Angular aperture

The angular aperture of a lens is the apparent angle of the lens aperture as seen from the focal point:

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

where

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:

$\mathrm{NA} = \sin a/2 = \sin \arctan \left( \frac {D} {2 f} \right)$

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

$\mathrm{NA} \approx a/2$