## 2 Materials and Methods
### Designing an EO Beam Steerer Based on Polycrystalline PMN-PT
Therefore, the primary and secondary electro-optic effects on a free crystal (\(\sigma=0\)) can be calculated as  
\[B_{ij}(E)-B_{ij}(0)=\Delta B_{ij}=r_{ijk}E_{k}+s_{ijkl}E_{k}E_{l}+\left(pd_{ kj}\right)E_{k} \tag{25}\]  
Because the relative impermeability tensor is symmetric for inactive and lossless medium, the \(i\) and \(j\) subscripts can be permuted. This will reduce the number of independent elements of \(r_{ijk}\) from 27 to 18 and from 81 to 36 for \(s_{ijkl}\). The same thing is applied for piezoelectric tensor except for some factor of 2. Therefore, the change in indicatrix can be written for convenience as follows [34]:  
\[B_{ij}(E)-B_{ij}(0)=\Delta B_{ij}=r_{ij}E_{j}+s_{ijk}E_{j}E_{k}+(pd_{ki})E_{k} \tag{26}\]  
where  
\[p_{mm}=\pi_{mr}c_{rn}(\text{m},\text{n}=1,2,\dots,6) \tag{27}\]  
\[\pi_{mr}=\pi_{ijkl}\text{ (n}=1,2,3) \tag{28}\]  
\[B_{ij}x_{i}x_{j}=1 \tag{20}\]0  
As discussed, although each individual grain of PMN-PT ceramics is not centrosymmetric, the polycrystalline nature makes the PMN-PT ceramic isotropic and, thus, polarization independent. Therefore, the \(r_{ij}\) and \(d_{ki}\) in Equation (26) are zero, and the Kerr effect is the main factor in altering the refractive index. By contracting the indices as 11:1, 22:2, 33:3, 23:4, 13:5, and 12:6, the impermeability tensor becomes  
\[B=\left( \begin{array}{ccc}B_{1}&B_{6}&B_{5}\\ B_{6}&B_{2}&B_{4}\\ B_{5}&B_{4}&B_{3}\end{array} \right) \tag{30}\]  
The Kerr effect tensor for an isotropic medium is given as [1]  
\[s=\left( \begin{array}{cccccc}s_{11}&s_{12}&s_{12}&0&0&0\\ s_{12}&s_{11}&s_{12}&0&0&0\\ s_{12}&s_{12}&s_{11}&0&0&0\\ 0&0&0&0.5(s_{11}-s_{12})&0&0\\ 0&0&0&0&0.5(s_{11}-s_{12})&0\\ 0&0&0&0&0.5(s_{11}-s_{12})\end{array} \right) \tag{31}\]  
When a polycrystalline PMN-PT gets under a slowly varying external electric field, its molecules get aligned with the field, and the isotropic ceramic becomes birefringent, behaving like a uniaxial crystal in which the electrical field defines the optical axis. The change in impermeability tensor by applying an electric field on the PMN-PT ceramic will be  
\[\left( \begin{array}{c}\Delta B_{1}\\ \Delta B_{2}\\ \Delta B_{3}\\ \Delta B_{4}\\ \Delta B_{5}\\ \Delta B_{6}\end{array} \right)=\left( \begin{array}{cccccc}s_{11}&s_{12}&s_{ 12}&0&0&0\\ s_{12}&s_{11}&s_{12}&0&0&0\\ s_{12}&s_{12}&s_{11}&0&0&0\\ 0&0&0&0.5(s_{11}-s_{12})&0&0\\ 0&0&0&0&0.5(s_{11}-s_{12})\end{array} \right)\left( \begin{array}{c}E_{x}^{2} \\ E_{y}^{2}\\ E_{z}^{2}\\ 2E_{y}E_{z}\\ 2E_{x}E_{y}\end{array} \right) \tag{32}\]  
Assuming the external electric field is in the direction of the z-axis (\(E=E_{z}\)\(\hat{z}\)), the indicatrix of PMN-PT ceramic will be defined as  
\[x^{2}\bigg{(}\frac{1}{n^{2}}+s_{12}E^{2}\bigg{)}+y^{2}\bigg{(}\frac{1}{n^{2}} +s_{12}E^{2}\bigg{)}+z^{2}\bigg{(}\frac{1}{n^{2}}+s_{11}E^{2}\bigg{)}=1 \tag{33}\]  
\[\frac{x^{2}+y^{2}}{n_{0}^{2}}+\frac{z^{2}}{n_{e}^{2}}=1 \tag{34}\]  
where \(n_{0}\) is the refractive index associated with ordinary light, and \(n_{e}\) is the refractive index seen by an extraordinary light propagating through the PMN-PT. Hence, if the light is polarized perpendicular to the external electric field direction, the index change it sees will be  
\[\Delta n=n_{o}-n=-0.5n^{3}s_{12}E^{2} \tag{35}\]  
The EO scanner designed in this study had the light propagate perpendicular to the external electric field direction and was polarized parallel to the optical axis defined by that field.