# What are the material restrictions for a $\chi^{3}$ non-linear optical material?

For instance, I know that $$\chi^{2}$$ materials need to lack inversion symmetry, but what about $$\chi^{3}$$ materials?

Basically, $$\chi^{(3)}$$ is a rank four tensor with 81 components. You then apply the Neuman principle saying that symmetries of the crystal that supports $$\chi^{(3)}$$ should also be the symmetries of the tensor. Then you look for independent components of this tensor, which boils down to looking for trivial irreducible sub-representations in the representation of the symmetry group of the crystal over that rank four tensor.

So lets say the representation of your crystal symmetry group ($$G$$) over vectors is given by: $$M\left(a\right): \mathbb{R}^3\to \mathbb{R}^3$$, where $$a\in G$$

Then the induced representation for $$\chi^{(3)}$$ is $$K\left(a\right)=M\left(a\right)\otimes M\left(a\right) \otimes M\left(a\right) \otimes M\left(a\right)$$. If you want to know the independent components of $$\chi^{(3)}$$, simply define the projection operator, to project into trivial irreps:

$$P=\frac{1}{\#G}\sum_{a\in G} K\left(a\right)$$

$$P:\mathbb{R}^3 \otimes \mathbb{R}^3 \otimes \mathbb{R}^3 \otimes \mathbb{R}^3\to\mathbb{R}^3 \otimes \mathbb{R}^3 \otimes \mathbb{R}^3 \otimes \mathbb{R}^3$$

The eigenvectors of this projection operator will be the allowed components of $$\chi^{(3)}$$, if you only want to know the number of allowed components $$n_{allowed}$$, it is given by the trace of the projection operator:

$$n_{allowed}=Tr\left(P\right)=\frac{1}{\#G}\sum_{a\in G} Tr\left(K\left(a\right)\right)=\frac{1}{\#G}\sum_{a\in G} Tr\left(M\left(a\right)\right)^4$$

Have a look in a suitable nonlinear optics book, e.g. Popov, Svirko, Zheludev "Susceptibility Tensors for Nonlinear Optics", Appendix F. They list all the allowed components of $$\chi^{(3)}$$ for all crystallographic classes. There is no class where $$\chi^{(3)}$$ is not allowed, but in some cases the number of allowed components can be as low as 2 (Cubic system, class 432) or even 1 (isotropic medium).