## 2 Materials and Methods
### Designing an EO Beam Steerer Based on Polycrystalline PMN-PT
Therefore, the index change seen by the light can be described as  
\[\Delta n=n_{e}-n=-0.5n^{3}s_{11}E^{2} \tag{36}\]  
The birefringence (\(\Delta n\)) imposes relative phase retardation (\(\Gamma\)) to the light propagating through that medium by the length of \(L\) as  
\[\Gamma=\frac{2\pi\Delta nL}{\lambda}=-\frac{\pi n^{3}s_{11}E^{2}L}{\lambda} \tag{37}\]therefore, the electric field that causes \(\pi\) radians of phase retardation will be  
\[E_{\pi}=\sqrt{\frac{\lambda}{n^{3}s_{11}L}} \tag{38}\]  
The external electric field (\(E\)) is approximated as the applied voltage divided by the thickness of the PMN-PT ceramic (\(E\) = V/d), which was 7.5 mm (d = 7.5 mm) thick. The \(s_{11}\) is the Kerr coefficient of polycrystalline PMN-PT, which varies by wavelength and temperature. This coefficient can be defined by measuring the EO steerer's deflection angle associated with each wavelength at different temperatures. The polycrystalline PMN-PT is isotropic and polarization-independent. Unlike single ferroelectric crystals, no crystalline orientation adjustment and no domain-engineered multi-prisms are required to provide the OPD for linearly polarized light propagating through the crystal. The most simple structure of an EO steerer we can build with the PMN-PT ceramic is a triangular prism with uniform electrodes deposited on two opposite sides, as seen in Figure 11.  
Before applying the voltage on the electrodes, according to [PERSON]'s law, we have  
\[\theta+A=\sin^{-1}(nSinA) \tag{39}\]  
After applying the voltage, the isotropic medium becomes uniaxial. The light is polarized along the external electric field. Therefore the deflection angle change will be  
\[\Delta\theta=\frac{\Delta nsinA}{\sqrt{1-n^{2}sin^{2}A}} \tag{40}\]  
where \(\Delta n\) is given by Equation (36). To prevent a total internal reflection, \(\left.\right|nSinA\left.\right|\) must be smaller than 1. Instead of having a triangular prism with uniform electrodes, a rectangular prism with triangular electrodes can be considered. The non-uniform electrodes should be deposited on the two opposite sides of the rectangular PMN-PT ceramic (7.5 mm \(\times\) 7.5 mm \(\times\) 40 mm) to impose a non-uniform electric field distribution and index change to defect the light. Figure 12 depicts the arbitrary electrodes on the rectangular PMN-PT ceramic. Figure 13 shows the refraction of the light at the dielectric interfaces. Because the electric field should be nonuniformly distributed to steer the light, the number of electrode interfaces (\(N\)) should be equal to or larger than 1 (\(N\geq 1\)). Suppose the number of interfaces (\(N\)) was small and the PMN-PT ceramic length was long. In that case, the incident angle of an interface might get wide enough to enable a total internal reflection, at least at some voltages applied on the electrode. Therefore, the number of interfaces and the electrode geometry should be designed to avoid the total internal reflections and partial back-reflections. Figure 11: A triangular prism-shaped EO scanner. (W is the width, L is the length, and H is the height of the triangular prism). When the light hits the electrode interface, the region with a lower refractive index, no total internal reflection should occur. Therefore, we should have  
\[\frac{x^{2}+y^{2}}{n_{0}^{2}}+\frac{z^{2}}{n_{e}^{2}}=1 \tag{34}\]0  
Once a ray is reflected back from an electrode interface, it is considered a loss because even if it is partially reflected back again, it will not get steered to the desired angle. Therefore, the total partial back-reflections through the medium (TPBR) should be minimized. \[\frac{x^{2}+y^{2}}{n_{0}^{2}}+\frac{z^{2}}{n_{e}^{2}}=1 \tag{34}\]1  
Finally, the \(\Delta n\) and the ceramic's length should be large enough to give the desired steering angle (\(\theta\)).