I found myself often needing to calculate the collection efficiency of convex/Fresnel lenses and the hotspot+corona/spill breakdown of reflectors, so I whipped up this quick calculator to do it for me. Hope someone also finds it useful.
The core of the calculator is just a spherical cap area calculation, which follows from the (neat) result that: for a Lambertian emitter in 3 dimensions, any sphere centered directly in front of the emitter, with radius equal to the distance from center to emitter, has constant illuminance (lux) across its entire surface.
The calculator assumes that the LES is small relative to the secondary optic, but does not essentially depend on its exact size/shape.
Separating hotspot from corona is a harder task because both are captured by the reflector. I will try to come up with a decent approximation.
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.
using tools that simplify complex calculations is always a win. i recently founds the water calculator to be super useful for keeping my hydration in check. it’s great that you’ve developed a calculator for lens and reflector efficience - definitely can make these processes easier. having both these tools on hand makes calculating efficiency and personal health adjustments much smoother. keep up the good work with the updates and improvements
Thanks! The 10% estimation error only applies to computing the total flux in a reflector’s hotspot or corona, because forward reflectors are inherently more complex than convex lenses, and the hotspot/spill dichotomy doesn’t hold due to presence of the corona.
For a convex lens, the situation is much simpler: light from the emitter is either wasted or ends up in the hotspot; this dichotomy makes accurate estimation easy. If you’re working with more complicated lens systems, chances are that you can design things in such a way that only the first collection stage is lossy, which makes computation simple.
Welcome to BLF! I hope you find your stay here worth your while.