Angular aperture

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

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

The angular aperture ${\displaystyle a}$ of a thin lens with focal point at F and an aperture of diameter ${\displaystyle D}$.

where

${\displaystyle f}$ is the focal length

${\displaystyle D}$ 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:

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

In the paraxial approximation, with a small aperture, ${\displaystyle D<f}$:

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