Angular aperture

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The angular aperture a {\displaystyle a} {\displaystyle a} of a thin lens with focal point at F and an aperture of diameter D {\displaystyle D} {\displaystyle D}.

The angular aperture a {\displaystyle a} {\displaystyle a} of a lens is the angular size of the lens aperture as seen from the focal point:

a = 2 arctan ⁡ ( D / 2 f ) = 2 arctan ⁡ ( D 2 f ) {\displaystyle a=2\arctan \left({\frac {D/2}{f}}\right)=2\arctan \left({\frac {D}{2f}}\right)} {\displaystyle a=2\arctan \left({\frac {D/2}{f}}\right)=2\arctan \left({\frac {D}{2f}}\right)}

where

f {\displaystyle f} {\displaystyle f} is the focal length
D {\displaystyle D} {\displaystyle 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:

N A = sin ⁡ a / 2 = sin ⁡ arctan ⁡ ( D 2 f ) {\displaystyle \mathrm {NA} =\sin a/2=\sin \arctan \left({\frac {D}{2f}}\right)} {\displaystyle \mathrm {NA} =\sin a/2=\sin \arctan \left({\frac {D}{2f}}\right)}

In the paraxial approximation, with a small aperture, D < f {\displaystyle D<f} {\displaystyle D<f}:

N A ≈ a / 2 {\displaystyle \mathrm {NA} \approx a/2} {\displaystyle \mathrm {NA} \approx a/2}

References

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

See also