In the previous chapter, explicit formulae for the nonlinear susceptibilities were derived. The susceptibilities exhibit various types of symmetry which are of fundamental importance in nonlinear optics: permutation symmetry, time-reversal symmetry, and symmetry in space. (Another kind of symmetry–the relationship between the real and imaginary parts of the susceptibilities – is described in Appendix 8.) The time-reversal and permutation symmetries are fundamental properties of the susceptibilities themselves, whereas the spatial symmetry of the susceptibility tensors reflects the structural properties of the nonlinear medium. All of these have important practical implications. In this chapter we outline the essential features and some practical consequences.

Permutation symmetry

The permutation-symmetry properties of the nonlinear susceptibilities have already been encountered in earlier chapters. Intrinsic permutation symmetry, first described in §§2.1 and 2.2, implies that the nth-order susceptibility is invariant under all n! permutations of the pairs (α1, ω1), (α2, ω2), …, (αn, ωn). Intrinsic permutation symmetry is a fundamental property of the nonlinear susceptibilities which arises from the principles of time invariance and causality, and which applies universally. In some circumstances a susceptibility may also possess a more general property, overall permutation symmetry, in which the susceptibility is invariant when the permutation includes the additional pair (µ, -ωσ); i.e., the nth-order susceptibility is invariant under all (n + 1)! permutations of the pairs (µ, ωσ), (α1, ω1), …, (αn, ωn).