Kleinman symmetry

Assuming an instantaneous response we can consider the second order polarisation to be given by ${\displaystyle P(t)=\epsilon _{0}\chi ^{(2)}E^{2}(t)}$ for ${\displaystyle E}$ the applied field onto a nonlinear medium.

For a lossless medium with spatial indices ${\displaystyle i,j,k}$ we already have full permutation symmetry, where the spatial indices and frequencies are permuted simultaneously according to

${\displaystyle {\begin{aligned}&\chi _{ijk}^{(2)}(\omega _{3};\omega _{1}+\omega _{2})=\chi _{jki}^{(2)}(\omega _{1};-\omega _{2}+\omega _{3})=\chi _{kij}^{(2)}(\omega _{2};\omega _{3}-\omega _{1})\\=&\chi _{ikj}^{(2)}(\omega _{3};\omega _{2}+\omega _{1})=\chi _{kji}^{(2)}(\omega _{2};-\omega _{1}+\omega _{3})=\chi _{jik}^{(2)}(\omega _{1};\omega _{3}-\omega _{2})\end{aligned}}}$

In the regime where all frequencies ${\displaystyle \omega _{i}\ll \omega _{0}}$ for resonance ${\displaystyle \omega _{0}}$ then this response must be independent of the applied frequencies, i.e. the susceptibility should be dispersionless, and so we can permute the spatial indices without also permuting the frequency arguments:

${\displaystyle {\begin{aligned}&\chi _{ijk}^{(2)}(\omega _{3};\omega _{1}+\omega _{2})=\chi _{jki}^{(2)}(\omega _{3};\omega _{1}+\omega _{2})=\chi _{kij}^{(2)}(\omega _{3};\omega _{1}+\omega _{2})\\=&\chi _{ikj}^{(2)}(\omega _{3};\omega _{1}+\omega _{2})=\chi _{kji}^{(2)}(\omega _{3};\omega _{1}+\omega _{2})=\chi _{jik}^{(2)}(\omega _{3};\omega _{1}+\omega _{2})\end{aligned}}}$

This is the Kleinman symmetry condition.