# Angular aperture

The angular aperture of a thin lens with focal point at F and an aperture of diameter $d$.

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

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

where

$f$ is the focal length
$D$ 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:

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

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

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

## References

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