LED collimating and beam diameter/divergence

Hey all,

I want to make an LED flashlight (luxeon rebel 4.5x3mm) that can produce an 80mm diameter beam from about 450cm away using a 20mm lens.

From what i've read, I can use a plano-convex lens to collimate light from a light source. If I use a 20mm lens, a perfect light point at the lens's focal length would produce a 20mm diameter beam correct? However, since it's not a point, the light would slightly diverge. I wanted to use that to my advantage to create a beam as mentioned above.

I read you can mess with the focal length as well as placing the LED in away from the focal length in order to produce a larger beam diameter at a certain distance away, but I couldn't figure out the calculations. I was wondering if anyone can help me figure this out.

Thanks so much!

One parameter I'm not seeing is the effective size of the emitter. For tightest focus the lens would be projecting an image of the emitter to the desired size at distance.
(object_size/distance_1) = (image_size/distance_2)
There's also a relation between those distances and the lens focal length that I'd have to look up.