This is a quick writeup to make precise the terms used in describing the beam coming from a forward reflector, and to give some simple formulas to estimate their size.
A calculator that outputs the exact numbers already exists, but its computation is opaque and complicated. Furthermore, it is often useful to have a short, understandable formula on hand for rough estimates.
I. Definitions and illustrations
If you look closely at a reflector beam, you will see 4 distinct regions within the beam:
These regions can be unambiguously defined by standing far away from the light, and looking into the reflector. Clearly, the bigger the lit region inside the reflector, the more intense the corresponding part of the beam (at your location).
This perspective change leads to the following correspondence:
- Hotspot: the region where you see the entire rim of the reflector lit up.
- Primary corona: the region where you see a connected lit region in the reflector, but the rim of the reflector is not fully lit.
- Secondary corona: the lit region of the reflector is disconnected, like a pair of wings.
- Spill: the region where the reflector is not lit at all, and only the bare emitter is visible.
Observe that the primary corona has a somewhat sharp outer cutoff, whereas the secondary corona fades smoothly into spill. In fact, the secondary corona is not visible at all under any exposure that does not overexpose the hotspot. For this reason, it is natural to consider the colloquial/ambiguous term “corona” to refer to the primary corona.
IIa. Estimation formulas
Some considerations
It is useful to first consider how each of the regions is influenced by the geometry of the reflector and the emitter. In particular: the hotspot and (primary) corona both depend on the size of the emitter and the geometry of the reflector. They become wider with a smaller reflector or a larger emitter, and vice versa.
The spill is essentially unaffected by emitter size, for small emitters, and does not depend on the size of the reflector, just its aspect ratio. This can be seen by dilating or contracting the reflector about the emitter: it does not change the visibility of any ray coming directly from the center of the emitter. Another way to see this is to think of the spill as an infinite cone, which does not change its shape when dilated or contracted.
Now, it is useful to set notation for the relevant quantities:
- D = diameter of the reflector;
- H = height/depth of the reflector;
- d = diameter of the (circular) emitter;
- A = D/H the aspect ratio of the reflector.
Now we are ready to present estimates of the hotspot, corona, and spill divergence angles. These are full (diametrical) divergence angles, and given in degrees.
Spill angle ≈ 2*arctan(A/2) + 50d/D
For throwy lights (large reflector relative to emitter, C8 SFT42R and above):
Hotspot angle ≈ 230dH/(4H²+D²)
Corona angle ≈ 31d/D * (2+sqrt(4+A²))/A
For floody lights (small reflector relative to emitter, S6 SFT42R and below):
Hotspot angle ≈ 48d/D
Corona angle ≈ 160dH/D²
Notes on the formulas
The spill angle is derived with simple trigonometry, using the right triangle formed by the emitter and a radius of the reflector. A compensation term approximately equal to the hotspot angle is added to include the region where the emitter is partially blocked by the bezel.
The hotspot+corona estimates for throwy lights are derived using small-angle simplification of the formulas from the calculator linked above, with the irrational constants rounded in a direction that minimizes estimation error of most common lights.
The hotspot angle is roughly the angle subtended by the emitter from the perspective of a point on the rim of the reflector. The corona angle is the maximum angle subtended by the emitter over all points on the reflector’s surface; this is a considerably harder quantity to analyze. As a result, the estimates for corona are less accurate.
The formulas for floody lights are further simplified with assumptions that favor floodier lights. Some functions are replaced with simple linear or reciprocal approximations on an interval, and the numerical constants are selected to work well with most common lights.
IIb. Examples
To illustrate the accuracy of these formulas, we consider the following 4 common lights spanning a range of beam profiles and reflector geometries:
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Convoy S2+ SFT42R. (D, H, d) = (18, 12, 2.52). The calculator gives hotspot, corona, spill angles as (7.3, 15.0, 81.1). Our floody estimates give (6.7, 14.9, 80.7). Error < 9% for all 3 angles.
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Convoy S6 SFT42R. (D, H, d) = (19, 19, 2.52). Calculator: (5.9, 22.1, 59.0). Floody estimates: (6.36, 21.2, 59.8). Error < 8%.
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Convoy C8+ SFT42R. (D, H, d) = (40, 33, 2.52). Calculator: (3.15, 7.19, 65.6). Throwy estimates: (3.21, 6.99, 65.6). Error < 3%.
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Convoy L21B SFT42R. (D, H, d) = (58, 50, 2.52). Calculator: (2.14, 4.98, 62.4). Throwy estimates: (2.17, 5.01, 62.4). Error < 2%.
The approximation errors are with 10% for all parts of the beam across a wide range of lights, which makes the approximations decent for their simplicity.
