I've enjoyed reading and thinking about this thread and I think I do actually have something valuable to contribute, hopefully!
Firstly, I just wanted to say I found it really useful to copy what saablaster did and used a mask on the torch to see how the torch's beam is made up of lots (infinite) different images stacked on top of each other. Half an hour of playing in the dark with this gave me a much better feel for how its working. Using this method I was actually able to figure out why my LD20 R4 has a bit of a donut in the hot spot – it seems to be caused by some parallel lines on the surface of the xp-g, possibly wires or something, which I can clearly see when using a mask like saablaster did. When you rotate the image of the die in the hotspot around, the lines mean a bit less light hits the centre of the hotspot. You can see these lines in photos of the actual xp-g itself, so it was cool to see them projected onto the wall using a mask.
Ok, now to the thing I wanted to contribute. We all know leds (more or less) emit light in a conical shaped arc out from the emitter, with a higher amount of light going out the centre of the cone with lesser intensity as you go towards the edge of the cone. There is a graph of the relatively intensities depending on angle in the xp-g datasheet on the cree website (
http://www.cree.com/products/pdf/XLampXP-G.pdf) . Originally I felt this meant that most of the light was actually going straight out the front of the torch without bouncing off the reflector, because we all know more comes out the front.
But, there are two reasons why the amount of light escaping out the front without reflecting isn't anywhere near as much as I'd originally thought.
Firstly, the measurements in Cree's graph are for
relative intensity at the different angles, so even at a significant (say 45 degrees) off centre angle the intensity is about 75% of the intensity that is going straight out the front. So less intensity, to be sure, but still a lot of light.
Secondly, the 3D cone of light that the led puts out can be divided into two sections. The first section is the unmodified cone of light emitted (ie assume no reflector). The second section is a smaller cone inside the larger one, which represents light that escapes straight out the front of the lens without bouncing off the reflector. I set out to calculate the relative areas of the curved base of these cones (the "front" of them when looking straight into the lens). The nature of these cones that means that small central cone that escapes has a much smaller base surface area compared to the overall output cone.
I worked with some rough inputs based roughly on my LD20 R4. I roughly estimated the angle of the cone from the emitter to escape without hitting the reflector to be about 40 degrees across (using a protractor and line of sight) (ie 20 degrees either side of centred).
Below is a chart of how much of the total output travels in each angular sector from the emitter. Eg the 10 degree sector is all the light coming out from the central 20 degrees (10 off centre) of the emitter. Looking in the front of the torch the sectors would be a bunch of concentric circles rippling out from the led, however note these circles are really lines on a sphere with the sphere's centre being the led. The green rows are the ones where I estimate the LD20 will let the light straight out the front instead of reflecting. If someone wants to check my maths that would be nice actually, I can send a copy of the spreadsheet
The "Cree relative %Intensity in this sector" column is from me looking at the graph on the xp-g specifications and should be accurate to within a couple of %. "Nominal units of output" is a nominal measurement of how much light goes out through that sector, based on relative intensity and the % of total emitter output area. Not 100% on my maths in that column. It doesn't matter what the units are, they are just used for relative calculations between the sectors.
Anyway using those inputs, the results are that the area of the base of the cone of light escaping out the front without reflection is only about 7% of the area of the large cone of light the emitter actually puts out. However about 12% of the total emitted light goes out through this area as the emitter output is higher straight out the front.
So end of the story is - on my LD20, only a smallish proportion of the emitted light goes out the front without being reflected first.
Hope I've made sense to some people! And that the maths is right.
Ps. The formula I used for calculating the 3d surface area of a piece of a sphere marked out by a cone coming from the centre of the sphere with a given angle is:
A = 2 * pi * r * r * (1 – cos(a)) (where r is radius of the sphere and a is the angle of the cone). Just pick any r (as long as you use the same one), then adjust "a" as you need - you're only after a ratio here not an absolute value.
Pps. I haven't taken into account the size of the emitter to simply this a lot. Its not a precise calculation, for that you'd have to do maths way over my head as there's an infinite number of cones centred on the infinite number of spots on the emitter's surface. However its fine for a rough calculation which is all I/we need.