Angular aperture

The angular aperture $a$ of a lens is the angular size of the lens aperture as seen from the focal point:

a = 2 \arctan (\frac{D / 2}{f}) = 2 \arctan (\frac{D}{2 f})

where

f

D

is the diameter of the aperture.

Relation to numerical aperture

In a medium with an index of refraction close to 1, such as air, the angular aperture is approximately equal to twice the numerical aperture of the lens.^[1]

Formally, the numerical aperture in air is:

N A = \sin a / 2 = \sin \arctan (\frac{D}{2 f})

In the paraxial approximation, with a small aperture, $D < f$ :

N A \approx a / 2

↑ Albert Abraham Michelson (1995). Studies in Optics. Courier Dover. p. 32. ISBN 0-486-68700-7.