The c - axis oriented single crystal of $LiNbO_3$ and $LiNbO_3$ : 5mol%MgO was success-fully grown by Floating zone method using halogen lamp as a heat source. The effects of the sintering condition of the feed rod and the atmosphere gas during the crystal growth on the be havior of the feed rod/melt interface were studied for growing crystal with the high quality, and then, the optimum growth conditions were determined by studying the experimental param eters, such as gas flow rate, pulling rate, rotation speeds of the feed rod and the seed. The grown crystals were analyzed using the chemical etching to observe the tch pattern and the ICP (Inductively Coupled Plasma) to determine the composition uniformity and the impurity content of Fe. The effects of additive (5 mol % MgO) on the transmittance and refractive index was, also, analyzed. In order to compare the nonlinear optical oharacteristics of $LiNbO_3$ with those of the other optical materials, the nonlinear optical refractive index ($n_2$) was calcu l lated using the measured refractive index.