Please, if possible, find spme correlation between the surface area of the reflector and the intensity per surface area of the projection…
Photos attatched for examples of an optimally focused yuji in a 2d mag:
Looking at reflector
Horizontally projected beam profile
You might be able to see that there is a distinct partition within the hot spot, as well as a ring that splits the spill, where the ring is almost as intense as the corona.
I will try my best!
Yuji in a 2d mag (originally LED or incan?) is an interesting situation. The Yuji is not a Lambertian emitter and will mess with relative intensities, but the division into hotspot/corona/spill should be largely unchanged. Also curious where the ring in the spill is from…
Have all of these variables been considered?
Hole size at bottom of reflector.
Exact interface of LED and the bottom of the reflector.
Dome versus no dome.
With some reflector sizes a larger hole means a much larger hotspot and less throw. The reflector on the GT mini has a fairly wide barely beveled polished area around the LED.
And it’s one of the tightest hot spots that I have. This is a slightly exaggerated drawing.
The camera shows it forming because of the rim/lens reflection. Pic incoming in a minute…it’s too blown out, I can’t get a good pic even after reducing the exposure…
But basically, it seems to start at the bottom edge of the reflector closest to the led, then seems to refocus or amplify right at the edge where the reflector meets the lens, almost as if some extra refraction/lensing is going on. Looks very similar to the sun peeking around the earth.
There have been new developments at the Map of Flashlights as to hotspot angles, including the near-final model description.
This is good news for the ray-tracing model.
For the special case of single smooth reflectors - if you know their specific h, D, and d values - a hotspot boundary derived from the ray-tracing model must coincide with some value of the max intensity fraction that I use to define where the hotspot ends, and thus potentially improve predictions for individual lights in this subset (as long as it complies with the etendue conservation).
A win-win situation!
Thank you very much for the update! I’m sorry that I was away for a while, mostly traveling and grading exams. Looks like I’ve missed many exciting updates!
(1) For the Map: do you have an explanation somewhere of how the hotspot size values are obtained? If so, I may have missed it. I would be very interested in comparing it with my ray-tracing model.
(2) I built an exact simulation of a parabolic reflector in 2D, from which you can get what the simulated beam looks like and some critical measurements like hotspot/corona/spill angles. I will post it in a separate thread and edit this comment to include the link. Play around with it and let me know what you think!
EDIT: 2D Reflector Simulator
Hey thanks for sending this! A couple items I’d like to ask more about:
(1) How do you justify the assumption of Gaussian beam profile? Almost every beam profile (esp. mules and aspherics) is well-known to be non-Gaussian, though I suppose that Gaussian is a reasonable distribution for modeling any distribution that is unimodal and symmetric, which describes most beams. I suppose I could use my exact simulation to determine the accuracy of the Gaussian approximation.
(2) The 50%-of-total-output threshold is a sensible definition of hotspot, and will be very close to my purely geometric definition for most lights. However, assuming Gaussian beam profile, the right FWHM angle is not 2.35 times the standard deviation, but possibly around 1.1774 times the standard deviation. The 2.35 comes from a one-dimensional Gaussian distribution, but beams in 3D space are approximated by a 2-dimensional Gaussian, which is slightly more radially concentrated. I would suggest going through all of the calculations (e.g., etendue) and checking for dimensional inconsistencies like this.
BTW, here’s a new version dedicated to computing hotspot/corona/spill angles. I like geometric approaches because they are essentially dimension-independent.
I evidently don’t know how to explain it, but it’s not about an error in etendue value calculations.
Single-rays tracing models, all of them, can show that a path for each of the rays exists, not how tightly all other rays can be packed within the envelope defined by those tracers. The latter is limited by etendue and hidden in such models. One solution is tracing infinitesimal bundles of rays and Monte Carlo ray-tracing, but you can’t do it by hand.
Envelope ray-tracing that ignores etendue will always underestimate hotspot angle (it calculates the absolute lowest limit for the angle - etendue limit is on top of it, and this is not because of imperfections, it’s fundamental). The geometry of the reflector, mostly H, will determine by how much. It’s never zero and it may be trivial but for some perfectly realistic reflector setups could be several percent.
What I suggest is not that etendue conservation is a bad approach. What I suggest is that dimensionality errors from the 2D Gaussian density calculations can easily propagate through etendue calculations, if one does not check every step carefully.
Regarding the issue of etendue: my model makes no claims about intensity or about how tightly rays are packed. It simply calculates some angles that arise from the geometry of the reflector and emitter. My definitions of hotspot/corona/spill are fundamentally not based on intensity, but on geometry; one can ignore everything about lights and interpret my model as a geometry problem about angles in parabolas. I’m not sure how I can explain this more explicitly; if you do, please let me know.
You can argue that my model does not apply to your definition of hotspot, which is true because we use different definitions of the term “hotspot”.
EDIT: another way of seeing this is that in my model, very shallow reflectors can produce hotspots that are arbitrarily small fractions (say 1%) of total output. Thus, the total flux going from emitter to hotspot is demonstrably not conserved, so the naive form of conservation of etendue as you previously stated–which relies on conservation of total flux–cannot be applied.
Since it predicts radial intensity profiles, it does. It’s just hidden. The etendue limits what these profiles can possibly be. It’s about what’s possible to do with light, not with single rays. All naive edge/envelope tracing models suffer from this, hence ray-bundles tracing and Monte Carlo.
Also, no matter where you predict the ‘edge’ of the hotspot should fall, the only way to find it within the beam is by fraction of intensity to which you think it corresponds, for any distribution. The problem is not with the definition of the edge of the hotspot - any will do, but the issue above will not vanish.
Sure, one can interpret the purely geometric problem in terms of radial intensity profiles, but the math should remain correct regardless of how you want to interpret it.
This is a very general claim, and I don’t see a way to back it up. A little close-minded, perhaps?
This is unclear. Could you explain more?
I’ll try one more time to be as explicit as possible. Examine the following diagram:
I find it disheartening that you denounce the validity of my model in vague, general terms, but refuse to (1) point out a single error in my derivation, (2) produce a single counterexample, or (3) make reference to what my definitions and claims are. As a referee for math journals, I am required to do all of the above when I want to reject a submission. I also find it rather surprising, given your mathematical sophistication. The above renders this discussion empty and useless; perhaps we should table this subject and move on to something more productive.
It’s all good and time well spent, and I gained many insights thanks to our discussions and derivation details checking. Don’t take it personally.
Your ray is certainly there, I just don’t think it defines the hotspot boundary I a meaningful way outside of your model. I think it will correspond to a wide range of intensity cutoff fractions for different reflector geometries, which may not make it as helpful as it seems to calculate the hotspot sizes. And this etendue thing is not a trick - it does govern how tightly light can redistribute itself between the tracer rays in reality, and it needs to be accounted for to represent it.
I agree with all of the above, and certainly feel grateful that I have a much better understanding now than before this discussion, despite the inability to reach an agreement. I was being overly negative in the previous post, and apologize for that.
That is very reasonable criticism, and it remains to be shown (beyond personal observation) that my definition of “hotspot” agrees with what most people would perceive as the hotspot.
Sadly it’s hard to find/measure real-world data for unusual reflector geometries (e.g., very wide or very narrow), but this simulation (which has been around for a long time) should give some insight. For more common geometries, maybe I could try reddit to get some data to test our models on.
I absolutely agree that the conservation of etendue is a real thing. It’s just that I don’t believe every derivation must be done through etendue calculations; instead, I believe that any correct derivation must produce a result that is consistent with conservation of etendue, even without using the principle. The same way that simple physics problems can be solved without appealing to conservation of energy or the second law of thermodynamics: any correct solution–no matter how it’s found–must be consistent with them.
I chose the ray-tracing route over the etendue route for my derivations simply because the math is easier. To use etendue in full effectiveness/generality involves not just multiplying emitter area and emitting angle, but extremely ugly integration. On the other hand, dealing with reflector geometry requires little more than basic trigonometry, and some occasional root-finding.
Porting the discussion from the other thread–I remember having a couple objections to the Gaussian model, and would appreciate hearing your take.
As before–angle-space is bounded, so the Gaussian distribution is quite literally ill-defined on this space. To overcome this ill-definedness, some sort of modification like truncation (e.g., Gaussian distribution conditioned on a disk of radius 90 degrees) or projection is necessary.
I see on the site that you’re truncating at 90 degrees. That, to me, seems to introduce another issue: when the truncation is fixed, Gaussians with different variances look completely different after truncation. A very narrow Gaussian is almost unchanged, while a very wide Gaussian almost turns into the uniform distribution on the interval:
Reflector beams have very distinct regions known as hotspot, corona, and spill, and so do many TIRs. A Gaussian beam profile does not meaningfully capture these distinct regions. (I see that this is just part of the tradeoff between simplicity/generality and accuracy.)
The truncated tails are renormalised - the modeled profile will not look like above even for flooders.
Could you elaborate on how that is done, or point me to existing documentation?
The flux in the cut-off tails is redistributed into the truncated (super) Gaussian in proportion to the angular intensity so that the total lumens remain constant and accounted for.
This does not change the shape of the truncated Gaussian–it just scales the shape by a constant. For this reason, the discontinuous cutoff mentioned above remains an issue.
My plot above normalized by peak intensity purely for visualization purposes, with the understanding that the area enclosed (by the solid of revolution) should remain the same. If I normalized by area instead, one of the curves would become hard to visually distinguish from the horizontal axis, hence the choice to normalize by peak intensity instead.
For a mule with Acuity of around one, the regular truncated and renormalised gaussian model introduces a jump at the +/- 90 deg edges of less than 9% of max intensity and it rapidly decreases as either the Acuity Index or the exponent of the supergaussian increases. I think it’s super good enough for the purpose, especially that I’m mostly interested in describing the reality of the intensity distribution near the centre of the beam well enough to compare lights. The input measurements have already higher relative uncertainty than that.
