In optics, the absorption of photons by a material can be described by considering the material's susceptibility. For linear absorption (involving a single photon), we think about the imaginary part of the complex linear susceptibility, $\mathrm{Im}[\chi^{(1)}]$, while its real part, $\mathrm{Re}[\chi^{(1)}]$, describes refraction.
In nonlinear optics, it's straightforward to consider odd-order susceptibilities: $\mathrm{Im}[\chi^{(3)}]$ is related to 2-photon absorption, while $\mathrm{Re}[\chi^{(3)}]$ is related to third-harmonic generation. This logic generalizes to $n$th order processes, as long as $n$ is odd: $n$th harmonic generation is described by $\mathrm{Re}[\chi^{(n)}]$ (true for even and odd $n$) and $n$-photon absorption is described by $\mathrm{Im}[\chi^{(2n - 1)}]$.
However, even orders are tricky, since $2n-1$ is always an odd number. For example, $\mathrm{Re}[\chi^{(2)}]$ describes second-harmonic generation, which is straightforward, but $\mathrm{Im}[\chi^{(2)}]$ does not seem to have a physical meaning, since two-photon absorption is already described by $\mathrm{Im}[\chi^{(3)}]$ and single-photon absorption is described by $\mathrm{Im}[\chi^{(1)}]$. Since there is no absorption role for $\mathrm{Im}[\chi^{(2)}]$ to fill, what does it describe?
