Complex part of second-order susceptibility in nonlinear optics

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?