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 opt ...

as seen from the

focal point Focal point may refer to: * Focus (optics) * Focus (geometry) * Conjugate points, also called focal points * Focal point (game theory) * Unicom Focal Point, a portfolio management software tool * Focal point review, a human resources process for ...

: :

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

where :

f

is the

focal length The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative foca ...

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 for ...

of the

Relation to numerical aperture

In a medium with an

index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...

close to 1, such as air, the angular aperture is approximately equal to twice the

numerical aperture In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the proper ...

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 optica ...

, with a small aperture,

D :

: \mathrm \approx a/2

References

{{reflist

Relation to numerical aperture

References

See also