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).
To enable efficient nonlinear wavelength conversion, both photon energy conservation and photon momentum conservation are to be satisfied simultaneously in the nonlinear optical process, as shown in figure 2. The requirement on the photon momentum conservation is also called “phase-matching” constraint. If such a phase matching condition is not satisfied, the generated photons will destructively interfere with each other, limiting the number of generated photons that finally exit the crystal. However, the refractive index of an optical material is dependent on the wavelength of light propagating through the material, thus the phase matching condition might not be satisfied due to the relative phase delay among mixing photons, as shown in figure 3.