= Nonlinear optical susceptibility
{title2=$\chi^{(2)}$}
Choose one driving <electric field> component and one responding <electric polarization> component. A schematic local, instantaneous expansion is
$$
P=\epsilon_0\left(\chi^{(1)}E+\chi^{(2)}E^2+\chi^{(3)}E^3+\cdots\right).
$$
Here $\epsilon_0$ is the <vacuum permittivity>, $\chi^{(1)}$ is the linear <electric susceptibility>, and the higher coefficients describe nonlinear response. In SI, $P$ has units $\mathrm{C\,m^{-2}}$, $E$ has units $\mathrm{V\,m^{-1}}$, and $\chi^{(2)}$ has units $\mathrm{m\,V^{-1}}$. The selected scalar coefficient depends on <polarization of waves> and <crystal> orientation; it is not a universal material number independent of geometry.
For a chosen <monochromatic wave> $E(t)=E_0\cos\omega t$, the quadratic contribution is exactly
$$
P^{(2)}(t)=\frac{\epsilon_0\chi^{(2)}E_0^2}{2}\left(1+\cos2\omega t\right).
$$
The <cosine> identity separates a constant contribution, associated with <optical rectification>, from an oscillation at twice the <angular frequency>, the source for <second-harmonic generation>. A term oscillating at $2\omega$ is a local driving source; efficient outgoing <light> additionally requires propagation and <phase matching>.
For a material with <optical dispersion>, the scalar instantaneous expansion is only a teaching model. The actual quadratic response is a <tensor> with output and input <frequency> arguments: $\chi^{(2)}_{ijk}(2\omega;\omega,\omega)$ describes the harmonic, whereas $\chi^{(2)}_{ijk}(0;\omega,-\omega)$ describes the constant response. Their coefficients need not coincide. The <cosine> calculation establishes which <frequencies> are available, without assigning measured coefficients to any <crystal>.
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