Relationship Beam Tightness - Lens/LED-Die Diameter

Hi All,
I asked a similar question direct to Ra, but perhaps this might of interest in more general form:
Does anone know some (perhaps simplified) mathematic relationship between the size (diameter of LED die) of the light emitting object and the size (diameter) of an asperical lens mounted in front of it regarding divergence of the resulting beam?

I'm asking this because I'm planning to do LED based lamp with a big asperical lens (> 10cm) and want to estimate what tightness (beam divergence) can be achieved with such a setup.

Did some of you calculate what opening angle the beam of Asperic Lens equipped flashlights is?
The same would be interesting for reflector based lights.



Oh, I forgot my personal result:

17mm / FL unknown (but LED 9mm away from lens) asperical lens with CREE Q5: 7-8 degrees

55mm Dia /38mm (LED 25mm away from lens back) FL with CREE Q5: 2-3 degrees (need to measure more acurate)

Regards,

Holzleim

Oops, nearly 11 months late. I'm going to assume this is for Mr. Beambastic which you finished, but I guess this can still be used for future reference.

Wikipedia says this...

divergence = diameter (narrowest point of beam before lens, in this case the led) / focal length

I was researching this because I saw a fairly cheap 395mm x 395mm with a 220mm focal length fresnel lens on a website and I wondered how insane the throw would be. Based on the formula...