What is the relationship between Mie scattering amplitudes, incident intensity and outgoing intensity?

Wikipedia’s Mie scattering Mathematics discusses the scattering amplitudes $S_1(theta), S_2(theta)$ for each outgoing polarization of an incident EM plane wave on a uniform sphere.

It defines the scattered intensities as

$$i_1(theta) = |S_1(theta)|^2$$

$$i_2(theta) = |S_2(theta)|^2$$

and then defines the scattering intensity

$$I(theta) = I_0 frac{lambda^2}{8 pi^2 r^2} left(i_1(theta) + i_2(theta)right)$$

where r is the radius of the sphere.

The incoming intensity $I_0$ would likely have units of power per unit area, whereas I’d expect the scattered intensity or radiance $I(theta)$ to have units of power per unit solid angle, but if that were true the units don’t work.

Could $I_0$ possibly be the power incident on the geometrical cross-section $pi r^2$?

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I’m doing an approximate calculation of something similar to what’s described in this question. I’ve found a python script that generates the scattering amplitudes $S_1(theta), S_2(theta)$, I’ll have a laser pointer with $I$ watts/m^2 and a detector at some angle $theta$ with solid angle $Delta Omega$ and I’d like to calculate the scattered intensity reaching that detector for a given single particle, but I’m stuck on how to correctly convert from $S$ to scattered power into $Delta Omega$.

The answer there says:

Wikipedia just pointed me to an English translation of the original paper by Mie, which I didn’t know existed. I haven’t yet had a chance to read it, so I don’t know how useful it is.

but I’m not able to access that link.