This is a very interesting problem that I've been struggling to solve for a week, so I decided to ask for some orientation, as I think it could also be of interest for the community.

Let's consider a plane wave that incides on a material such that it induces a charge distribution with only quadropole moments $$Q_{33}=-2Q_{22}=-Q_{11}=Q_0$$, while all the other components $$Q_{ij}$$ are zero. Moreover, no dipolar terms (both magnetic and electric) are induced, so we have a pure quadrupole irradiating system.

It's evident that, if the incoming wave is non-polarized, we will still have a difracted polarized wave due to the quadrupolar nature of the difracted radiation.

I'm trying to find the cross section of the difracted wave for each one of the polarizations (parallel and perpendicular to the plane of incidente)

My idea was to find the diffracted way through means of the quadropole radiation equations given by Jackson's Chapter 9,

$$a_E(l,m)=\frac{ck^{l+2}}{i(2l+1)!}\left(\frac{l+1}{l}\right)^{1/2}(Q_{lm}+Q'_{lm})$$

$$a_M(l,m)=\frac{ik^{l+2}}{(2l+1)!}\left(\frac{l+1}{l}\right)^{1/2}(M_{lm}+M'_{lm})$$

From there, one can find the total power and then divide by the total intensity of the incident wave to find the total cross section,

$$\sigma=\frac{P_{diffracted}}{I_{incident}}$$

However, not only this approach doesn't allow me to reduce the harmonic spheric terms, but I have no idea how to separate the polarization components of the final cross section.