The angular aperture of a

lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements ...

is the

angular size The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is ...

of the lens

aperture In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane. An ...

as seen from the focal point: :

a = 2 \arctan \left( \frac   \right) = 2 \arctan \left( \frac   \right)

where :

f

is the focal length :

D

is the

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...

of the

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. Formally, the numerical aperture in air is: :

\mathrm = \sin a/2 = \sin \arctan \left( \frac   \right)

In the

paraxial approximation In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the optical ...

, with a small aperture,

D :

: \mathrm \approx a/2

References

{{reflist

Relation to numerical aperture

References

See also