Editing Numerical aperture (section)

=== Numerical aperture versus f-number ===
[[Image:Numerical aperture for a lens.svg|thumb|250px|right|For a [[thin lens]] here, the numerical aperture is <math>\text{NA} = n\sin \theta</math> and the angular aperture is <math>a = 2 \theta</math>. For a thick lens, these parameters are with respect to a [[Cardinal point (optics)#Principal planes and points|principal plane]]; the front principal plane if the parameters are for light gathering capability, and the rear principal plane if for light focusing capability.]]

Numerical aperture is not typically used in [[photography]]. Instead, the angular aperture <math>a = 2 \theta</math> of a [[photographic lens|lens]] (or an imaging mirror) is expressed by the [[f-number]], written {{f/|''N''}}, where {{mvar|N}} is the f-number given by the ratio of the [[focal length]] {{mvar|f}} to the diameter of the [[entrance pupil]] {{math|''D''}}:

<math display="block">N = \frac{f}{D}.</math>

This ratio is related to the image-space numerical aperture when the lens is focused at infinity.<ref name="Greivenkamp"/>  Based on the diagram at the right, the image-space numerical aperture of the lens is:

<math display="block">\text{NA}_\text{i} = n \sin \theta = n \sin \left[ \arctan \left( \frac{D}{2f} \right) \right] \approx n \frac{D}{2f},</math>

thus {{math|1= ''N'' ≈ {{sfrac|1|2NA<sub>i</sub>}}}}, assuming normal use in air ({{math|''n'' {{=}} 1}}).

The approximation holds when the numerical aperture is small, but it turns out that for well-corrected optical systems such as camera lenses, a more detailed analysis shows that {{math|''N''}} is almost exactly equal to {{math|1/(2NA<sub>i</sub>)}} even at large numerical apertures. As Rudolf Kingslake explains, "It is a common error to suppose that the ratio [{{math|''D''/2''f''}}] is actually equal to {{math|tan ''θ''}}, and not {{math|sin ''θ''}} ...  The tangent would, of course, be correct if the principal planes were really plane.  However, the complete theory of the [[Abbe sine condition]] shows that if a lens is corrected for [[Coma (optics)|coma]] and [[spherical aberration]], as all good photographic objectives must be, the second principal plane becomes a portion of a sphere of radius {{mvar|f}} centered about the focal point".<ref>{{cite book | title = Lenses in photography: the practical guide to optics for photographers | author = Rudolf Kingslake | publisher = Case-Hoyt, for Garden City Books | year = 1951 | pages = 97–98}}</ref> In this sense, the traditional thin-lens definition and illustration of f-number is misleading, and defining it in terms of numerical aperture may be more meaningful.