Determining the orientation of electric dipoles in Q-switched crystals with anisotropic and nonlinear absorption properties is an important condition for measuring their absorption cross sections. Through the analysis of the space group to which the Co ²⁺∶MgAl₂O₄ crystal belongs and the position symmetry of the doped Co ²⁺ in the cell structure, it is determined that the transition direction of the electric dipole in the crystal is mainly distributed along the diagonal direction of four individuals in the cell, and the theoretical model of polarized incident light propagation in the Co ²⁺∶MgAl₂O₄ crystal is established. The nonlinear transmittance curve of Co ²⁺∶MgAl₂O₄ crystal cut along [100] direction is measured experimentally in the direction of polarized incident light. The experimental results are fitted by the established model. The ground state absorption cross section and excited state absorption cross section of Co ²⁺∶MgAl₂O₄ crystal in 1.5 μm wave band are (3.6±0.3)×10 ^-19 cm ² and (4.5±0.4)×10 ^-20 cm ², respectively.