# Optimal Control of Passive Cascaded Liquid Crystal Polarization Gratings
## 2 Theory
### Vertical Incident Liquid Crystal Adjustable Phase Delayer Phase Delay Theory  
Figure 4 shows a schematic of the LCVR structure. The transparent electrode ITO is typically plated on the inside of two parallel quartz glass (S\({}_{1}\)O\({}_{2}\)) plates, followed by coating with a transparent material, PI. Finally, liquid crystal molecules were filled between the two parallel plates and enclosed in a bracket. When the drive voltage exceeds the critical voltage \(V_{C}\) of the LCVR, the LC molecules start to spin away from the plate, and the rotation angle depends on the drive voltage at both ends of the plate. In addition, the liquid crystal molecules do not rotate when the driving voltage is below the critical value \(V_{C}\), or when the rotation angle is 0. Consequently, the driving voltage that the liquid crystal controller applies to the two ends of the LCVR can be altered to alter the phase delay.  
The relationship between the rotation angle \(\varphi\) of the liquid crystal molecule and the drive voltage \(V\) can be expressed as follows [12]:  
\[\beta(V)=\left\{\begin{array}{c}q_{C};V\leq V_{C}\\ \frac{\pi}{2}-2\tan^{-1}\exp\left[-(\frac{V-V_{C}}{V_{0}})\right]+q_{C};V>V_{ C}\end{array}\right. \tag{7}\]  
where \(V_{o}\) is the voltage constant, \(q_{C}\) is the initial rotation angle, \(V\) and \(V_{c}\) are the drive and critical voltages of the liquid crystal adjustable phase delayed, respectively.  
When incident light is vertical, LCVR is frequently utilized. In this scenario, the phase delay of the light changes with the driving voltage and wavelength of the incident light, and the relationship between the two can be written as follows:  
\[\delta=\frac{2\pi\cdot d\cdot\left[n_{ef}(\lambda,\varphi)-n_{0}\right]}{\lambda} \tag{8}\]  
Here, \(n_{ef}(\lambda,\varphi)\) represents the effective refractive index, which is related to the dispersion characteristics of the liquid crystal adjustable phase delayed and the liquid crystal driving voltage, \(n_{0}\) is the refractive index of o light, d is the thickness of the liquid crystal  
Figure 4: Schematic diagram of a typical liquid crystal adjustable phase delayed.  
layer, when the light is incident, the thickness d is equal to the optical path, and \(\lambda\) is the wavelength of the incident light.  
\(n_{\textit{eff}}(\lambda,\varphi)\) is also related to the e-light refractive index \(n_{e}\), o-light refractive index \(n_{0}\), and tilt angle \(\varphi\) of liquid crystal molecules. The relationship between them can be expressed as follows [13; 14]:  
\[\frac{1}{n_{\textit{eff}}^{2}(\lambda,\varphi)}=\frac{\cos^{2}\varphi}{n_{e}^{ 2}}+\frac{\sin^{2}\varphi}{n_{0}^{2}} \tag{9}\]  
From vertical Equations (8) and (9), the phase delay LCVR quantity can be expressed as follows:  
\[\delta=\frac{2\pi\cdot d}{\lambda}\cdot\frac{n_{0}n_{e}}{\sqrt{n_{e}^{2}+(n_{ 0}^{2}-n_{e}^{2})\cos^{2}\varphi}} \tag{10}\]  
The amount of delay \(\delta\) can be approximated as follows:  
\[\delta=\frac{2\pi\cdot d}{\lambda}\cdot(n_{e}-n_{0})\cos^{2}\varphi \tag{11}\]  
\(\Delta n(\lambda)\) is the birefringence of the liquid crystal layer, \(\Delta n(\lambda)=n_{e}(\lambda)-n_{0}(\lambda)\), which is wavelength dependent and can be represented by the dispersion equation; its simplified form can be expressed as follows [15]:  
\[\Delta n(\lambda)=a+\frac{b}{\lambda^{2}}+\frac{c}{\lambda^{4}} \tag{12}\]  
where \(a\), \(b\), and \(c\) represent the regression coefficients of the dispersion equation, and \(n_{e}\) and \(n_{0}\) are related to wavelength. Therefore, \(\delta^{\prime}\) can be expressed as follows:  
\[\begin{array}{l}\delta^{\prime}=\frac{2\pi\cdot d}{\lambda}\cdot\Delta n( \lambda)\cdot\cos^{2}\varphi\\ =\frac{2\pi\cdot d}{\lambda}\cdot(a+\frac{b}{\lambda^{2}}+\frac{c}{\lambda^{4 }})\cdot\cos^{2}\varphi\end{array} \tag{13}\]