Editing Numerical aperture (section)

===Working (effective) f-number===
The f-number describes the light-gathering ability of the lens in the case where the marginal rays on the object side are parallel to the axis of the lens. This case is commonly encountered in photography, where objects being photographed are often far from the camera.  When the object is not distant from the lens, however, the image is no longer formed in the lens's [[focal plane]], and the f-number no longer accurately describes the light-gathering ability of the lens or the image-side numerical aperture.  In this case, the numerical aperture is related to what is sometimes called the "[[working f-number]]" or "effective f-number".

The working f-number is defined by modifying the relation above, taking into account the magnification from object to image:

<math display="block">\frac{1}{2 \text{NA}_\text{i}} = N_\text{w} = \left(1 - \frac{m}{P}\right) N, </math>

where {{math|''N''<sub>w</sub>}} is the working f-number, {{math|''m''}} is the lens's [[magnification]] for an object a particular distance away, {{math|''P''}} is the [[pupil magnification]], and the {{abbr|NA|numerical aperture}} is defined in terms of the angle of the marginal ray as before.<ref name="Greivenkamp">{{cite book |last= Greivekamp |first= John E. |date= 2004 |title= Field Guide to Geometrical Optics |publisher= SPIE |series= SPIE Field Guides |volume= FG01 |isbn= 0-8194-5294-7 |url= https://books.google.com/books?id=1YfZNWZAwCAC&pg=PA29 |page= 29 }}</ref><ref>{{cite book |last1= Arecchi |first1= Angelo V. |last2= Messadi |first2= Tahar |last3= Koshel |first3= R. John |date= 2007 |title= Field Guide to Illumination |name-list-style= amp |publisher= SPIE |isbn= 978-0-8194-6768-3 |page= 48 |url = https://books.google.com/books?id=Ax0RqdqeDG0C&q=working-f-number+aperture&pg=PP14 }}</ref>  The magnification here is typically negative, and the pupil magnification is most often assumed to be 1 — as Allen R. Greenleaf explains, "Illuminance varies inversely as the square of the distance between the exit pupil of the lens and the position of the plate or film.  Because the position of the exit pupil usually is unknown to the user of a lens, the rear conjugate focal distance is used instead; the resultant theoretical error so introduced is insignificant with most types of photographic lenses."<ref>{{cite book |last= Greenleaf |first= Allen R. |date= 1950 |title= Photographic Optics |publisher= The Macmillan Company |page= 24 |url= https://books.google.com/books?id=L4dtAAAAIAAJ&q=intitle:optics+inauthor:greenleaf+%22position+of+the+exit+pupil%22 }}</ref>

In photography, the factor is sometimes written as {{math|1 + ''m''}}, where {{math|''m''}} represents the [[absolute value]] of the magnification; in either case, the correction factor is 1 or greater. The two equalities in the equation above are each taken by various authors as the definition of working f-number, as the cited sources illustrate.  They are not necessarily both exact, but are often treated as if they are.

Conversely, the object-side numerical aperture is related to the f-number by way of the magnification (tending to zero for a distant object):

<math display="block">\frac{1}{2 \text{NA}_\text{o}} = \frac{m - P}{mP} N. </math>