Got it, thanks for the explanation! I see that you defined the hotspot to be the half-intensity cutoff, which is probably a good definition to use.
I tried an alternate derivation via ray tracing, and ended up with a formula that suggests a tighter hotspot than yours. Could you take a look and see if it seems sensible?
A nice surprise is that the “beam acuity index” happens to be a very convenient quantity to work with throughout the derivation.