Rutile type of $TiO_{2}$ single crystal was grown by floating zone method in order to obtain the highly transparent and contamination-free single crystal. The linear refractive index of perpendicular and parallel to c-axis was measured as a function o f wavelength from 500 to 1000nm. The optical energy band gap was estimated as 2.99 eV from the absorption spectrum. The theoretical $chi^{(3)}$} value of $TiO_{2}$ was discussed in comparison with that of $SiO_{2}$ quartz single crystal on the basis of semiempirical model. On the other hand, the second-hyperpolarizability, ${gamma}({Ti}^{4+})$ was calculated in order to describe the effect of $Ti_{4}$ ion on the third order nonlinear optical properties.