Universal theory of third-order nonlinear polarization in isotropic media

Vector expression of third-order nonlinear polarization in isotropic media is derived, which is suitable for multiple beams incidence with different frequencies． This expression is applied to elucidate multiple well-known and distinct nonlinear optical phenomena, thereby validating its universality and accuracy． This offers a unified description of third-order nonlinear optical effects in isotropic media． By extending its application to hexagonal materials, this expression offers straightforward solutions to a variety of complex cases． This study is crucial for a systematic understanding of third-order nonlinear optical effects in both isotropic media and hexagonal materials, with the potential to enhance application of nonlinear optics in fields such as frequency conversion, polarization detection, and quantum information．