Updated the calculator to show the reflector hotspot/corona breakdown.
The proportion of lumens in hotspot will be a slight overestimate, and the corona a complementary underestimate, but not by more than 10% relative, i.e., off by a factor of 1.1.
Rough sketch of the derivation
- Compute hotspot (solid) angle for an arbitrary small emitter diameter.
- Compute central hotspot intensity for the same emitter.
- Assume that the hotspot has uniform intensity. This is almost true–for most reflector/emitter combinations, the hotspot does not seem to vary in intensity by more than 10%. This is supported by empirical measurements and heuristics based on this calculator. This deviation is the reason for the estimation error mentioned above. Reflectors with a shallow aspect ratio are more prone to error.
- Multiply together (1) and (2), and simplify using asymptotic approximations under the assumption that the emitter is very small relative to the reflector. The emitter size terms in (1) and (2) cancel out, and these approximations end up slightly correcting for the aforementioned estimation errors.
Based on the calculator, a deep reflector (Diameter=Height) has a hotspot-corona-spill breakdown of roughly 15-65-20. For shallow (D=2H), it’s roughly 20-30-50.
One interesting implication is that: compared to a forward reflector, a well-designed TIR or convex lens system of the same aperture diameter can project a much larger hotspot of equal intensity.