It has been known for a long time that given a fixed diameter, a deeper reflector out-throws a shallow one. However, I couldn’t seem to find an explicit quantification of this relationship.
Here’s a plot I made, assuming that the emitter is perfectly Lambertian–in this case the difference in intensity can be attributed entirely to the fact that a deeper reflector has less reflective material removed at the base, where the emitter lies. The horizontal axis is depth/diameter ratio, and the vertical axis is intensity relative to the maximum achievable for a fixed diameter. Reflector intensity vs depth | Desmos
My plot appears to be in agreement with another plot I found here on slide 16:
In any case, the following would be a good takeaway:
For a fixed diameter, deeper reflectors do throw better, but not by much past a certain point.
For a depth/diameter ratio of 0.5, increasing depth alone never gets you more than 20% more intensity. Almost all reflectors have said ratio exceeding 0.5, so the conclusion applies.
For a depth/diameter ratio of 1, increasing depth alone never gets you more than 6% more intensity.
The above plot can be interpreted as intensity vs depth for a fixed diameter. For completeness, I also wanted to generate a plot of intensity versus diameter, for a fixed depth: Reflector intensity vs diameter | Desmos
And here are the plots together, first intensity vs depth, then intensity vs diameter:
Most reflectors have a depth/diameter ratio between 0.5 and 1, which corresponds to the green regions. It can be observed that in the green region, diameter is a much more effective contributor to intensity than depth.
In fact, the following holds numerically over the green regions:
Every 1% increase in diameter buys you >1.7% increase in intensity, holding depth constant.
Every 1% increase in depth buys you <0.3% increase in intensity, holding diameter constant.
Yeh, a deeper reflector basically just collects that much more spill and pushes it into the hotspot, and pushes that spill-circle forward.
But I think it also makes die placement, and die “smallness”, that much more critical.
But there’s a difference between just keeping the same diameter and shaping the reflector to be deeper and pointier, vs just continuing the same shallow reflector and making it longer and wider.
Yep, hotspot and corona! Lights with extremely deep reflectors (say Maglite 2AA LED or Ultrafire T6) produce coronas that are huge compared to the hotspot, almost half as wide as the spill. This is consistent with results obtained via ray-tracing simulations here.
Certainly! The latter (extending a shallow parabola) also makes it wider in absolute terms, which gives it a significant boost in throw.
The existing plot can be interpreted as intensity vs depth for a fixed diameter. For completeness, I updated the OP with a new plot of intensity versus diameter, for a fixed depth: Reflector intensity vs diameter | Desmos
Here are the plots together, first intensity vs depth, then intensity vs diameter:
A takeaway is that increasing diameter is much more effective than increasing depth for maximizing throw, for most reflector shapes (green regions). More quantitatively, the following holds over the green regions:
Every 1% increase in diameter buys you >1.7% increase in intensity, holding depth constant.
Every 1% increase in depth buys you <0.3% increase in intensity, holding diameter constant.
Yeh, and a wider reflector also widens the hotspot accordingly, essentially diluting the intensity.
I hate tradeoffs. They make my head hurt.
I wonder how much spread when far from ideal. Eg, instead of a perfect point-source, also take into account aberrations from light coming off the edges and especially corners of the emitter.
This is what intuition suggests, and is indeed what happens point-blank when the beam is just exiting the light. However, if the target is far away, the wider reflector actually gives you a tighter and more intense hotspot! The beam is initially thicker but less divergent, so over a long distance the lower divergence outweighs the initial thickness.
It might help to have a schematic drawing of what a wide reflector beam (red) and a small reflector beam (blue) look like, point-blank (left) and at a distance (right):
Incidentally, what you stated is exactly the reason why a super thrower like BLF GT can’t burn something point-blank, while a much floodier light like a S2+ triple can.
The fact that emitters are not perfect point sources is the reason flashlight beams aren’t perfectly parallel (and don’t have infinite candela) like the red beam from the above illustration. Without going into the weeds: for a fixed reflector, the size of the hotspot at a distance is roughly proportional to the size of the emitter.
If you’re interested in the weeds, here’s a simulation I built that lets you input the reflector geometry and emitter size, and outputs a simulation of what the beam looks like at a distance, hotspot and corona and spill. Calculator for reflector beam profile
I’ve thought about this too, but these calculations seem incredibly complex to me. We need to consider several factors:
The LED’s radiation pattern. A real LED is very different from a Lambertian light source.
The shape of the emitting surface. Although, I imagine square LEDs are not much different from round ones.
The ratio of the emitting surface size to the reflector diameter.
Your graphs are clearly a rough approximation, since the first one tends to 1 and the second one tends to infinity, which doesn’t happen in the real world. If the diameter-to-depth ratio is too large, no light will go to the hotspot. If it’s too small, the reflector will turn into a long, narrow tube, and the light will bounce chaotically off its walls.
So, for a given diameter, emitting surface size, and radiation pattern, there must be an optimal depth that provides maximum intensity. As I said, I think calculating this mathematically is an incredibly difficult task, so we have to rely on empirical data.
Once you do them, chances are that the calculations are simpler than you’d initially assume! One incredibly helpful perspective to have is to imagine yourself standing far away from the light, looking at the beam. From this perspective, the light/reflector looks like this:
and intensity you receive is proportional to the apparent area of the reflector from your point of view, reducing the complexity of a paraboloid to a simple difference-of-circles area calculation.
In fact, the intensity in candela equals the luminance (cd/mm^2) of the emitter times the apparent area of the optic (mm^2), assuming a lossless optical system. In lossy systems, the loss is usually a fixed multiplicative constant and thus does not alter a relationship that has been normalized/relativized.
Different yes, but perhaps not very, at least according to the SFT42R’s datasheet angular distribution, which is typical of domeless emitters:
This turns out to not matter very much, because a small neighborhood of the hotspot center (where the maximum intensity is achieved) corresponds to a small neighborhood of the center of the emitting surface. If you stand sufficiently far from the reflector and look into it, you will be unable to tell the shape of the emitting surface from its virtual image in the reflector–the whole reflector is just filled uniformly with the yellow virtual image of the phosphor, in a small neighborhood of the emitting surface’s center. What happens far from the center is not observable from the hotspot center, and thus does not influence it.
Another way to see this (albeit slightly oversimplified) is to think of the reflector as a convex lens, where each point in the projected image corresponds to exactly one point on the LED. That’s why you can get extremely sharp projections of the emitting surface like the one below:
This determines beam profile, but not intensity without specifying the output, or equivalently, luminance (cd/mm^2 or lm/mm^2) of the emitter. The reason is simple: the same light with the same geometry has different throw on different modes.
This reasoning doesn’t hold because both of my graphs describe relationships between finite numbers–if you put a finite number in, you get a finite number out.
If you instead plot A against r in A=πr^2 (area of a circle as a function of radius), you get a curve that “tends to infinity” even faster. Does this invalidate the formula, or make it a “rough approximation” unsuitable for describing real-world circles?
This intuition is excellent, and somewhat accurate too. However, it is possible for the reflector to collect only a tiny portion of the output, and still generate an intense hotspot.
Recall that intensity (candela) is defined as output per solid angle (lm/sr), so if the beam is narrow enough (solid angle small enough), the intensity can still be very high despite low output. This is indeed what happens with a wide but shallow reflector.
For an overly deep reflector, you indeed get higher-order chaotic reflections; fortunately, this does not interfere with calculations based on the perspective described earlier in the post–all this chaos ends up messing up the corona/spill only.
In any case, these considerations are not particularly relevant as almost all reflectors come in a narrow range of aspect ratios–reflectors with extreme aspect ratios end up producing throw that is suboptimal for their size.
For an ideal Lambertian emitter, there is no optimal depth–the intensity keeps increasing, but in a bounded way, as you increase depth indefinitely while fixing the diameter. The deeper the reflector gets, the less you truncate at the focal plane. The assertion that “there must be an optimal depth” seems unjustified.
I do not make any claims for a radiation pattern that is far from Lambertian. This is because my calculations rely on a special property of Lambertian emitters: the luminance (cd/mm^2), i.e., brightness per visible area, is the same no matter which angle you look at it from. In this sense, the Lambertian emitter is the most natural flat analogue we have of a spherical uniform emitter, which appears the same (and thus has the same luminance) in all directions.
Thanks for the explanation! I think the model you proposed works perfectly and greatly simplifies the reasoning. My attempt at a summary:
The only parameters that matter for throw are:
luminance at the focal point. Any point outside the focal point is irrelevant, since the light emitted by it doesn’t reach the eye of our infinitely distant observer due to the optical properties of the paraboloid. Therefore, the shape of the emitting surface is completely irrelevant for throw. This also means that multi-die LEDs like the SFT70 shouldn’t be used in throwers, as they have reduced luminance in the center.
The projected area of the reflector surface onto a plane orthogonal to its axis, which is equal to the difference in the areas of the circles that cut the paraboloid at the top and bottom, in the focal plane.
So my assumption about the existence of an optimal depth-to-diameter ratio was incorrect, although it does have some logic in the real world, where distances are finite. No one wants their beam to be a huge corona with a tiny point of intensity in the center or a dim, narrow ring with no light at all in the middle.
I think your summary is both clear and accurate–thank you very much!
In the real world, of course, reflectors sometimes fail to be perfect paraboloids due to manufacturing imprecision. In this case, the correspondence I proposed earlier no longer holds:
and consequently, what happens outside the very center of the emitting surface starts to matter more. An example of this phenomenon is the defective batch of L21B reflectors, which allow the larger (but less intense) SFT40 to out-throw the smaller but more intense SFT25R. If the manufacturing quality is good, this should not be an issue, and the existing analysis holds.
And indeed, the projected area in the direction you described is the only relevant parameter for the secondary optic, assuming that the optic is non-diffusive. I also agree that multi-die emitters should not be paired with such optics, which produce a beam that has reduced luminance in the center in addition to undesirable artifacts.
That’s a very sensible consideration, and also the reason I dislike overly deep reflectors. They sacrifice a lot of near-field visibility for a tiny boost in throw, which could have been achieved instead by widening or dilating the reflector, which does not sacrifice spill.
And an overly wide reflector would indeed exhibit the latter problem you described. If depth is fixed, the intensity grows with diameter in a curious way: initially quadratic (scaling with the square of the diameter as one would expect for an area), but gradually transitioning to a linear regime where most of the parabola has been truncated away, leaving only a narrow ring along the boundary as you described.
nice analysis. in practice the base cutout is the silent killer here — most production reflectors lose a chunk of output right at the base opening where the emitter sits, and that’s the exact zone your plot says matters most.
also why dedomed emitters throw so much further: smaller apparent source = reflector sees something closer to a point, so more of the cone lands in the collimated beam instead of spill. geometry is only half the story, the other half is how ugly the real emitter looks at the focal point lol
The base cutout is indeed the reason for the intensity loss, though it turns out to contribute very little–the above plots indicate that almost all reflectors lose less than 20% intensity from this cutout; some lose as little as 6%. The relevant quantity for throw is total projected area of the reflector, regardless of the shape of the area and where it is located. A previous comment has a bit more detail on why projected area is relevant.
In fact, one can argue that removing material near the center loses less throw than elsewhere. If you have a disk and remove 50% of the radius at the center, you still retain 75% of the total area. But if you removed 50% of the radius at the outer rim instead, you’d only retain 25%. This is also the reason why expanding the diameter slightly gains much more throw than making the reflector arbitrarily deep, as my plots indicate.
Your intuition is correct! To be more precise, the size of the emitter doesn’t substantially change the proportion of its output that ends up in the hotspot/corona or spill; what it does change is the divergence of the hotspot and corona. A smaller apparent source projects a beam that has a tighter divergence angle in its hotspot and corona.
@QReciprocity42 Next approximation project, the revival of RECOIL THROWERS! also, non-standard reflector shapes, such as a 4-“cornered” reflector that best approximates the projected emitter image? (think cuboid, but with bulging walls)
Indeed! Enderman has also already built a simulation for recoil reflectors, which makes them already well-understood. Plus, recoil reflectors are generally much harder to source (and use) than forward reflectors, for a variety of reasons. I don’t think there’s much I can contribute to the recoil-thrower literature in a way that would significantly benefit the community here.
