Angular aperture: Difference between revisions

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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

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}{2f}}\right)

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

\mathrm {NA} \approx a/2

References

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

@@ Line 1: / Line 1: @@
+[[Image:Angular aperture.svg|right|The angular aperture of a [[thin lens]] with focal point at '''''F''''' and an aperture of diameter <math>D</math>.|thumb|250px]]
-{{unreferenced||date=June 2006}}
-The '''angular aperture''' of a [[lens (optics)|lens]] is the apparent [[angle]] of the lens [[aperture]] as seen from the [[Focus (optics)|focal point]]:
+The '''angular aperture''' of a [[lens (optics)|lens]] is the [[angular size]] of the lens [[aperture]] as seen from the [[Focus (optics)|focal point]]:
-:<math>a = 2 \arctan \left( \frac {A} {2 f} \right)</math>
+:<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>A</math> is the diameter of the aperture.
+:<math>D</math> is the [[diameter]] of the [[aperture]].
-The angular aperture is approximately equal to twice the [[numerical aperture]] of the lens.
+== 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.<ref>{{cite book|title=Studies in Optics |author=Albert Abraham Michelson|year= 1995|publisher=Courier Dover|pages=32|isbn=0-486-68700-7|url= https://books.google.com/books?id=m2tUZ4_8WGMC&q=%22Angular+aperture%22&pg=PA32}}</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>
+==References==
+{{reflist}}
 ==See also==
 *[[f-number]]
 *[[Numerical aperture]]
+*Acceptance angle, half the angular aperture
+*[[Field of view]]
-[[Category:Optics]]
+[[Category:Geometrical optics]]
 [[Category:Angle]]
-[[es:Apertura angular]]