## 2 Materials and Methods
### Designing an EO Beam Steerer Based on Polycrystalline PMN-PT  
The light steerer designed in this study was based on the EO effect in a 7.5 mm \(\times\) 7.5 mm \(\times\) 40 mm PMN-PT ceramic, which is considered isotropic, as discussed before. The propagation of an optical wave in a crystal can be described in terms of the relative impermeability tensor as \(B_{\bar{y}}=1/n_{\bar{y}}{}^{2}=\epsilon_{0}/\epsilon_{\bar{y}}\), where \(\epsilon_{0}\) is the permittivity of free space and \(\epsilon_{\bar{y}}\) is the permittivity tensor. The \(B_{\bar{y}}\) is a symmetric tensor because the \(\epsilon_{\bar{y}}\) is a symmetric \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **LiNbO\({}_{3}\) (x-cut)** & **LiNbO\({}_{3}\) (z-cut)** \\ \hline Material Capacitance (pF) & 113.20 & 48.55 \\ Open-Cell Capacitance (pF) & 1.35 & 1.63 \\ Calculated Dielectric Constant & 83.73 & 29.75 \\ Accepted Dielectric Constant & 85.2 & 28.7 \\ Percent Error (\%) & 1.73 & 3.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Measured capacitance values and resulting dielectric constants calculations for multiple orientations of lithium niobate (LiNbO\({}_{3}\)) single crystals. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **LiNbO\({}_{3}\) (x-cut)** & **LiNbO\({}_{3}\) (z-cut)** \\ \hline Area (cm\({}^{2}\)) & 1.6445 & 1.0 \\ Thickness (mm) & 1.0 & 0.5 \\ Calculated Capacitance (pF) & 124.0 & 50.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Expectation values for the capacitance measurements of x-cut and z-cut lithium niobate (LiNbO\({}_{3}\)) single crystal samples. tensor for a lossless and inactive medium. The two directions of dielectric polarization and the refractive indices can be calculated by indicatrix or index ellipsoid as [1]  
\[B_{ij}x_{i}x_{j}=1 \tag{20}\]  
where \(i\) and \(j\) are 1, 2, and 3 and represent the principal axes of a crystal which are the directions in the crystal where the electric field (E) and displacement vectors (D) are parallel. If we consider \(x,y,\) and \(z\) respectively for the subscripts of 1, 2, and 3, the ellipsoid index can be expanded as follows:  
\[\frac{x^{2}}{n_{x}^{2}}+\frac{y^{2}}{n_{y}^{2}}+\frac{z^{2}}{n_{z}^{2}}=1 \tag{21}\]  
where \(n_{x}\), \(n_{y}\), and \(n_{z}\) are the refractive indices for the incident light polarized in the directions of \(x,y,\) and \(z\), respectively. The quantum theory of solids says the indicatrix depends on the charge distribution in the crystal. The external electric field (\(E\)) and stress (\(\sigma\)) can change that distribution and result in a change in the shape, size, and orientation of the index ellipsoid specified by electro-optic and photoelastic effects on \(\Delta B_{ij}\) as follows [34]:  
\[B_{ij}(E)-B_{ij}(0)=\Delta B_{ij}=r_{ijk}E_{k}+s_{ijkl}E_{k}E_{l}+\pi_{ijkl} \sigma_{kl} \tag{22}\]  
where \(r_{ijk}\) and \(s_{ijkl}\) are the [PERSON] electro-optic and [PERSON] electro-optic tensors, and \(\pi_{ijkl}\) is the piezo-optical tensor. Note that the stress field is related to the strain tensor (\(T_{rs}\)) as per [PERSON]'s law [34];  
\[\sigma_{kl}=c_{klrs}T_{rs} \tag{23}\]  
and  
\[\pi_{ijkl}=p_{ijrs}\Omega_{rslk} \tag{24}\]  
where \(p_{ijrs}\) and \(\Omega_{rslk}\) are elasto-optical tensor and compliances, respectively. Although the direct photoelastic effect vanishes when no direct stress is applied to the crystal (free crystal), mechanical constraints still can exist because of the converse piezoelectric effect. That is because the electric field will cause a strain by the converse piezoelectric effect, and therefore, the indicatrix will get modified by the photoelastic effect, which is called the secondary effect. The strain produced by an electric field is calculated as \(d_{kj}\)\(E_{k}\), where \(d_{kj}\) is the piezoelectric tensor.