A fiber laser with wavelength of 1070 nm is employed to cut the sapphire substrate. The reasons for the generation of powder materials on the sapphire substrate surface and the collapse edge during the laser cutting sapphire substrate are researched, and the relation between the laser processing parameters (laser fluence, cutting velocity, repetition frequency, the pressure of assistant gas) and the collapse edge size on both sides of sapphire substrate is analyzed. The results show that the interaction between laser and sapphire is mainly thermal effect, which makes the sapphire material melt and evaporate. With the absorption of plasma which is produced by nitrogen for laser, the phenomenon of keyhole appears inside the sapphire structure and the length of keyhole have a significant effect on cutting quality. When the sapphire which thickness is 0.31 mm cut by laser, the collapse edge size on the sapphire substrate surface is 3 mm, and collapse edge size under the surface is 8 mm with the optimized laser processing parameters，that laser fluence is 5.7 ×103 J/cm2 , cutting velocity is 6 mm/s, repetition frequency is 1.8 kHz, the pressure of assistant gas is 0.9 MPa.