness of the cascaded LCPGs is affected by two factors. First, the adjustable 1/2 wave plate's real polarization state change, as per the original 0\({}^{\circ}\) incidence design, is no longer 1/2 wavelength. Second, the diffraction efficiency of the polarization grating is decreased because of the increase in incidence angle. [PERSON] created a liquid crystal polarization grating with a similar structure to the double twist layer, which makes the liquid crystal polarization grating in the 20\({}^{\circ}\) incidence field of view of the first level diffraction efficiency of more than 90%, thereby improving the diffraction efficiency. In fact, after design optimization and process improvement, the diffraction efficiency of the 0\({}^{\circ}\) beam into the monolithic polarization grating can reach 97% [4]. The relation curve between diffraction efficiency and incidence angle is shown in  
Figure 1: Structure of a monolithic LCPG. The polarization state of the incident light has an impact on the diffraction efficiency of the LCPG, in addition to the liquid crystal material and structure. The liquid crystal variable retarder is usually used for polarization modulation of incident light, and the ideal liquid crystal variable retarder ideal phase modulation amount is  
\[\varphi(V_{\mathit{i}\mathit{i}\mathit{t}\mathit{e}}) =\mathrm{d}(n_{e}-n_{0})\frac{2\pi}{\lambda} \tag{3}\] \[=\pi\]  
When the angle of incidence is angled in the actual cascade situation, the incident adjustable half-beam wave angle is variable, and the following actual phase is produced:  
\[