Why spot and beam uniformity are always not equal

Published by Optics for Hire.

For many LED optic projects a spot size and beam angle can be specified with a single uniformity value. But in cases of beam angles above 30 degrees things can be tricky. Please see the images below for a visual explanation.

Or if you prefer to understand the math...

Illuminance is changed by formula E=(I/r*r)*cos(A), where E - illuminance, I - luminous intensity, r - distance to spot, A - angle of ray.

usually distance r is constant. If dependence I from A is constant, we have illuminance falloff at high values of A, because cos (A) multiplier has smaller value.