Complex part of second-order susceptibility in nonlinear optics

Question

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?

Answer 1

The main argument to ignore second order (and most even) nonlinear processes is because historically, centrosymmetric systems have negligible even order susceptibilities ($\chi^n$ where n=2,4...) and the tensor vanishes due to symmetry (this is well documented in Boyd's nonlinear optics book).

Here, Burris and McIlrath https://opg.optica.org/josab/fulltext.cfm?uri=josab-2-8-1313&id=3665 also use the same argument for constructing the imaginary part (although

$2\eta\kappa = 4\pi Im \{ \chi^{(1)} + \chi^{(3)}E^2 + ... \}$

This argument comes from Loudon's Quantum Theory of Light in the same paper:

Even terms in χ(n) make no contribution to the susceptibility for uniform excitation in an isotropic medium owing to inversion symmetry.

While this argument may not be that interesting, I have not been able to find a more compelling argument as for what is the physical meaning of the imaginary part of the second order nonlinear susceptibility tensor.

On one hand, you have a few older articles where it is understood as simply the phase relation of two intensities on a surface: https://www.sciencedirect.com/science/article/pii/S0039602897003646.