Poling condition in 2 mol% MgO-doped $LiNbO_3$ crystals is investigated by varying the amplitude of poling electric field from 3V/cm to 7V/cm. It is found that 5V/cm is the best value for the formation of single domain by analysing the characteristics of the second harmonic generation(SHG) as the function of temperature. The temperature dependence of the phase-matching profile of SHG for the crystal applied by a spatially modulated electric field is observed to be very different from the simple sinc function. The distorted profile can be consistently fitted with the numerical simulations. This shows that the crystal homogeneity can be tested by the SHG temperature phase-matching profile. In addition, the thermo birefringence coefficient and electro birefringence coefficient of SHG were measured from the temperature dependence of phase-matching profile and shifts of phase-matching temperature by appling electric field along c-axis.