Expression of third-order effective nonlinear susceptibility for third-harmonic generation in crystals

Third-harmonic-generation processes in crystals are governed by the fourth-rank tensor ((3))(Xijkl), which reflects the crystal symmetry. In this case, the third-order nonlinear susceptibility tensor can be contracted to the compact matrix form ((3))(Xim). The matrices ((3))(Xim) for isotropic media and all 32 crystallographic point groups are presented. With these matrices, the analytic expressions of third-order effective nonlinear susceptibility can be easily derived.