- f Number Arithmetic
- Understanding Camera Lenses – focal length & f-number – includes how to interpret f-numbers when purchasing a lens
- Large format photography—how to select the f-stop
In optics, the f-number (sometimes called focal ratio, f-ratio, or relative aperture) of an optical system expresses the diameter of the entrance pupil in terms of the effective focal length of the lens; in simpler terms, the f-number is the focal length divided by the aperture diameter. It is a dimensionless number that is a quantitative measure of lens speed, an important concept in photography.
Diagram of decreasing apertures, that is, increasing f-numbers, in one-stop increments; each aperture has half the light gathering area of the previous one. The actual size of the aperture will depend on the focal length of the lens. (*)
The f-number f/#, often notated as N, is given by
where f is the focal length, and D is the diameter of the entrance pupil. By convention, "f/#" is treated as a single symbol, and specific values of f/# are written by replacing the number sign with the value. For example, if the focal length is 16 times the pupil diameter, the f-number is f/16, or N = 16. The greater the f-number, the less light per unit area reaches the image plane of the system.
The literal interpretation of the f/N notation for f-number N is as an arithmetic expression for the effective aperture diameter (input pupil diameter), the focal length divided by the f-number: D = f / N.
The pupil diameter is proportional to the diameter of the aperture stop of the system. In a camera, this is typically the diaphragm aperture, which can be adjusted to vary the size of the pupil, and hence the amount of light that reaches the film or image sensor. Other types of optical system, such as telescopes and binoculars may have a fixed aperture, but the same principle holds: the greater the focal ratio, the fainter the images created (measuring brightness per unit area of the image). Note that the common assumption in photography that the pupil diameter is equal to the aperture diameter is not correct for all types of camera lens. A 100mm lens with an aperture setting of f/4 will have a pupil diameter of 25mm. A 135mm lens with a setting of f/4 will have a pupil diameter of 33.8mm or 34mm. The 135mm lens' f/4 opening is larger than that of the 100mm lens though both will transmit the same amount of light to the film or sensor. A focal ratio of f/16 tells us that the physical aperture inside the camera lens has a pupil diameter equal to one sixteenth of that lens' focal length and this applies to all lenses using this designation.
A Canon 7 mounted with a 50mm lens capable of an exceptional f/ 0.95 (*)
Stops, f-stop conventions, and exposure
The term 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 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 one stop meaning a factor of two, or one-half. The one-stop unit is also known as the EV (exposure value) unit. On a camera, the f-number is usually 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 √2 or about 1.414, and hence a halving of the area of the pupil.
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 √2 (1.414): 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. The values of the ratios are rounded off to these particular conventional numbers, to make them easy to remember and write down.
The slash indicates division. For example, f/16 means that the pupil diameter is equal to the focal length divided by sixteen; that is, if the camera has an 80 mm lens, all the light that reaches the film passes through a virtual disk known as the entrance pupil that is 5 mm (80 mm/16) in diameter. The location of this virtual disk inside the lens depends on the optical design. It may simply be the opening of the aperture stop, or may be a magnified image of the aperture stop, formed by elements within the lens.
Shutter speeds are arranged in a similar scale, so that one step in the shutter speed scale corresponds to one stop in the aperture scale. 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 speed is set twice as fast (i.e. the shutter is open half as long); the film will usually respond equally to these equal amounts of light, since it has the property known as reciprocity. Alternatively, one could use a film that is half as sensitive to light, with the original shutter speed.
Photographers sometimes express other exposure ratios in terms of 'stops'. If we ignore the f-number markings, the f-stops make a logarithmic scale of exposure intensity. Given this interpretation, you can then think of taking a half-step along this scale, to make an exposure difference of "half a stop".
A 35mm 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
Most old cameras had an aperture scale graduated in full stops but the aperture is continuously variable allowing to select any intermediate aperture.
Click-stopped aperture became a common feature in the 1960s; the aperture scale was usually marked in full stops, but many lenses had a click between two marks, allowing a gradation of one half of a 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 also seen on some cameras. 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.
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 may not be an integral power of (i.e. 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 .
Modern electronically-controlled interchangeable lenses, such as those from Canon and Sigma 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
Including aperture value AV:
Typical one-half-stop f-number scale
Typical one-third-stop f-number scale
Notice that sometimes a number is ambiguous; for example, f/1.2 may be used in either a half-stop or a one-third-stop system; sometimes f/1.3 and f/3.2 and other differences are used for the one-third stop scale.
Since all lenses absorb some portion of the light passing through them (particularly zoom lenses containing many elements), T-stops are sometimes used instead of f-stops for exposure purposes, especially for motion picture camera lenses. The practice became popular in cinematographic usage before the advent of zoom lenses, where fixed focal length lenses were calibrated to T-stops: This allowed the turret-mounted lenses to be changed without affecting the overall scene brightness. Lenses were bench-tested individually for actual light transmission and assigned T stops accordingly (The T in T-stop stands for transmission), but modern cinematographic lenses now usually tend to be factory-calibrated in T-stops. T-stops measure the amount of light transmitted through the lens in practice, and are equivalent in light transmission to the f-stop of an ideal lens with 100% transmission. Since all lenses absorb some quantity of light, the T-number of any given aperture on a lens will always be greater than the f-number. In recent years, advances in lens technology and film exposure latitude have reduced the criticality of t-stop values. Remember: F-stops are for geometry, T-stops are for transmission.
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 a shutter speed close to the reciprocal of the ISO speed of the film; for example, using ISO 200 film, an aperture of f/16 and a shutter speed of 1/200 second. The f-number may then be adjusted downwards for situations with lower light.
Effects on image quality
Depth of field increases with f-number, as illustrated in the photos below. This means that photos taken with a low f-number will tend to have one subject in focus, with the rest of the image out of focus. This is frequently useful for nature photography or certain special effects. 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 format of the film or sensor used to capture the image. 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. Therefore, reduced-depth-of-field effects, like those shown here, will require smaller f-numbers (and thus more complex optics) than do larger format cameras.
Picture sharpness also varies with f-number. The optimal f-stop varies with the lens characteristics. For modern standard lenses having 6 or 7 elements, the sharpest image is often obtained around f/5.6–f/8, while for older standard lenses having only 4 elements (Tessar formula) stopping to f/11 will give the sharpest image. The reason the sharpness is best at medium f-numbers is that the sharpness at high f-numbers is constrained by diffraction, whereas at low f-numbers limitations of the lens design known as aberrations will dominate. 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-stops. 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. To measure the actual resolution of the lens at the different f-numbers it is necessary to use a standardized measurement chart like the 1951 USAF Resolution Test Chart.
Photojournalists have a saying, "f/8 and be there," meaning that being on the scene is more important than worrying about technical details. The aperture of f/8 gives adequate depth of field, assuming a 35 mm or DSLR camera, minimum shutter-speed, and ISO film rating within reasonable limits subject to lighting.
Varying the f-number varies the amount of light that is let through the lens. If the f-number is too low (for the combination of shutter speed, ISO film speed, and illumination), the image may be over-exposed, resulting in blown-out highlight areas. Conversely, if the f-number is too high the image may be under-exposed, resulting in image noise and loss of shadow detail.
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.  Toxic substances and poisons (like Atropine) can significantly reduce this range. Pharmaceutical products such as eye drops may also cause similar side-effects.
Diagram of the focal ratio of a simple optical system where f is the focal length and D is the diameter of the objective
Focal ratio in telescopes
In astronomy, the f-number is commonly referred to as the focal ratio (or f-ratio). It is still defined as the focal length f of an objective divided by its diameter D or by the diameter of an aperture stop in the system.
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.
The f-number accurately describes the light-gathering ability of a lens only for objects an infinite distance away. This limitation is typically ignored in photography, where objects are usually not extremely close to the camera, relative to the distance between the lens and the film. 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 Nw is given by
where N is the uncorrected f-number, "NA" is the numerical aperture of the lens, and m is the lens's magnification for an object a particular distance away. (Note that the magnification m here is negative for the common case where the image is inverted.) 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.
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 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.
In 1874, John Henry Dallmeyer called the ratio 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.
Although he did not yet have access to Ernst Abbe's theory of stops and pupils , which was made widely available by Siegfried Czapski in 1893, 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.
This point is further emphasized by Czapski in 1893. 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."
J. H. Dallmeyer's son, Thomas Rudolphus Dallmeyer, inventor of the telephoto lens, followed the intensity ratio terminology in 1899.
Aperture numbering systems
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."  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." 
Here is the situation as seen in 1899:
Piper in 1901 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. The R.P.S. had changed their name and moved off of the U.S. system some time between 1895 and 1902. Modern conventions have rounded the numbers from f/5.66 to f/5.6, f/11.13 to f/11, and f/44.72 to f/45. This is only for ease of writing – the actual ratio of aperture size to focal length is still based off the doubling or halving of the amount of light getting through the lens.
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 as the hooked italic f as in f/#. Notations for f-numbers were also quite variable in the early part of the twentieth century. They were sometimes written with a capital F, sometimes with a dot (period) instead of a slash, and sometimes set as a vertical fraction.
The 1961 ASA standard PH2.12-1961 American Standard General-Purpose Photographic Exposure Meters (Photoelectric Type) specifies that "The symbol for relative apertures shall be f/ or f : followed by the effective f-number." Note that they show the hooked italic f not only in the symbol, but also in the term f-number, which today is more commonly set in an ordinary non-italic face.
* Circle of confusion
* Depth of field
* Exposure value
* Optical telescope
* Pinhole camera
* Printer points
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