View source for F-number

{{Short description|Measure of lens speed}}
{{other uses}}
{{lowercase title}} 
{{Use dmy dates|date=January 2020||cs1-dates=y}}
[[File:Aperture diagram.svg|right|thumb|320px|Diagram of decreasing [[aperture]]s, that is, increasing f-numbers, in one-stop increments; each aperture has half the light-gathering area of the previous one.]]

An '''f-number''' is a measure of the light-gathering ability of an optical system such as a [[camera lens]]. It is calculated by dividing the system's [[focal length]] by the diameter of the [[entrance pupil]] ("clear [[aperture]]").<ref name="ReferenceA">Smith, Warren ''Modern Optical Engineering'', 4th Ed., 2007 McGraw-Hill Professional, p. 183.</ref><ref>{{cite book |first=Eugene |last=Hecht |year=1987 |title=Optics |edition=2nd |publisher=Addison Wesley |isbn=0-201-11609-X |page=152}}</ref><ref>{{cite book |first=John E. |last=Greivenkamp |year=2004 |title=Field Guide to Geometrical Optics |publisher= [[SPIE#SPIE Press|SPIE]] |location=Bellingham, Wash|series= SPIE Field Guides vol. FG01 |isbn= 9780819452948 |oclc= 53896720 |page=29}}</ref> The f-number is also known as the '''focal ratio''', '''f-ratio''', or '''f-stop''', and it is key in determining the [[depth of field]], [[diffraction]], and [[Exposure (photography)|exposure]] of a photograph.<ref>Smith, Warren ''Modern Lens Design'' 2005 McGraw-Hill.</ref> The f-number is [[dimensionless number|dimensionless]] and is usually expressed using a lower-case [[Ƒ|hooked f]] with the format {{f/}}''N'', where ''N'' is the f-number.

The f-number is also known as the '''inverse relative aperture''', because it is the [[Multiplicative inverse|inverse]] of the '''relative aperture''', defined as the aperture diameter divided by focal length.<ref>ISO, Photography—Apertures and related properties pertaining to photographic lenses—Designations and measurements, ISO 517:2008</ref> The relative aperture indicates how much light can pass through the lens at a given focal length. A lower f-number means a larger relative aperture and more light entering the system, while a higher f-number means a smaller relative aperture and less light entering the system. The f-number is related to the [[numerical aperture]] (NA) of the system, which measures the range of angles over which light can enter or exit the system. The numerical aperture takes into account the [[refractive index]] of the medium in which the system is working, while the f-number does not.

== Notation ==

The f-number {{mvar|N}} is given by:

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

where {{mvar|f}} is the [[focal length]], and {{mvar|D}} is the diameter of the entrance pupil (''effective aperture''). It is customary to write f-numbers preceded by "{{f/}}", which forms a mathematical expression of the entrance pupil's diameter in terms of {{mvar|f}} and {{mvar|N}}.<ref name="ReferenceA"/> For example, if a [[Lens (optics)|lens's]] focal length were {{val|100|u=mm}} and its entrance pupil's diameter were {{val|50|u=mm}}, the f-number would be 2. This would be expressed as {{nowrap|"{{f/|2}}"}} in a lens system. The aperture diameter would be equal to {{math|''f''/2}}.

Camera lenses often include an adjustable [[diaphragm (optics)|diaphragm]], which changes the size of the [[aperture stop]] and thus the entrance pupil size. This allows the user to vary the f-number as needed. The entrance pupil diameter is not necessarily equal to the aperture stop diameter, because of the magnifying effect of lens elements in front of the aperture.

Ignoring differences in light transmission efficiency, a lens with a greater f-number projects darker images. The brightness of the projected image ([[illuminance]]) relative to the brightness of the scene in the lens's field of view ([[luminance]]) decreases with the square of the f-number. A {{val|100|u=mm}} focal length {{f/|4}} lens has an entrance pupil diameter of {{val|25|u=mm}}. A {{val|100|u=mm}} focal length {{f/|2}} lens has an entrance pupil diameter of {{val|50|u=mm}}. Since the area is proportional to the square of the pupil diameter,<ref>See [[Area of a circle]].</ref> the amount of light admitted by the {{f/|2}} lens is four times that of the {{f/|4}} lens. To obtain the same [[Exposure (photography)|photographic exposure]], the exposure time must be reduced by a factor of four.

A {{val|200|u=mm}} focal length {{f/|4}} lens has an entrance pupil diameter of {{val|50|u=mm}}. The {{val|200|u=mm}} lens's entrance pupil has four times the area of the {{val|100|u=mm}} {{f/|4}} lens's entrance pupil, and thus collects four times as much light from each object in the lens's field of view. But compared to the {{val|100|u=mm}} lens, the {{val|200|u=mm}} lens projects an image of each object twice as high and twice as wide, covering four times the area, and so both lenses produce the same illuminance at the focal plane when imaging a scene of a given luminance.

== Stops, f-stop conventions, and exposure ==
[[Image:Canon 7 with 50mm f0.95 IMG 0374.JPG|thumb|A [[Canon 7]] mounted with a {{val|50|u=mm}} lens capable of {{f/|0.95}}]]
[[Image:lens aperture side.jpg|thumb|A {{val|35|u=mm}} lens set to {{f/|11}}, as indicated by the white dot above the f-stop scale on the aperture ring. This lens has an aperture range of {{f/|2}} to {{f/|22}}.]]

The word ''stop'' is sometimes confusing due to its multiple meanings. A stop can be a physical object: an opaque part of an optical system that blocks certain rays. The ''[[aperture stop]]'' is the aperture setting that limits the brightness of the image by restricting the input pupil size, while a ''field stop'' is a stop intended to cut out light that would be outside the desired field of view and might cause flare or other problems if not stopped.

In photography, stops are also a ''unit'' used to quantify ratios of light or exposure, with each added stop meaning a factor of two, and each subtracted stop meaning a factor of one-half. The one-stop unit is also known as the EV ([[exposure value]]) unit. On a camera, the aperture setting is traditionally adjusted in discrete steps, known as '''''f-stops'''''. Each "'''stop'''" is marked with its corresponding f-number, and represents a halving of the light intensity from the previous stop. This corresponds to a decrease of the pupil and aperture diameters by a factor of 1/{{sqrt|2}} or about 0.7071, and hence a halving of the area of the pupil.

Most modern lenses use a standard f-stop scale, which is an approximately [[geometric sequence]] of numbers that corresponds to the sequence of the [[exponentiation|powers]] of the [[square root of 2]]:  {{f/|1}}, {{f/|1.4}}, {{f/|2}}, {{f/|2.8}}, {{f/|4}}, {{f/|5.6}}, {{f/|8}}, {{f/|11}}, {{f/|16}}, {{f/|22}}, {{f/|32}}, {{f/|45}}, {{f/|64}}, {{f/|90}}, {{f/|128}}, etc. Each element in the sequence is one stop lower than the element to its left, and one stop higher than the element to its right. The values of the ratios are rounded off to these particular conventional numbers, to make them easier to remember and write down.
The sequence above is obtained by approximating the following exact geometric sequence:

<math display=block>f/1 = \frac{f}{(\sqrt{2})^0},\ f/1.4 = \frac{f}{(\sqrt{2})^1},\ f/2 = \frac{f}{(\sqrt{2})^2},\ f/2.8 = \frac{f}{(\sqrt{2})^3},\ \ldots</math>
In the same way as one f-stop corresponds to a factor of two in light intensity, [[shutter speed]]s are arranged so that each setting differs in duration by a factor of approximately two from its neighbour. Opening up a lens by one stop allows twice as much light to fall on the film in a given period of time. Therefore, to have the same exposure at this larger aperture as at the previous aperture, the shutter would be opened for half as long (i.e., twice the speed). The film will respond equally to these equal amounts of light, since it has the property of ''[[Reciprocity (photography)|reciprocity]]''. This is less true for extremely long or short exposures, where there is [[reciprocity failure]]. Aperture, shutter speed, and film sensitivity are linked: for constant scene brightness, doubling the aperture area (one stop), halving the shutter speed (doubling the time open), or using a film twice as sensitive, has the same effect on the exposed image. For all practical purposes extreme accuracy is not required (mechanical shutter speeds were notoriously inaccurate as wear and lubrication varied, with no effect on exposure). It is not significant that aperture areas and shutter speeds do not vary by a factor of precisely two.

Photographers sometimes express other [[Exposure (photography)|exposure]] ratios in terms of 'stops'. Ignoring the f-number markings, the f-stops make a [[logarithmic scale]] of exposure intensity. Given this interpretation, one can then think of taking a half-step along this scale, to make an exposure difference of a "half stop".

=== Fractional stops ===
{{multiple image
 | width = 120
 | image1 = Povray focal blur animation.gif
 | alt1 = Changing a camera's aperture in half-stops
 | image2 = Povray focal blur animation mode tan.gif
 | alt2 = Changing a camera's aperture from zero to infinity
 | footer = Computer simulation showing the effects of changing a camera's aperture in half-stops (at left) and from zero to infinity (at right)
}}

Most twentieth-century cameras had a continuously variable aperture, using an [[iris diaphragm]], with each full stop marked. Click-stopped aperture came into common use in the 1960s; the aperture scale usually had a click stop at every whole and half stop.

On modern cameras, especially when aperture is set on the camera body, f-number is often divided more finely than steps of one stop. Steps of one-third stop ({{1/3}} EV) are the most common, since this matches the ISO system of [[film speed]]s. Half-stop steps are used on some cameras. Usually the full stops are marked, and the intermediate positions click but are not marked. As an example, the aperture that is one-third stop smaller than {{f/|2.8}} is {{f/|3.2}}, two-thirds smaller is {{f/|3.5}}, and one whole stop smaller is {{f/|4}}. The next few f-stops in this sequence are:

<math display=block>f/4.5,\ f/5,\ f/5.6,\ f/6.3,\ f/7.1,\ f/8,\ \ldots</math>

To calculate the steps in a full stop (1 EV) one could use

<math display=block>(\sqrt{2})^{0},\ (\sqrt{2})^{1},\ (\sqrt{2})^{2},\ (\sqrt{2})^{3},\ (\sqrt{2})^{4},\ \ldots</math>

The steps in a half stop ({{1/2}} EV) series would be

<math display=block>(\sqrt{2})^{\frac{0}{2}},\ (\sqrt{2})^{\frac{1}{2}},\ (\sqrt{2})^{\frac{2}{2}},\ (\sqrt{2})^{\frac{3}{2}},\ (\sqrt{2})^{\frac{4}{2}},\ \ldots</math>

The steps in a third stop ({{1/3}} EV) series would be

<math display=block>(\sqrt{2})^{\frac{0}{3}},\ (\sqrt{2})^{\frac{1}{3}},\ (\sqrt{2})^{\frac{2}{3}},\ (\sqrt{2})^{\frac{3}{3}},\ (\sqrt{2})^{\frac{4}{3}},\ \ldots</math>

As in the earlier DIN and ASA film-speed standards, the ISO speed is defined only in one-third stop increments, and shutter speeds of digital cameras are commonly on the same scale in reciprocal seconds. A portion of the ISO range is the sequence

<math display=block>\ldots 16/13^\circ,\ 20/14^\circ,\ 25/15^\circ,\ 32/16^\circ,\ 40/17^\circ,\ 50/18^\circ,\ 64/19^\circ,\ 80/20^\circ,\ 100/21^\circ,\ 125/22^\circ,\ \ldots</math>

while shutter speeds in reciprocal seconds have a few conventional differences in their numbers ({{frac|15}}, {{frac|30}}, and {{frac|60}} second instead of {{frac|16}}, {{frac|32}}, and {{frac|64}}).

In practice the maximum aperture of a lens is often not an [[integer|integral]] power of {{sqrt|2}} (i.e., {{sqrt|2}} to the power of a whole number), in which case it is usually a half or third stop above or below an integral power of {{sqrt|2}}.

Modern electronically controlled interchangeable lenses, such as those used for SLR cameras, have f-stops specified internally in {{frac|8}}-stop increments, so the cameras' {{1/3}}-stop settings are approximated by the nearest {{frac|8}}-stop setting in the lens.{{citation needed|date=December 2021}}

==== Standard full-stop f-number scale ====
Including [[APEX system|aperture value]] AV:
<math display=block>N = \sqrt{2^{\text{AV}}}</math>

Conventional and calculated f-numbers, full-stop series:
{|class="wikitable" style="text-align:center"
! scope="row" | AV
| −2 || −1 || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16
|-bgcolor="#CCFFCD"
! scope="row" | ''N''
| 0.5 || 0.7 || 1.0 || 1.4 || 2 || 2.8 || 4 || 5.6 || 8 || 11 || 16 || 22 || 32 || 45 || 64 || 90 || 128 || 180 || 256
|-
! scope="row" | calculated
| 0.5 || 0.707... || 1.0 || 1.414... || 2.0 || 2.828... || 4.0 || 5.657... || 8.0 || 11.31... || 16.0 || 22.62... || 32.0 || 45.25... || 64.0 || 90.51... || 128.0 || 181.02... || 256.0
|}

==== Typical one-half-stop f-number scale ====
{|class="wikitable" style="text-align:center"
! scope="row" | AV
| −1 || −{{frac|2}} || 0 || {{frac|2}} || 1 || {{frac|1|1|2}} || 2 || {{frac|2|1|2}} || 3 || {{frac|3|1|2}} || 4 || {{frac|4|1|2}} || 5 || {{frac|5|1|2}} || 6 || {{frac|6|1|2}} || 7 || {{frac|7|1|2}} || 8 || {{frac|8|1|2}} || 9 || {{frac|9|1|2}} || 10 || {{frac|10|1|2}} || 11 || {{frac|11|1|2}} || 12 || {{frac|12|1|2}} || 13 || {{frac|13|1|2}} || 14
|-bgcolor="#FFFFCC"
! scope="row" | ''N''
|style="background:#CCFFCC;"| 0.7 || 0.8 ||style="background:#CCFFCC;"| 1.0 || 1.2 ||style="background:#CCFFCC;"| 1.4 || 1.7 ||style="background:#CCFFCC;"| 2 || 2.4 ||style="background:#CCFFCC;"| 2.8 || 3.3 ||style="background:#CCFFCC;"| 4 || 4.8 ||style="background:#CCFFCC;"| 5.6 || 6.7 ||style="background:#CCFFCC;"| 8 || 9.5 ||style="background:#CCFFCC;"| 11 || 13 ||style="background:#CCFFCC;"| 16 || 19 ||style="background:#CCFFCC;"| 22 || 27 ||style="background:#CCFFCC;"| 32 || 38 ||style="background:#CCFFCC;"| 45 || 54 ||style="background:#CCFFCC;"| 64 || 76 ||style="background:#CCFFCC;"| 90 || 107 ||style="background:#CCFFCC;"| 128
|}

==== Typical one-third-stop f-number scale ====
{|class="wikitable" style="text-align:center"
! scope="row" | AV
| −1 || −{{frac|2|3}} || −{{frac|3}} || 0 || {{frac|3}} || {{frac|2|3}} || 1 || {{frac|1|1|3}} || {{frac|1|2|3}} || 2 || {{frac|2|1|3}} || {{frac|2|2|3}} || 3 || {{frac|3|1|3}} || {{frac|3|2|3}} || 4 || {{frac|4|1|3}} || {{frac|4|2|3}} || 5 || {{frac|5|1|3}} || {{frac|5|2|3}} || 6 || {{frac|6|1|3}} || {{frac|6|2|3}} || 7 || {{frac|7|1|3}} || {{frac|7|2|3}} || 8 || {{frac|8|1|3}} || {{frac|8|2|3}} || 9 || {{frac|9|1|3}} || {{frac|9|2|3}} || 10 || {{frac|10|1|3}} || {{frac|10|2|3}} || 11 || {{frac|11|1|3}} || {{frac|11|2|3}} || 12 || {{frac|12|1|3}} || {{frac|12|2|3}} || 13
|-bgcolor="#e5d1cb"
! scope="row" | ''N''
|style="background:#CCFFCC;"| 0.7 || 0.8 || 0.9 ||style="background:#CCFFCC;"| 1.0 || 1.1 || 1.2 ||style="background:#CCFFCC;"| 1.4 || 1.6 || 1.8 ||style="background:#CCFFCC;"| 2 || 2.2 || 2.5 ||style="background:#CCFFCC;"| 2.8 || 3.2 || 3.5 ||style="background:#CCFFCC;"| 4 || 4.5 || 5.0 ||style="background:#CCFFCC;"| 5.6 || 6.3 || 7.1 ||style="background:#CCFFCC;"| 8 || 9 || 10 || style="background:#CCFFCC;"|11 || 13 || 14 ||style="background:#CCFFCC;"| 16 || 18 || 20 ||style="background:#CCFFCC;"| 22 || 25 || 29 ||style="background:#CCFFCC;"| 32 || 36 || 40 ||style="background:#CCFFCC;"| 45 || 51 || 57 ||style="background:#CCFFCC;"| 64 || 72 || 80 ||style="background:#CCFFCC;"| 90
|}

Sometimes the same number is included on several scales; for example, an aperture of {{f/|1.2}} may be used in either a half-stop<ref>
{{cite book
 | url = https://books.google.com/books?id=YjAzP4i1oFcC&pg=PA136
 | title = Set lighting technician's handbook: film lighting equipment, practice, and electrical distribution
 | author = Harry C. Box
 | edition = 3rd
 | publisher = Focal Press
 | year = 2003
 | isbn = 978-0-240-80495-8
 }}</ref>
or a one-third-stop system;<ref>
{{cite book
 | url = https://books.google.com/books?id=DvYMl-s1_9YC&pg=PA19
 | title = Underwater photography
 | author = Paul Kay
 | publisher = Guild of Master Craftsman
 | year = 2003
 | isbn = 978-1-86108-322-7
 }}</ref>
sometimes {{f/|1.3}} and {{f/|3.2}} and other differences are used for the one-third stop scale.<ref>
{{cite book
 | url = https://books.google.com/books?id=IWkpoJKM_ucC&pg=PA145
 | title = Manual for cinematographers
 | author = David W. Samuelson
 | edition = 2nd
 | publisher = Focal Press
 | year = 1998
 | isbn = 978-0-240-51480-2
 }}</ref>

==== Typical one-quarter-stop f-number scale ====
{|class="wikitable" style="text-align:center"
! scope="row" | AV
| 0 || {{frac|4}} || {{frac|2}} || {{frac|3|4}} || 1 || {{frac|1|1|4}} || {{frac|1|1|2}} || {{frac|1|3|4}} || 2 || {{frac|2|1|4}} || {{frac|2|1|2}} || {{frac|2|3|4}} || 3 || {{frac|3|1|4}} || {{frac|3|1|2}} || {{frac|3|3|4}} || 4 || {{frac|4|1|4}} || {{frac|4|1|2}} || {{frac|4|3|4}} || 5 
|-bgcolor="#5D8AA8"
! scope="row" | ''N''
|style="background:#CCFFCC;"| 1.0 || 1.1 ||style="background:#FFFFCC;"| 1.2 || 1.3 ||style="background:#CCFFCC;"| 1.4 || 1.5 ||style="background:#FFFFCC;"| 1.7 || 1.8 ||style="background:#CCFFCC;"| 2 || 2.2 ||style="background:#FFFFCC;"| 2.4 || 2.6 ||style="background:#CCFFCC;"| 2.8 || 3.1 ||style="background:#FFFFCC;"| 3.3 || 3.7 ||style="background:#CCFFCC;"| 4 || 4.4 ||style="background:#FFFFCC;"| 4.8 || 5.2 ||style="background:#CCFFCC;"| 5.6
|}
{|class="wikitable" style="text-align:center"
! scope="row" | AV
| 5 || {{frac|5|1|4}} || {{frac|5|1|2}} || {{frac|5|3|4}} || 6 || {{frac|6|1|4}} || {{frac|6|1|2}} || {{frac|6|3|4}} || 7 || {{frac|7|1|4}} || {{frac|7|1|2}} || {{frac|7|3|4}} || 8 || {{frac|8|1|4}} || {{frac|8|1|2}} || {{frac|8|3|4}} || 9 || {{frac|9|1|4}} || {{frac|9|1|2}} || {{frac|9|3|4}} || 10
|-bgcolor="#5D8AA8"
! scope="row" | ''N''
|style="background:#CCFFCC;"| 5.6 || 6.2 ||style="background:#FFFFCC;"| 6.7 || 7.3 ||style="background:#CCFFCC;"| 8 || 8.7 ||style="background:#FFFFCC;"| 9.5 || 10 ||style="background:#CCFFCC;"| 11 || 12 ||style="background:#FFFFCC;"| 14 || 15 ||style="background:#CCFFCC;"| 16 || 17 ||style="background:#FFFFCC;"| 19 || 21 ||style="background:#CCFFCC;"| 22 || 25 ||style="background:#FFFFCC;"| 27 || 29 ||style="background:#CCFFCC;"| 32
|}

=== H-stop ===
<!-- This section header is used in redirects -->
An '''H-stop''' (for hole, by convention written with capital letter H) is an f-number equivalent for effective exposure based on the area covered by the holes in the [[diffusion disc]]s or [[sieve aperture]] found in [[Rodenstock Imagon]] lenses.

=== T-stop ===
<!-- This section header is used in redirects -->
A '''T-stop''' (for transmission stops, by convention written with capital letter T) is an f-number adjusted to account for light transmission efficiency (''[[transmittance]]''). A lens with a T-stop of {{mvar|N}} projects an image of the same brightness as an ideal lens with 100% transmittance and an f-number of {{mvar|N}}. A particular lens's T-stop, {{mvar|T}}, is given by dividing the f-number by the square root of the transmittance of that lens:
<math display=block>T = \frac{N}{\sqrt{\text{transmittance}}}.</math>
For example, an {{f/|2.0}} lens with transmittance of 75% has a T-stop of 2.3:
<math display=block>T = \frac{2.0}{\sqrt{0.75}} = 2.309...</math>
Since real lenses have transmittances of less than 100%, a lens's T-stop number is always greater than its f-number.<ref>[https://www.dxomark.com/glossary/transmission-light-transmission/ Transmission, light transmission] {{Webarchive|url=https://web.archive.org/web/20210508111318/https://www.dxomark.com/glossary/transmission-light-transmission/ |date=2021-05-08  }}, DxOMark</ref>

With 8% loss per air-glass surface on lenses without coating, [[History of photographic lens design#Anti-reflection coating|multicoating]] of lenses is the key in lens design to decrease transmittance losses of lenses. Some reviews of lenses do measure the T-stop or transmission rate in their benchmarks.<ref>[https://www.dxomark.com/sigma-85mm-f1-4-art-lens-review-new-benchmark/ Sigma 85mm F1.4 Art lens review: New benchmark] {{Webarchive|url=https://web.archive.org/web/20180104073126/https://www.dxomark.com/sigma-85mm-f1-4-art-lens-review-new-benchmark/ |date=2018-01-04  }}, DxOMark</ref><ref>[https://www.lenstip.com/129.1-article-Colour_rendering_in_binoculars_and_lenses.html Colour rendering in binoculars and lenses - Colours and transmission] {{Webarchive|url=https://web.archive.org/web/20180104013937/https://www.lenstip.com/129.1-article-Colour_rendering_in_binoculars_and_lenses.html |date=2018-01-04  }}, LensTip.com</ref> T-stops are sometimes used instead of f-numbers to more accurately determine exposure, particularly when using external [[light meter]]s.<ref name=KMPCF>{{cite web |publisher=[[Eastman Kodak]] |url=http://www.kodak.com/US/en/motion/support/h2/intro01P.shtml |title=Kodak Motion Picture Camera Films |date= November 2000 |access-date=2 September 2007 |archive-url=https://web.archive.org/web/20021002095739/http://www.kodak.com/US/en/motion/support/h2/intro01P.shtml |archive-date=2002-10-02}}</ref> Lens transmittances of 60%–95% are typical.<ref>{{Cite web |url=http://forums.dpreview.com/forums/post/33785655 |title=Marianne Oelund, "Lens T-stops", dpreview.com, 2009 |access-date=2013-01-11  |archive-date=2012-11-10  |archive-url=https://web.archive.org/web/20121110221724/http://forums.dpreview.com/forums/post/33785655 |url-status=live }}</ref> T-stops are often used in cinematography, where many images are seen in rapid succession and even small changes in exposure will be noticeable. Cinema camera lenses are typically calibrated in T-stops instead of f-numbers.<ref name=KMPCF/> In still photography, without the need for rigorous consistency of all lenses and cameras used, slight differences in exposure are less important; however, T-stops are still used in some kinds of special-purpose lenses such as [[Smooth Trans Focus]] lenses by [[Minolta]] and [[Sony]].

=== ASA/ISO numbers ===
[[Photographic film]]'s and electronic camera sensor's [[photosensitivity|sensitivity to light]] is often specified using [[Film speed|ASA/ISO numbers]]. Both systems have a linear number where a doubling of sensitivity is represented by a doubling of the number, and a logarithmic number. In the ISO system, a 3° increase in the logarithmic number corresponds approximately to a doubling of sensitivity. Doubling or halving the sensitivity is equal to a difference of one T-stop in terms of light transmittance.

=== Gain ===
[[File:Panasonic_Iris-Gain_relationship.png|thumb|300px|right|Iris/gain relationship on Panasonic camcorders as described in the HC-V785 operating manual]]
Most electronic cameras allow the user to adjust the amplification of the signal coming from the image sensor. This amplification is usually called '''[[Gain (electronics)|gain]]''' and is measured in decibels. A {{val|6|u=dB}} of gain is roughly equivalent to one T-stop in terms of light transmittance. Many camcorders have a unified control over the lens f-number and gain. In this case, starting from (arbitrarily defined) zero gain and a fully open iris, one can either increase the f-number by reducing the iris size while gain remains zero, or increase the gain while the iris remains fully open.

=== Sunny 16 rule ===
An example of the use of f-numbers in photography is the ''[[sunny 16 rule]]'': an approximately correct exposure will be obtained on a sunny day by using an aperture of {{f/|16}} and the shutter speed closest to the [[Multiplicative inverse|reciprocal]] of the ISO speed of the film; for example, using ISO 200 film, an aperture of {{f/|16}} and a shutter speed of {{frac|200}} second. The f-number may then be adjusted downwards for situations with lower light. Selecting a lower f-number is "opening up" the lens. Selecting a higher f-number is "closing" or "stopping down" the lens.

== Effects on image sharpness ==
[[Image:Jonquil flowers merged.jpg|thumb|400px|Comparison of {{f/|32}} (top-left half) and {{f/|5}} (bottom-right half)]]
[[File:Blumen im Sommer.jpg|thumb|400px|Shallow focus with a wide open lens]]

[[Depth of field]] increases with f-number, as illustrated in the image here. This means that photographs taken with a low f-number (large aperture) will tend to have subjects at one distance in focus, with the rest of the image (nearer and farther elements) out of focus. This is frequently used for [[nature photography]] and [[portrait photography|portraiture]] because background blur (the aesthetic quality known as '[[bokeh]]') can be aesthetically pleasing and puts the viewer's focus on the main subject in the foreground. The [[depth of field]] of an image produced at a given f-number is dependent on other parameters as well, including the [[focal length]], the subject distance, and the [[film format|format]] of the film or sensor used to capture the image. Depth of field can be described as depending on just angle of view, subject distance, and [[entrance pupil]] diameter (as in [[Moritz von Rohr|von Rohr's method]]). As a result, smaller formats will have a deeper field than larger formats at the same f-number for the same distance of focus and same [[angle of view]] since a smaller format requires a shorter focal length (wider angle lens) to produce the same angle of view, and depth of field increases with shorter focal lengths. Therefore, reduced–depth-of-field effects will require smaller f-numbers (and thus potentially more difficult or complex optics) when using small-format cameras than when using larger-format cameras.

Beyond focus, image sharpness is related to f-number through two different optical effects: [[Optical aberration|aberration]], due to imperfect lens design, and [[diffraction]] which is due to the wave nature of light.<ref>{{cite book | title = Basic Photography | author = Michael John Langford | isbn = 0-240-51592-7 | year = 2000 | publisher = [[Focal Press]] | url-access = registration | url = https://archive.org/details/basicphotography00lang }}</ref> The blur-optimal f-stop varies with the lens design. For modern standard lenses having six or seven elements, the [[sharpness (visual)|sharpest]] image is often obtained around {{f/|5.6}}–{{f/|8}}, while for older standard lenses having only four elements ([[Zeiss Tessar|Tessar formula]]) stopping to {{f/|11}} will give the sharpest image.{{citation needed|date=July 2015}} The larger number of elements in modern lenses allow the designer to compensate for aberrations, allowing the lens to give better pictures at lower f-numbers. At small apertures, depth of field and aberrations are improved, but [[diffraction]] creates more spreading of the light, causing blur.

Light falloff is also sensitive to f-stop. Many wide-angle lenses will show a significant light falloff ([[vignetting]]) at the edges for large apertures.

[[Photojournalist]]s have a saying, "[[f/8 and be there|{{f/|8}} and be there]]", meaning that being on the scene is more important than worrying about technical details. Practically, {{f/|8}} (in 35&nbsp;mm and larger formats) allows adequate depth of field and sufficient lens speed for a decent base exposure in most daylight situations.<ref>{{cite book|last=Levy|first=Michael|title=Selecting and Using Classic Cameras: A User's Guide to Evaluating Features, Condition & Usability of Classic Cameras|publisher=Amherst Media, Inc|year=2001|page=163|isbn=978-1-58428-054-5}}</ref>

== Human eye ==
[[File:Pupillary light reflex.jpg|thumb|382x382px|The human pupil in its constricted (3&nbsp;mm) and fully dilated (9&nbsp;mm) states. At 9&nbsp;mm, the effective f-number is approximately {{f/|1.6}}.]]
Computing the f-number of the [[human eye]] involves computing the physical aperture and focal length of the eye. Typically, the pupil can dilate to be as large as 6–7&nbsp;mm in darkness, which translates into the maximal physical aperture. Some individuals' pupils can dilate to over 9&nbsp;mm wide.

The f-number of the human eye varies from about {{f/|8.3}} in a very brightly lit place to about {{f/|2.1}} in the dark.<ref>{{cite book | first=Eugene|last=Hecht|year=1987|title=Optics|edition=2nd|publisher=[[Addison Wesley]]|isbn=0-201-11609-X}} Sect. 5.7.1</ref> Computing the focal length requires that the light-refracting properties of the liquids in the eye be taken into account. Treating the eye as an ordinary air-filled camera and lens results in an incorrect focal length and f-number.

== {{anchor|Focal ratio}}Focal ratio in telescopes ==
[[Image:Focal ratio.svg|right|thumb|250px|Diagram of the [[focal ratio]] of a simple optical system where <math>f</math> is the [[focal length]] and <math>D</math> is the diameter of the [[Objective (optics)|objective]]]]

In astronomy, the f-number is commonly referred to as the ''focal ratio'' (or ''f-ratio'') notated as <math>N</math>. It is still defined as the [[focal length]] <math>f</math> of an [[Objective (optics)|objective]] divided by its diameter <math>D</math> or by the diameter of an [[aperture]] stop in the system:

<math display=block>N = \frac fD \quad \xrightarrow {\times D} \quad f = ND</math>

Even though the principles of focal ratio are always the same, the application to which the principle is put can differ. In [[photography]] the focal ratio varies the focal-plane illuminance (or optical power per unit area in the image) and is used to control variables such as [[depth of field]]. When using an [[optical telescope]] in astronomy, there is no depth of field issue, and the brightness of stellar point sources in terms of total optical power (not divided by area) is a function of absolute aperture area only, independent of focal length. The focal length controls the [[Field of view#Astronomy|field of view]] of the instrument and the scale of the image that is presented at the focal plane to an [[eyepiece]], film plate, or [[Charge-coupled device|CCD]].

For example, the [[Southern Astrophysical Research Telescope|SOAR]] four-meter telescope has a small field of view (about {{f/|16}}) which is useful for stellar studies. The [[Large Synoptic Survey Telescope|LSST]] 8.4&nbsp;m telescope, which will cover the entire sky every three days, has a very large field of view. Its short 10.3&nbsp;m focal length ({{f/|1.2}}) is made possible by an error correction system which includes secondary and tertiary mirrors, a three element refractive system and active mounting and optics.<ref name=RefDesign>{{Cite journal |title=LSST Reference Design |publisher=LSST Corporation |author1=Charles F. Claver |date=19 March 2007 |url=http://lsst.org/files/docs/LSST-RefDesign.pdf |pages=45–50 |access-date=10 January 2011 |display-authors=etal |archive-url=https://web.archive.org/web/20090306173830/http://lsst.org/files/docs/LSST-RefDesign.pdf |archive-date=6 March 2009 |url-status=dead }}</ref>

== Camera equation (G#) ==

The camera equation, or G#, is the ratio of the [[radiance]] reaching the camera sensor to the [[irradiance]] on the focal plane of the [[camera lens]]:<ref name="g-number">{{cite book |last1=Driggers |first1=Ronald G. |title=Encyclopedia of Optical Engineering: Pho-Z, pages 2049-3050 |date=2003 |publisher=CRC Press |isbn=978-0-8247-4252-2 |url=https://books.google.com/books?id=rcrGlrguj1YC |access-date=18 June 2020 |language=en}}</ref>

<math display=block>G\# = \frac {1 + 4 N^2} {\tau \pi} \,,</math>

where {{mvar|τ}} is the transmission coefficient of the lens, and the units are in inverse [[steradian]]s (sr<sup>−1</sup>).

== Working f-number ==<!-- This section header is used in redirects -->
The f-number accurately describes the light-gathering ability of a lens only for objects an infinite distance away.<ref name="Greivenkamp">{{cite book | first=John E. | last=Greivenkamp | year=2004 | title=Field Guide to Geometrical Optics | publisher=SPIE | others=SPIE Field Guides vol. '''FG01''' | isbn=0-8194-5294-7 }} p. 29.</ref> This limitation is typically ignored in photography, where f-number is often used regardless of the distance to the object. In [[optical design]], an alternative is often needed for systems where the object is not far from the lens. In these cases the '''working f-number''' is used. The working f-number {{mvar|N<sub>w</sub>}} is given by:<ref name="Greivenkamp"/>

<math display=block>N_w \approx {1 \over 2 \mathrm{NA}_i} \approx \left(1+\frac{|m|}{P}\right)N\,,</math>

where {{mvar|N}} is the uncorrected f-number, {{math|NA<sub>''i''</sub>}} is the image-space [[numerical aperture]] of the lens, <math>|m|</math> is the [[absolute value]] of the lens's [[magnification]] for an object a particular distance away, and {{mvar|P}} is the [[pupil magnification]]. Since the pupil magnification is seldom known it is often assumed to be 1, which is the correct value for all symmetric lenses.

In photography this means that as one focuses closer, the lens's effective aperture becomes smaller, making the exposure darker. The working f-number is often described in photography as the f-number corrected for lens extensions by a [[bellows factor]]. This is of particular importance in [[macro photography]].

== History ==
The system of f-numbers for specifying relative apertures evolved in the late nineteenth century, in competition with several other systems of aperture notation.

=== Origins of relative aperture ===
In 1867, Sutton and Dawson defined "apertal ratio" as essentially the reciprocal of the modern f-number. In the following quote, an "apertal ratio" of "{{frac|24}}" is calculated as the ratio of {{convert|6|in|mm}} to {{convert|1/4|in|mm}}, corresponding to an {{f/|24}} f-stop:

<blockquote>In every lens there is, corresponding to a given apertal ratio (that is, the ratio of the diameter of the stop to the focal length), a certain distance of a near object from it, between which and infinity all objects are in equally good focus. For instance, in a single view lens of 6-inch focus, with a {{1/4}} in. stop (apertal ratio one-twenty-fourth), all objects situated at distances lying between 20 feet from the lens and an infinite distance from it (a fixed star, for instance) are in equally good focus. Twenty feet is therefore called the 'focal range' of the lens when this stop is used. The focal range is consequently the distance of the nearest object, which will be in good focus when the ground glass is adjusted for an extremely distant object. In the same lens, the focal range will depend upon the size of the diaphragm used, while in different lenses having the same apertal ratio the focal ranges will be greater as the focal length of the lens is increased. The terms 'apertal ratio' and 'focal range' have not come into general use, but it is very desirable that they should, in order to prevent ambiguity and circumlocution when treating of the properties of photographic lenses.<ref name="Sutton">Thomas Sutton and George Dawson, ''A Dictionary of Photography'', London: Sampson Low, Son & Marston, 1867, (p. 122).</ref></blockquote>

In 1874, [[John Henry Dallmeyer]] called the ratio <math>1/N</math> the "intensity ratio" of a lens:

<blockquote>The ''rapidity'' of a lens depends upon the relation or ratio of the aperture to the equivalent focus. To ascertain this, divide the ''equivalent focus'' by the diameter of the actual ''working aperture'' of the lens in question; and note down the quotient as the denominator with 1, or unity, for the numerator. Thus to find the ratio of a lens of 2 inches diameter and 6 inches focus, divide the focus by the aperture, or 6 divided by 2 equals 3; i.e., {{1/3}} is the intensity ratio.<ref name="Dallmeyer">John Henry Dallmeyer, ''Photographic Lenses: On Their Choice and Use – Special Edition Edited for American Photographers'', pamphlet, 1874.</ref></blockquote>

Although he did not yet have access to [[Ernst Abbe]]'s theory of stops and pupils,<ref>{{Cite book| url =  https://archive.org/details/principlesandme01soutgoog/page/n493 | page = 537 | title = The Principles and Methods of Geometrical Optics: Especially as applied to the theory of optical instruments | publisher = Macmillan | last1 = Southall | first1 = James P. C. | year = 1910}}</ref> which was made widely available by [[Siegfried Czapski]] in 1893,<ref name="Czapski">Siegfried Czapski, ''Theorie der optischen Instrumente, nach Abbe,'' Breslau: Trewendt, 1893.</ref> Dallmeyer knew that his ''working aperture'' was not the same as the physical diameter of the aperture stop:

<blockquote>It must be observed, however, that in order to find the real ''intensity ratio'', the diameter of the actual working aperture must be ascertained. This is easily accomplished in the case of single lenses, or for double combination lenses used with the full opening, these merely requiring the application of a pair of compasses or rule; but when double or triple-combination lenses are used, with stops inserted ''between'' the combinations, it is somewhat more troublesome; for it is obvious that in this case the diameter of the stop employed is not the measure of the actual pencil of light transmitted by the front combination. To ascertain this, focus for a distant object, remove the focusing screen and replace it by the collodion slide, having previously inserted a piece of cardboard in place of the prepared plate. Make a small round hole in the centre of the cardboard with a piercer, and now remove to a darkened room; apply a candle close to the hole, and observe the illuminated patch visible upon the front combination; the diameter of this circle, carefully measured, is the actual working aperture of the lens in question for the particular stop employed.<ref name="Dallmeyer"/></blockquote>

This point is further emphasized by Czapski in 1893.<ref name="Czapski"/> According to an English review of his book, in 1894, "The necessity of clearly distinguishing between effective aperture and diameter of physical stop is strongly insisted upon."<ref>Henry Crew, "Theory of Optical Instruments by Dr. Czapski," in ''Astronomy and Astro-physics'' XIII pp. 241–243, 1894.</ref>

J. H. Dallmeyer's son, [[Thomas Rudolphus Dallmeyer]], inventor of the telephoto lens, followed the ''intensity ratio'' terminology in 1899.<ref>Thomas R. Dallmeyer, ''Telephotography: An elementary treatise on the construction and application of the telephotographic lens'', London: Heinemann, 1899.</ref>

=== Aperture numbering systems ===
[[File:No1-A Autographic Kodak Jr.jpg|thumb|right | A 1922 Kodak with aperture marked in U.S. stops. An f-number conversion chart has been added by the user.]]

At the same time, there were a number of aperture numbering systems designed with the goal of making exposure times vary in direct or inverse proportion with the aperture, rather than with the square of the f-number or inverse square of the apertal ratio or intensity ratio. But these systems all involved some arbitrary constant, as opposed to the simple ratio of focal length and diameter.

For example, the ''Uniform System'' (U.S.) of apertures was adopted as a standard by the [[Royal Photographic Society|Photographic Society of Great Britain]] in the 1880s. Bothamley in 1891 said "The stops of all the best makers are now arranged according to this system."<ref>C. H. Bothamley, ''Ilford Manual of Photography'', London: Britannia Works Co. Ltd., 1891.</ref> U.S. 16 is the same aperture as {{f/|16}}, but apertures that are larger or smaller by a full stop use doubling or halving of the U.S. number, for example {{f/|11}} is U.S. 8 and {{f/|8}} is U.S. 4. The exposure time required is directly proportional to the U.S. number. [[Eastman Kodak]] used U.S. stops on many of their cameras at least in the 1920s.

By 1895, Hodges contradicts Bothamley, saying that the f-number system has taken over: "This is called the {{f/|''x''}} system, and the diaphragms of all modern lenses of good construction are so marked."<ref>John A. Hodges, ''Photographic Lenses: How to Choose, and How to Use'', Bradford: Percy Lund & Co., 1895.</ref>

Here is the situation as seen in 1899:
[[File:Diaphragm Numbers.gif|center|823 px]]

Piper in 1901<ref>C. Welborne Piper, ''A First Book of the Lens: An Elementary Treatise on the Action and Use of the Photographic Lens'', London: Hazell, Watson, and Viney, Ltd., 1901.</ref> discusses five different systems of aperture marking: the old and new [[Carl Zeiss|Zeiss]] systems based on actual intensity (proportional to reciprocal square of the f-number); and the U.S., C.I., and Dallmeyer systems based on exposure (proportional to square of the f-number). He calls the f-number the "ratio number", "aperture ratio number", and "ratio aperture". He calls expressions like {{f/|8}} the "fractional diameter" of the aperture, even though it is literally equal to the "absolute diameter" which he distinguishes as a different term. He also sometimes uses expressions like "an aperture of f 8" without the division indicated by the slash.

Beck and Andrews in 1902 talk about the Royal Photographic Society standard of {{f/|4}}, {{f/|5.6}}, {{f/|8}}, {{f/|11.3}}, etc.<ref>Conrad Beck and Herbert Andrews, ''Photographic Lenses: A Simple Treatise'', second edition, London: R. & J. Beck Ltd., c. 1902.</ref> The R.P.S. had changed their name and moved off of the U.S. system some time between 1895 and 1902.

=== Typographical standardization ===
[[File:Yashica-D front.jpg|thumb|[[Yashica#TLRs|Yashica-D TLR]] camera front view. This is one of the few cameras that actually says "F-NUMBER" on it.]]
[[File:Yashica-D top.jpg|thumb|From the top, the Yashica-D's aperture setting window uses the "f:" notation. The aperture is continuously variable with no "stops".]]

By 1920, the term ''f-number'' appeared in books both as ''F number'' and ''f/number''. In modern publications, the forms ''f-number'' and ''f number'' are more common, though the earlier forms, as well as ''F-number'' are still found in a few books; not uncommonly, the initial lower-case ''f'' in ''f-number'' or ''f/number'' is set in a hooked italic form: ƒ.<ref>[https://books.google.com/books?as_q=lens+aperture&num=50&as_epq=f-number Google search]</ref>

Notations for f-numbers were also quite variable in the early part of the twentieth century. They were sometimes written with a capital F,<ref>{{cite book| url=https://books.google.com/books?id=ypakouuKvwYC&pg=RA2-PA61| format=Google| access-date=12 March 2007| title=Airplane Photography| last=Ives| first=Herbert Eugene| publisher=J. B. Lippincott| location= Philadelphia| year=1920| pages=61| isbn=9780598722225}}</ref> sometimes with a dot (period) instead of a slash,<ref>{{cite book| url=https://books.google.com/books?id=V7MCVGREPfkC&q=aperture+lens+uniform-system+date:0-1930| access-date=12 March 2007| title=The Fundamentals of Photography| first=Charles Edward Kenneth| last=Mees| publisher=Eastman Kodak| year=1920| pages=28}}</ref> and sometimes set as a vertical fraction.<ref>{{cite book| url=https://books.google.com/books?id=AN6d4zTjquwC&pg=PA83| format=Google| access-date=12 March 2007| title=Photography for Students of Physics and Chemistry| first=Louis| last=Derr| location=London| publisher=Macmillan| year=1906| pages=83}}</ref>

The 1961 [[American National Standards Institute|ASA]] standard PH2.12-1961 ''American Standard General-Purpose Photographic Exposure Meters (Photoelectric Type)'' specifies that "The symbol for relative apertures shall be {{not a typo|ƒ/}} or {{not a typo|ƒ:}} followed by the effective ƒ-number." They show the hooked italic 'ƒ' not only in the symbol, but also in the term ''f-number'', which today is more commonly set in an ordinary non-italic face.

== See also ==
{{Portal|Physics|Film}}
* [[Circle of confusion]]
* [[Group f/64]]
* [[Photographic lens design]]
* [[Pinhole camera]]
* [[Preferred number]]

== References ==
{{Reflist|30em}}

== External links ==
{{Commons category|F-number}}
* [https://www.largeformatphotography.info/fstop.html Large format photography—how to select the f-stop]

{{photography}}

[[Category:Optical quantities]]
[[Category:Science of photography]]
[[Category:Dimensionless numbers of physics]]
[[Category:Logarithmic scales of measurement]]