Maths behind optimal beam expansion

I am curious about the maths involved in dot size and energy reaching the terminal beam point if a beam expander is involved. The expander I have is a LaserGlow X10DR 10x expander that I have adapted to be a universal fit. It has two adjustment points by twisting either the expander's lens or the expander base. So, the effect is that I can adjust both the beam's focus AND the expansion factor. Also, because I have such a long lens holder, it can go well beyond 10x expansion factor. I'd say easily 20x. The problem is, then, that the beam is clipped on the outside, because the exit aperture has a finite diameter. Does anyone have any ideas about how to calculate the way to get the most beam intensity (W/m^2) at the final (large) dot? I think it is possible to define the optimal beam expansion factor in terms of only the distance the beam will travel.

My question specifically is this:
Given a laser with divergence θ, and a beam expander with N-times magnification, I can work out that the spot size is proportional to θ/N.
However, at expansions above N=10x, the wings of the beam are clipped by the lens diameter and the housing of the expander. Below is a diagram of my situation.



How can I describe the energy per unit square meter reaching the target dot based on only N (expansion factor) and L (distance to target)? How would this be with a single mode diode which clips all around the dot versus a multimode where it only clips the width of the beam?

Thanks.