F-number Totally Explained

Everything about Focal Ratio totally explained

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's a quantitative measure of lens speed, an important concept in photography.

Notation

The f-number #, often notated as

N

, is given by ť

f/# = N = frac fD

where

f

is the focal length, and

D

is the diameter of the entrance pupil. By convention, "#" is treated as a single symbol, and specific values of # 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 16, or

N = 16

. The greater the f-number, the less light per unit area reaches the image plane of the system; the amount of light transmitted to the film (or sensor) decreases with the f-number squared. Doubling the f-number increases the necessary exposure time by a factor of four.
The literal interpretation of the

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 isn't 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 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.

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): 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64, 90, 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, 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's 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 (for example 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's 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".

Fractional stops

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's 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 ť 4.5, 5, 5.6, 6.3, 7.1, 8, etc.

To calculate the steps in a full stop (1 EV) one could use ť 2^0*0.5, 2^1*0.5, 2^2*0.5, 2^3*0.5, 2^4*0.5 etc.

The steps in a halv stop (1/2 EV) series would be ť 2^0/2*0.5, 2^1/2*0.5, 2^2/2*0.5, 2^3/2*0.5, 2^4/2*0.5 etc.

The steps in a third stop (1/3 EV) series would be ť 2^0/3*0.5, 2^1/3*0.5, 2^2/3*0.5, 2^3/3*0.5, 2^4/3*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 ť ... 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

sqrt approx (1-m)N

,

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.

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's 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; for example, 1/3 is the intensity ratio.

Although he didn't yet have access to Ernst Abbe's theory of stops and pupils (External Link

), which was made widely available by Siegfried Czapski in 1893, Dallmeyer knew that his working aperture wasn't 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's somewhat more troublesome; for it's obvious that in this case the diameter of the stop employed isn't 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.
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 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 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 8 the "fractional diameter" of the aperture, even though it's 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. Modern conventions have rounded the numbers from 5.66 to 5.6, 11.13 to 11, and 44.72 to 45. This is only for ease of writing – the actual ratio of aperture size to focal length is still based on the doubling or halving of the amount of light getting through the lens.

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 as the hooked italic f as in . 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.

Further Information

Get more info on 'Focal Ratio'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:

<a href="http://f-number.totallyexplained.com">F-number Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.