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 angular size 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

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}{2f}}\right)

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

\mathrm {NA} \approx a/2

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