Thank you TEEJ, I definitely tried but was unable to find anything in those last criticisms that could be used to improve this, or even anything that was correct.
Sven, your basic question is what makes the different focal lengths create different beam profiles.
It's explained
here, but here is a more thorough explanation...
Collimation (the degree to which the photons project in the same direction) determines throw. There are two fundamental geometric components that determine collimation: the size of the source and the distance of the source from the reflector surface. To clean this up, I'll refer to that distance as STRD (Source to Reflection Distance). Smaller source size and greater STRD produce greater collimation. Refer to this most basic illustration...
The beam is a distant reflected "Image" of the "Source" (RED), enlarged at distance based on STRD. The greater the source size and the smaller the STRD, the less collimation occurs and the greater the reflected image size.
The larger image (BLUE outlines) is the result of the shorter STRD.
The smaller image (YELLOW outlines) is the result of the longer STRD.
It's a direct relation...
Double the source size equals double the image size, half the throw, and 1/4 the image intensity.
Double the STRD equals half the image size, double the throw, and quadruple the image intensity.
Likewise, double the source intensity equals double the image intensity.
The final beam is a combination of all images generated from all reflections over the reflector surface.
As this illustration is a short focal length reflector, there is much STRD variance, generating a beam of much image size variance. Hence a small central spot and a large dim corona. The bright central spot is the result of the very large STRD from the source to the very front of the aperture. Although that large distance has great collimation, only a relatively small percentage of the source light actually hits that region of the reflector. The majority of source light has a very short STRD which has very little collimation, making for a very large dim corona.
On the other hand, a long focal length reflector has much more uniform STRD over it's reflection surface, generating a beam of much more uniform size. There is less centralized intensity but less light wasted as large dim corona.
That's it in a nutshell.
On a side note, there are several other geometric factors occurring which I calculate...
-The intensity of the source is not uniform across all emittance angles. An angular intensity profile is measured and mapped for each source.
-Since the source is not perfectly spherical, the size of the facing area of the source changes at every reflection angle. For HID lights, the beam calculator uses a formula for calculating the angular facing area of an ellipsoid defined by the source width and height.
-The "Image" distance is resolved as from the aperture rather than the source.
-The aperture diameter at each reflection point has a geometric affect on the image.
-Focus offset for flood.
-Vertex hole diameter
Here is a diagram of all of the full geometry calculated for just one reflection point with offset focus:
As you can see, it takes several geometric functions and laws to evaluate for just one reflection point on the surface of the reflector. This has to be done across the entire surface of the reflector. And this diagram is without a retro-reflector. Of course there are also non-geometric factors calculated including source lumen output, reflector reflectance, and lens transmittance.
EDIT - I forgot to mention something I discovered a few days back that others might find fun.. due to the beam uniformity as a result of uniform STRD, and the direction of focus offset, a metal cutout "light shield" could be used for the Nightsword to project an image like the batman logo in the clouds by slightly defocusing the beam. I thought this would be fun.