Angular aperture

The angular aperture <math>a</math> of a thin lens with focal point at F and an aperture of diameter <math>D</math>.

The angular aperture <math>a</math> of a lens is the angular size of the lens aperture as seen from the focal point:

<math>a = 2 \arctan \left( \frac {D/2} {f} \right) = 2 \arctan \left( \frac {D} {2f} \right)</math>

where

<math>f</math> is the focal length

<math>D</math> is the diameter of the aperture.

Relation to numerical apertureEdit

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.<ref>Template:Cite book</ref>

Formally, the numerical aperture in air is:

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

In the paraxial approximation, with a small aperture, <math>D<f</math>:

<math>\mathrm{NA} \approx a/2</math>