The f-number of an optical system (such as a camera lens) is the ratio of the system's focal length to the diameter of the entrance pupil.[1] It is a dimensionless number that is a quantitative measure of lens speed, and an important concept in photography. It is also known as the focal ratio, f-ratio, or f-stop.[2] It is the reciprocal of the relative aperture.[3] The f-number is commonly indicated using a hooked f with the format f/N, where N is the f-number. Contents 1 Notation 2 Stops, f-stop conventions, and exposure 2.1 Fractional stops 2.1.1 Standard full-stop f-number scale 2.1.2 Typical one-half-stop f-number scale 2.1.3 Typical one-third-stop f-number scale 2.1.4 Typical one-quarter-stop f-number scale 2.2 H-stop 2.3 T-stop 2.4 Sunny 16 rule 3 Effects on image sharpness
4 Human eye
5
7.1 Origins of relative aperture
7.2
8 See also 9 References 10 External links Notation[edit] The f-number N or f# is given by: N = f D
displaystyle N= frac f D where f displaystyle f is the focal length, and D displaystyle 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 diameter in terms of f and N.[1] For example, if a lens's focal length is 10 mm and its entrance pupil diameter is 5 mm, the f-number is 2, expressed by writing "f/2", and the aperture diameter is equal to f / 2 displaystyle f/2 , where f displaystyle f is the focal length. 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. Doubling the f-number decreases the relative brightness by a factor of four. To maintain the same photographic exposure when doubling the f-number, the exposure time would need to be four times as long. Most lenses have an adjustable diaphragm, which changes the size of the aperture stop and thus the entrance pupil size. 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. A 100 mm focal length f/4 lens has an entrance pupil diameter of 25 mm. A 200 mm focal length f/4 lens has an entrance pupil diameter of 50 mm. The 200 mm lens's entrance pupil has four times the area of the 100 mm 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 100 mm lens, the 200 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. A T-stop is an f-number adjusted to account for light transmission efficiency. Stops, f-stop conventions, and exposure[edit] A
A 35 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/2.0 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 / 2 displaystyle scriptstyle 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 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: f / 1 = f ( 2 ) 0 , f / 1.4 = f ( 2 ) 1 , f / 2 = f ( 2 ) 2 , f / 2.8 = f ( 2 ) 3 ⋯ displaystyle 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 cdots In the same way as one f-stop corresponds to a factor of two in light intensity, shutter speeds 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. This is less true for extremely long or short exposures, where we have 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 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 "half a stop". Fractional stops[edit] Computer simulation showing the effects of changing a camera's aperture in half-stops (at left) and from zero to infinity (at right) Most old cameras had a continuously variable aperture scale, 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 speeds. Half-stop steps are used on some cameras. Usually the full stops are marked, and the intermediate positions are clicked. 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: f/4.5, f/5, f/5.6, f/6.3, f/7.1, f/8, etc. To calculate the steps in a full stop (1 EV) one could use 20×0.5, 21×0.5, 22×0.5, 23×0.5, 24×0.5 etc. The steps in a half stop (1/2 EV) series would be 20/2×0.5, 21/2×0.5, 22/2×0.5, 23/2×0.5, 24/2×0.5 etc. The steps in a third stop (1/3 EV) series would be 20/3×0.5, 21/3×0.5, 22/3×0.5, 23/3×0.5, 24/3×0.5 etc. 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 ... 16/13°, 20/14°, 25/15°, 32/16°, 40/17°, 50/18°, 64/19°, 80/20°, 100/21°, 125/22°... while shutter speeds in reciprocal seconds have a few conventional differences in their numbers (1/15, 1/30, and 1/60 second instead of 1/16, 1/32, and 1/64). In practice the maximum aperture of a lens is often not an integral power of 2 displaystyle scriptstyle sqrt 2 (i.e., 2 displaystyle scriptstyle 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 2 displaystyle scriptstyle sqrt 2 . Modern electronically controlled interchangeable lenses, such as those used for SLR cameras, have f-stops specified internally in 1/8-stop increments, so the cameras' 1/3-stop settings are approximated by the nearest 1/8-stop setting in the lens. Standard full-stop f-number scale[edit] Including aperture value AV: N = 2 A V displaystyle N= sqrt 2^ AV Conventional and calculated f-numbers, full-stop series: AV −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 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 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[edit] AV −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 N 0.7 0.8 1.0 1.2 1.4 1.7 2 2.4 2.8 3.3 4 4.8 5.6 6.7 8 9.5 11 13 16 19 22 27 32 38 45 54 64 76 90 107 128 Typical one-third-stop f-number scale[edit] AV −1 −0.7 −0.3 0 0.3 0.7 1 1.3 1.7 2 2.3 2.7 3 3.3 3.7 4 4.3 4.7 5 5.3 5.7 6 6.3 6.7 7 7.3 7.7 8 8.3 8.7 9 9.3 9.7 10 10.3 10.7 11 11.3 11.7 12 12.3 12.7 13 N 0.7 0.8 0.9 1.0 1.1 1.2 1.4 1.6 1.8 2 2.2 2.5 2.8 3.2 3.5 4 4.5 5.0 5.6 6.3 7.1 8 9 10 11 13 14 16 18 20 22 25 29 32 36 40 45 51 57 64 72 80 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[4] or a one-third-stop system;[5] sometimes f/1.3 and f/3.2 and other differences are used for the one-third stop scale.[6] Typical one-quarter-stop f-number scale[edit] AV 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 N 1.0 1.1 1.2 1.3 1.4 1.5 1.7 1.8 2 2.2 2.4 2.6 2.8 3.1 3.4 3.7 4 4.4 4.8 5.2 5.6 Minolta 1.00 1.01 1.02 1.03 1.40 1.41 1.42 1.43 2.00 2.01 2.02 2.03 2.80 2.81 2.82 2.83 4.00 4.01 4.02 4.03 5.60 AV 5 5.25 5.5 5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10 N 5.6 6.2 6.7 7.3 8 8.7 9.5 10 11 12 14 15 16 17 19 21 22 25 27 29 32 Minolta 5.60 5.61 5.62 5.63 8.00 8.01 8.02 8.03 110 111 112 113 160 161 162 163 220 221 222 223 320 H-stop[edit]
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 discs or sieve aperture found in
T = f transmittance . displaystyle T= frac f sqrt text transmittance . For example, an f/2.0 lens with transmittance of 75% has a T-stop of 2.3: T = 2.0 0.75 = 2.309... displaystyle T= frac 2.0 sqrt 0.75 =2.309... Since real lenses have transmittances of less than 100%, a lens's
T-stop number is always greater than its f-number.[7]
With 8% loss per air-glass surface on lenses without 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.[8][9] T-stops are
sometimes used instead of f-numbers to more accurately determine
exposure, particularly when using external light meters.[10] Lens
transmittances of 60%–95% are typical.[11] 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.[10]
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
Comparison of f/32 (top-left corner) and f/5 (bottom-right corner) Shallow focus with a wide open lens
Diagram of the focal ratio of a simple optical system where f displaystyle f is the focal length and D displaystyle D is the diameter of the objective. In astronomy, the f-number is commonly referred to as the focal ratio (or f-ratio) notated as N displaystyle N . It is still defined as the focal length f displaystyle f of an objective divided by its diameter D displaystyle D or by the diameter of an aperture stop in the system: N = f D → × D f = N D displaystyle N= frac f D quad xrightarrow times D quad f=ND
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 of the instrument and the scale of
the image that is presented at the focal plane to an eyepiece, film
plate, or CCD.
For example, the SOAR 4-meter telescope has a small field of view
(~f/16) which is useful for stellar studies. The
N w ≈ 1 2 N A i ≈ ( 1 +
m
P ) N displaystyle N_ w approx 1 over 2mathrm NA _ i approx left(1+ frac m P right)N , where N is the uncorrected f-number, NAi is the image-space numerical aperture of the lens,
m
displaystyle m is the absolute value of lens's magnification for an object a particular distance away, and P is the pupil magnification.[16] 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, the working f-number is described as the f-number corrected for lens extensions by a bellows factor. This is of particular importance in macro photography. History[edit] 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[edit] 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 "1/24" is calculated as the ratio of 6 inches (150 mm) to 1⁄4 inch (6.4 mm), corresponding to an f/24 f-stop: 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.[17] In 1874,
1 / N displaystyle 1/N the "intensity ratio" of a lens: 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.[18] Although he did not yet have access to Ernst Abbe's theory of stops and pupils,[19] which was made widely available by Siegfried Czapski in 1893,[20] Dallmeyer knew that his working aperture was not the same as the physical diameter of the aperture stop: 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.[18] This point is further emphasized by Czapski in 1893.[20] 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."[21]
J. H. Dallmeyer's son, Thomas Rudolphus Dallmeyer, inventor of the
telephoto lens, followed the intensity ratio terminology in 1899.[22]
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 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."[23] 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.
Piper in 1901[25] discusses five different systems of aperture
marking: the old and new 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.[26] The R.P.S. had changed
their name and moved off of the U.S. system some time between 1895 and
1902.
Typographical standardization[edit]
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
Physics portal Film portal Circle of confusion Group f/64 Photographic lens design Pinhole camera Preferred number References[edit] ^ a b Smith, Warren Modern Optical Engineering, 4th Ed. 2007
McGraw-Hill Professional
^ Smith, Warren Modern Lens Design 2005 McGraw-Hill
^ ISO, Photography—Apertures and related properties pertaining to
photographic lenses—Designations and measurements, ISO 517:2008
^ Harry C. Box (2003). Set lighting technician's handbook: film
lighting equipment, practice, and electrical distribution (3rd ed.).
Focal Press. ISBN 978-0-240-80495-8.
^ Paul Kay (2003). Underwater photography. Guild of Master Craftsman.
ISBN 978-1-86108-322-7.
^ David W. Samuelson (1998). Manual for cinematographers (2nd ed.).
Focal Press. ISBN 978-0-240-51480-2.
^ Transmission, light transmission, DxOMark
^ Sigma 85mm F1.4 Art lens review: New benchmark, DxOMark
^ Colour rendering in binoculars and lenses - Colours and
transmission, LensTip.com
^ a b "Kodak Motion Picture
External links[edit] Wikimedia Commons has media related to F-number. f Number arithmetic
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