In the above figure, X(1) is linear susceptibility, while X(2) and X(3) are the second-order and third-order nonlinear susceptibility. The linear term X(1) determines the linear propagation of optical waves, such as refraction, reflection, diffraction, and dispersion; while the higher order terms X(2) and X(3) correspond to nonlinear effects for harmonics wave generation under strong electric field. The second order term vanishes in dielectric mediums with inversion symmetry structures but could be large in several nonlinear mediums with non-centrosymmetric crystal structures, such as Lithium Niobate (LiNbO3).
The second order nonlinear process involves three waves, leading to the different effect such as second harmonics generation (SHG), sum-frequency generation (SFG), difference frequency generation(DFG) and optical parametric generation (OPG). The energy conservation of above processes satisfy the photon energy conservation automatically, but the momentum between the photons might be different in the material. The mismatch of wavevector represents the phase coherence between the generated waves at different crystal position, which is the so-called phase-matching condition.