Third-Order Nonlinearities

Third-order optical nonlinearities involve the nonlinear susceptibility tensor χ⁽³⁾. This term governs third harmonic generation, optical Kerr effect, Raman effect, and Brillouin scattering. In general, χ⁽³⁾ will couple together four frequency components, that is, three fields interact to produce a fourth field,
.......(10.7)
where ω₄ = ω₃ + ω₂ + ω₁. χ⁽³⁾ is a four-rank tensor with 81 elements, because an array of 3×27 coefficients couple a single component of the polarization P to three independent components of three electric field vectors.
The third-order optical nonlinearity is thus a four-photon process. There can be up to three different input laser frequencies, but several of the frequencies can also be the same. In a lossless medium the susceptibility coefficients of χ⁽³⁾ are real. In this case, the primary nonlinear optical effects are the generation of new frequency components and the intensity-dependent change in the refractive index (Kerr effect). In third-order interactions involving absorption, the imaginary part of χ⁽³⁾ describes Raman and Brillouin scattering.