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In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
, the f-number of an optical system such as a
camera lens A camera lens (also known as photographic lens or photographic objective) is an optical lens or assembly of lenses used in conjunction with a camera body and mechanism to make images of objects either on photographic film or on other media cap ...
is the
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of the system's
focal length The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative foc ...
to the diameter of the
entrance pupil In an optical system, the entrance pupil is the optical image of the physical aperture stop, as 'seen' through the front (the object side) of the lens system. The corresponding image of the aperture as seen through the back of the lens system ...
("clear
aperture In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane. An ...
").Smith, Warren ''Modern Optical Engineering'', 4th Ed., 2007 McGraw-Hill Professional, p. 183. It is also known as the focal ratio, f-ratio, or f-stop, and is very important in
photography Photography is the art, application, and practice of creating durable images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as photographic film. It is emplo ...
. It is a
dimensionless number A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
that is a quantitative measure of
lens speed Lens speed refers to the maximum aperture diameter, or minimum f-number, of a photographic lens. A lens with a larger than average maximum aperture (that is, a smaller minimum f-number) is called a "fast lens" because it can achieve the same exposu ...
; increasing the f-number is referred to as ''
stopping down In photography, stopping down refers to increasing the numerical f-stop number (for example, going from 2 to 4), which decreases the size (diameter) of the aperture of a lens, resulting in reducing the amount of light entering the iris of a lens ...
''. The f-number is commonly indicated using a lower-case hooked f with the format ''N'', where ''N'' is the f-number. The f-number is the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of the relative aperture (the aperture diameter divided by focal length).


Notation

The f-number is given by: N = \frac \ where f is the
focal length The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative foc ...
, and D is the diameter of the entrance pupil (''effective aperture''). It is customary to write f-numbers preceded by "", which forms a mathematical expression of the entrance pupil diameter in terms of f and . For example, if a lens's focal length were 10 mm and its entrance pupil diameter were 5 mm, the f-number would be 2. This would be expressed as "2" in a lens system. The aperture diameter would be equal to f/2. Most lenses have an adjustable
diaphragm Diaphragm may refer to: Anatomy * Thoracic diaphragm, a thin sheet of muscle between the thorax and the abdomen * Pelvic diaphragm or pelvic floor, a pelvic structure * Urogenital diaphragm or triangular ligament, a pelvic structure Other * Diap ...
, which changes the size of the aperture stop and thus the entrance pupil size. This allows the practitioner to vary the f-number, according to needs. It should be appreciated that 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 Luminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through, is emitted from, or is reflected from a particular area, and falls with ...
) decreases with the square of the f-number. A 100 mm focal length 4 lens has an entrance pupil diameter of 25 mm. A 100 mm focal length 2 lens has an entrance pupil diameter of 50 mm. Since the area varies as the square of the pupil diameter, the amount of light admitted by the 2 lens is four times that of the 4 lens. To obtain the same photographic exposure, the exposure time must be reduced by a factor of four. A 200 mm focal length 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 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 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 In optics, 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 ("clear aperture").Smith, Warren ''Modern Optical Engineering'', 4th Ed., 2007 McGraw-Hill Pro ...
is an f-number adjusted to account for light transmission efficiency.


Stops, f-stop conventions, and exposure

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 In photography, exposure value (EV) is a number that represents a combination of a camera's shutter speed and f-number, such that all combinations that yield the same exposure have the same EV (for any fixed scene luminance). Exposure value is ...
) 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 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 In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ...
of numbers that corresponds to the sequence of the powers of the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princi ...
: 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64, 90, 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 = \frac,\ f/1.4 = \frac,\ f/2 = \frac,\ f/2.8 = \frac,\ \ldots In the same way as one f-stop corresponds to a factor of two in light intensity,
shutter speed In photography, shutter speed or exposure time is the length of time that the film or digital sensor inside the camera is exposed to light (that is, when the camera's shutter is open) when taking a photograph. The amount of light that rea ...
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''. 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 A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
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

Most twentieth-century cameras had a continuously variable aperture, using an
iris diaphragm In optics, a diaphragm is a thin opaque structure with an opening (aperture) at its center. The role of the diaphragm is to ''stop'' the passage of light, except for the light passing through the ''aperture''. Thus it is also called a stop (an ...
, 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 ( EV) are the most common, since this matches the ISO system of
film speed Film speed is the measure of a photographic film's sensitivity to light, determined by sensitometry and measured on various numerical scales, the most recent being the ISO system. A closely related ISO system is used to describe the relation ...
s. 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 2.8 is 3.2, two-thirds smaller is 3.5, and one whole stop smaller is 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,\ \ldots To calculate the steps in a full stop (1 EV) one could use (\sqrt)^,\ (\sqrt)^,\ (\sqrt)^,\ (\sqrt)^,\ (\sqrt)^,\ \ldots The steps in a half stop ( EV) series would be (\sqrt)^,\ (\sqrt)^,\ (\sqrt)^,\ (\sqrt)^,\ (\sqrt)^,\ \ldots The steps in a third stop ( EV) series would be (\sqrt)^,\ (\sqrt)^,\ (\sqrt)^,\ (\sqrt)^,\ (\sqrt)^,\ \ldots 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 \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 while shutter speeds in reciprocal seconds have a few conventional differences in their numbers (, , and second instead of , , and ). In practice the maximum aperture of a lens is often not an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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 used for SLR cameras, have f-stops specified internally in -stop increments, so the cameras' -stop settings are approximated by the nearest -stop setting in the lens.


Standard full-stop f-number scale

Including aperture value AV: N = \sqrt Conventional and calculated f-numbers, full-stop series:


Typical one-half-stop f-number scale


Typical one-third-stop f-number scale

Sometimes the same number is included on several scales; for example, an aperture of 1.2 may be used in either a half-stop or a one-third-stop system; sometimes 1.3 and 3.2 and other differences are used for the one-third stop scale.


Typical one-quarter-stop f-number scale


H-stop

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 In optics, a diaphragm is a thin opaque structure with an opening (aperture) at its center. The role of the diaphragm is to ''stop'' the passage of light, except for the light passing through the ''aperture''. Thus it is also called a stop (an ...
s or sieve aperture found in Rodenstock Imagon lenses.


T-stop

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 Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is t ...
''). A lens with a T-stop of projects an image of the same brightness as an ideal lens with 100% transmittance and an f-number of . A particular lens's T-stop, , is given by dividing the f-number by the square root of the transmittance of that lens: T = \frac. For example, an 2.0 lens with transmittance of 75% has a T-stop of 2.3: T = \frac = 2.309... Since real lenses have transmittances of less than 100%, a lens's T-stop number is always greater than its f-number. 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. T-stops are sometimes used instead of f-numbers to more accurately determine exposure, particularly when using external
light meter A light meter is a device used to measure the amount of light. In photography, a light meter (more correctly an exposure meter) is used to determine the proper exposure (photography), exposure for a photograph. The meter will include either a Di ...
s. Lens transmittances of 60%–95% are typical. 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. 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 The Smooth Trans Focus (STF) technology in photographic lenses uses an apodization filter to realize notably smooth bokeh with rounded out-of-focus highlights in both the foreground and background. This is accomplished by utilizing a concave neut ...
lenses by
Minolta was a Japanese manufacturer of cameras, camera accessories, photocopiers, fax machines, and laser printers. Minolta Co., Ltd., which is also known simply as Minolta, was founded in Osaka, Japan, in 1928 as . It made the first integrated aut ...
and
Sony , commonly stylized as SONY, is a Japanese multinational conglomerate corporation headquartered in Minato, Tokyo, Japan. As a major technology company, it operates as one of the world's largest manufacturers of consumer and professional ...
.


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 16 and the shutter speed closest to the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of the ISO speed of the film; for example, using ISO 200 film, an aperture of 16 and a shutter speed of 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

Depth of field The depth of field (DOF) is the distance between the nearest and the furthest objects that are in acceptably sharp focus in an image captured with a camera. Factors affecting depth of field For cameras that can only focus on one object dis ...
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 Nature photography is a wide range of photography taken outdoors and devoted to displaying natural elements such as landscapes, wildlife, plants, and close-ups of natural scenes and textures. Nature photography tends to put a stronger emphasis o ...
and
portraiture A portrait is a painting, photograph, sculpture, or other artistic representation of a person, in which the face and its expressions are predominant. The intent is to display the likeness, personality, and even the mood of the person. For this re ...
because background blur (the aesthetic quality known as '
bokeh In photography, bokeh ( or ; ) is the aesthetic quality of the blur produced in out-of-focus parts of an image. Bokeh has also been defined as "the way the lens renders out-of-focus points of light". Differences in lens aberrations and ...
') can be aesthetically pleasing and puts the viewer's focus on the main subject in the foreground. The
depth of field The depth of field (DOF) is the distance between the nearest and the furthest objects that are in acceptably sharp focus in an image captured with a camera. Factors affecting depth of field For cameras that can only focus on one object dis ...
of an image produced at a given f-number is dependent on other parameters as well, including the
focal length The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative foc ...
, the subject distance, and the
format Format may refer to: Printing and visual media * Text formatting, the typesetting of text elements * Paper formats, or paper size standards * Newspaper format, the size of the paper page Computing * File format, particular way that informatio ...
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 In an optical system, the entrance pupil is the optical image of the physical aperture stop, as 'seen' through the front (the object side) of the lens system. The corresponding image of the aperture as seen through the back of the lens system ...
diameter (as in 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 The angle of view is the decisive variable for the visual perception of the size or projection of the size of an object. Angle of view and perception of size The perceived size of an object depends on the size of the image projected onto the ...
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: aberration, due to imperfect lens design, and
diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
which is due to the wave nature of light. The blur-optimal f-stop varies with the lens design. For modern standard lenses having 6 or 7 elements, the sharpest image is often obtained around 5.6–8, while for older standard lenses having only 4 elements ( Tessar formula) stopping to 11 will give the sharpest image. 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 Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
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 In photography and optics, vignetting is a reduction of an image's brightness or saturation toward the periphery compared to the image center. The word '' vignette'', from the same root as ''vine'', originally referred to a decorative bord ...
) at the edges for large apertures.
Photojournalist Photojournalism is journalism that uses images to tell a news story. It usually only refers to still images, but can also refer to video used in broadcast journalism. Photojournalism is distinguished from other close branches of photography (such ...
s have a saying, " 8 and be there", meaning that being on the scene is more important than worrying about technical details. Practically, 8 (in 35 mm and larger formats) allows adequate depth of field and sufficient lens speed for a decent base exposure in most daylight situations.


Human eye

Computing the f-number of the
human eye The human eye is a sensory organ, part of the sensory nervous system, that reacts to visible light and allows humans to use visual information for various purposes including seeing things, keeping balance, and maintaining circadian rhythm. ...
involves computing the physical aperture and focal length of the eye. The pupil can be as large as 6–7 mm wide open, which translates into the maximal physical aperture. The f-number of the human eye varies from about 8.3 in a very brightly lit place to about 2.1 in the dark. 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.


Focal ratio in telescopes

In astronomy, the f-number is commonly referred to as the ''focal ratio'' (or ''f-ratio'') notated as N. It is still defined as the
focal length The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative foc ...
f of an objective divided by its diameter D or by the diameter of an
aperture In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane. An ...
stop in the system: N = \frac fD \quad \xrightarrow \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 Photography is the art, application, and practice of creating durable images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as photographic film. It is emplo ...
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 The depth of field (DOF) is the distance between the nearest and the furthest objects that are in acceptably sharp focus in an image captured with a camera. Factors affecting depth of field For cameras that can only focus on one object dis ...
. When using an
optical telescope An optical telescope is a telescope that gathers and focuses light mainly from the visible part of the electromagnetic spectrum, to create a magnified image for direct visual inspection, to make a photograph, or to collect data through elect ...
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 The field of view (FoV) is the extent of the observable world that is seen at any given moment. In the case of optical instruments or sensors it is a solid angle through which a detector is sensitive to electromagnetic radiation. Human ...
of the instrument and the scale of the image that is presented at the focal plane to an
eyepiece An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is named because it is usually the lens that is closest to the eye when someone looks through the device. The ...
, film plate, or CCD. For example, the SOAR 4-meter telescope has a small field of view (about ) which is useful for stellar studies. The LSST 8.4 m telescope, which will cover the entire sky every three days, has a very large field of view. Its short 10.3 m focal length () is made possible by an error correction system which includes secondary and tertiary mirrors, a three element refractive system and active mounting and optics.


Camera equation (G#)

The camera equation, or G#, is the ratio of the
radiance In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiati ...
reaching the camera sensor to the
irradiance In radiometry, irradiance is the radiant flux ''received'' by a ''surface'' per unit area. The SI unit of irradiance is the watt per square metre (W⋅m−2). The CGS unit erg per square centimetre per second (erg⋅cm−2⋅s−1) is often used ...
on the focal plane of the
camera lens A camera lens (also known as photographic lens or photographic objective) is an optical lens or assembly of lenses used in conjunction with a camera body and mechanism to make images of objects either on photographic film or on other media cap ...
: G\# = \frac \,, where is the transmission coefficient of the lens, and the units are in inverse
steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radian ...
s (sr−1).


Working f-number

The f-number accurately describes the light-gathering ability of a lens only for objects an infinite distance away. p. 29. 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 is given by: N_w \approx \approx \left(1+\frac\right)N\,, where is the uncorrected f-number, is the image-space
numerical aperture In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the proper ...
of the lens, , m, is the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of the lens's
magnification Magnification is the process of enlarging the apparent size, not physical size, of something. This enlargement is quantified by a calculated number also called "magnification". When this number is less than one, it refers to a reduction in si ...
for an object a particular distance away, and 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 A bellows or pair of bellows is a device constructed to furnish a strong blast of air. The simplest type consists of a flexible bag comprising a pair of rigid boards with handles joined by flexible leather sides enclosing an approximately airtigh ...
. This is of particular importance in
macro photography Macro photography (or photomacrography or macrography, and sometimes macrophotography) is extreme close-up photography, usually of very small subjects and living organisms like insects, in which the size of the subject in the photograph is grea ...
.


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 "" is calculated as the ratio of to , corresponding to an 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 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.Thomas Sutton and George Dawson, ''A Dictionary of Photography'', London: Sampson Low, Son & Marston, 1867, (p. 122).
In 1874,
John Henry Dallmeyer John Henry Dallmeyer (6 September 183030 December 1883), Anglo-German optician, was born at Loxten, Westphalia, the son of a landowner. On leaving school at the age of sixteen he was apprenticed to an Osnabrück optician, and in 1851 he came to Lo ...
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., is the intensity ratio.John Henry Dallmeyer, ''Photographic Lenses: On Their Choice and Use – Special Edition Edited for American Photographers'', pamphlet, 1874.
Although he did not yet have access to
Ernst Abbe Ernst Karl Abbe HonFRMS (23 January 1840 – 14 January 1905) was a German physicist, optical scientist, entrepreneur, and social reformer. Together with Otto Schott and Carl Zeiss, he developed numerous optical instruments. He was also a c ...
's theory of stops and pupils, which was made widely available by Siegfried Czapski in 1893,Siegfried Czapski, ''Theorie der optischen Instrumente, nach Abbe,'' Breslau: Trewendt, 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 Thomas Rudolphus Dallmeyer (16 May 185925 December 1906),"Obituary; Thomas Rudolphus Dallmeyer" (1907) ''The Photographic Journal'', Vol. 31, pp. 20–21, Royal Photographic Society, London English optician, was the son of John Henry Dallmeyer wh ...
, 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 The Royal Photographic Society of Great Britain, commonly known as the Royal Photographic Society (RPS), is one of the world's oldest photographic societies. It was founded in London, England, in 1853 as the Photographic Society of London with ...
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 16, but apertures that are larger or smaller by a full stop use doubling or halving of the U.S. number, for example 11 is U.S. 8 and 8 is U.S. 4. The exposure time required is directly proportional to the U.S. number.
Eastman Kodak The Eastman Kodak Company (referred to simply as Kodak ) is an American public company that produces various products related to its historic basis in analogue photography. The company is headquartered in Rochester, New York, and is incorpor ...
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 ''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 Zeiss or Zeiß may refer to: People *Carl Zeiss (1816–1888), German optician and entrepreneur *Emil Zeiß (1833–1910), German Protestant minister and painter Companies *Carl Zeiss AG, German manufacturer of optics, industrial measurem ...
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 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 4, 5.6, 8, 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.


Typographical standardization

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. 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 or 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

*
Circle of confusion In optics, a circle of confusion (CoC) is an optical spot caused by a cone of light rays from a lens not coming to a perfect focus when imaging a point source. It is also known as disk of confusion, circle of indistinctness, blur circle, or ...
* Group f/64 *
Photographic lens design The design of photographic lenses for use in still or cine cameras is intended to produce a lens that yields the most acceptable rendition of the subject being photographed within a range of constraints that include cost, weight and materials. ...
*
Pinhole camera A pinhole camera is a simple camera without a lens but with a tiny aperture (the so-called '' pinhole'')—effectively a light-proof box with a small hole in one side. Light from a scene passes through the aperture and projects an inverted image ...
*
Preferred number In industrial design, preferred numbers (also called preferred values or preferred series) are standard guidelines for choosing exact product dimensions within a given set of constraints. Product developers must choose numerous lengths, distanc ...


References


External links


Large format photography—how to select the f-stop
{{photography Optics Science of photography Dimensionless numbers Logarithmic scales of measurement