The linear electro-optic effect of LiNb $O_3$ single crystal is observed and its electro-optic coefficients and half wave voltage is determined by using an ellipsometer with a photoelectric detector. The 5641 Å line of Hg spectrum is used for electro-optic measurement whereas the conoscopy is carried out by using He-Ne laser beam. The optic axis of the crystal is determined through the conoscopic figure formed by a convergent laser beam between crossed polarizers. The simple is ground into a hexahedron with each plane perpendicular to the $X_1$, $X_2$, $X_3$ axis of the crystal, and the two parallel planes perpendicular to $X_1$ axis are polished. Light wave propagates along $X_1$, $X_2$ or $X_3$ axis.
The refractive indices for ordinary and extraordinary rays are measured using a microscope to give results $N_o$=2.3144, $N_e$=2.2244, respectively. When the electric field is applied along $X_1$ axis through semitransparently coated gold electrodes, no electro-optic effect appears, as predicted by the theory. In case the applied electric fields are along $X_2$ or $X_3$ axis, the half wave voltages are determined to be 7.46KV or 3.07KV, respectively. From these values the electro-optic coefficients $r_{22}$ and ($0.89r_{33}-r_{13}$) are determined to be $5.90\times 10^{-12}$m/v and $14.4\times 10^{-12}$m/v, respectively. The crystal used in this work is the single domain crystal and shows distinct electro-optic response in accordance with the applied electric field. The linearity of the effect is proved to be in good agreement with all predicted by the theory. This work provides the basis for further applications in the electro-optic laser Q-switching and the optical communication.
