Editing Numerical aperture (section)

==General optics==
[[File:marginal_vs_chief_ray.svg|thumb|Simple ray diagram showing typical chief and marginal rays]]
In most areas of optics, and especially in [[microscopy]], the numerical aperture of an optical system such as an [[objective lens]] is defined by

<math display="block">\mathrm{NA} = n \sin \theta,</math>

where {{math|''n''}} is the [[Refractive index|index of refraction]] of the medium in which the lens is working (1.00 for [[air]], 1.33 for pure [[water]], and typically 1.52 for [[immersion oil]];<ref>{{cite web|title=Immersion oil and the microscope |first=John J. |last=Cargille |year=1985 |edition=2nd |url=https://cargille.com/wp-content/uploads/2018/03/Immersion_Oil_and_the_Microscope.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://cargille.com/wp-content/uploads/2018/03/Immersion_Oil_and_the_Microscope.pdf |archive-date=2022-10-09 |url-status=live |access-date=2019-10-16}}</ref> see also [[list of refractive indices]]), and {{math|''θ''}} is the [[half-angle]] of the maximum cone of light that can enter or exit the lens. In general, this is the angle of the real [[marginal ray]] in the system. Because the index of refraction is included, the {{abbr|NA|numerical aperture}} of a [[Pencil (optics)|pencil of rays]] is an invariant as a pencil of rays passes from one material to another through a flat surface. This is easily shown by rearranging [[Snell's law]] to find that {{math|''n'' sin ''θ''}} is constant across an interface; {{math|1= NA = ''n''{{sub|1}}{{tsp}}sin ''θ''{{sub|1}} = ''n''{{sub|2}}{{tsp}}sin ''θ''{{sub|2}}}}.

In air, the [[angular aperture]] of the lens, <math>a = 2 \theta</math>, is approximately twice this value (within the [[paraxial approximation]]). The {{abbr|NA|numerical aperture}} is generally measured with respect to a particular object or image point and will vary as that point is moved. In microscopy, {{abbr|NA|numerical aperture}} generally refers to object-space numerical aperture unless otherwise noted.

In microscopy, {{abbr|NA|numerical aperture}} is important because it indicates the [[Angular resolution|resolving power]] of a lens. The size of the finest detail that can be resolved (the ''resolution'') is proportional to {{math|{{sfrac|''λ''|2NA}}}}, where {{math|''λ''}} is the [[wavelength]] of the light. A lens with a larger numerical aperture will be able to visualize finer details than a lens with a smaller numerical aperture. Assuming quality ([[diffraction-limited]]) optics, lenses with larger numerical apertures collect more light and will generally provide a brighter image but will provide shallower [[depth of field]].

Numerical aperture is used to define the "pit size" in [[optical disc]] formats.<ref name="Crutchfield Advisor">[http://www.crutchfieldadvisor.com/S-UNO5yLxzuZf/learningcenter/home/hd_blu.html?page=2 "High-def Disc Update: Where things stand with HD DVD and Blu-ray"] {{Webarchive|url=https://web.archive.org/web/20080110072357/http://www.crutchfieldadvisor.com/S-UNO5yLxzuZf/learningcenter/home/hd_blu.html?page=2 |date=2008-01-10 }} by Steve Kindig, ''Crutchfield Advisor''. Accessed 2008-01-18.</ref>

Increasing the magnification and the numerical aperture of the objective reduces the working distance, i.e. the distance between front lens and specimen.

=== 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.

===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>