The electrical and optical properties of the annealed $LiNbO_{3}$ single crystal with congruently melting composition and MgO or ZnO doped $LiNbO_{3}$ single crystal grown by the FZ method. The electrical and optical properties such as electrical conductivity, dielectric constant (Curie temperature), electro-mechanical coupling factor, optical transmittance and refractive indices of the grown crystals were measured and the nonlinear refractive indices of the grown crystals were calculated theoretically. The doping effects of MgO and ZnO were investigated by comparing the electrical and optical properties of the undoped $LiNbO_{3}$ single crystal and those of the $LiNbO_{3}$ single crystals doped with MgO or ZnO.