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 Focal ratio in telescopes 6 Working f-number 7 History 7.1 Origins of relative aperture 7.2 Aperture numbering systems 7.3 Typographical standardization 8 See also 9 References 10 External links

Notation 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.

Effects on image sharpness Comparison of f/32 (top-left corner) and f/5 (bottom-right corner) Shallow focus with a wide open lens Depth of field increases with f-number, as illustrated in the image here. This means that photographs taken with a low f-number (large aperture) will tend to have subjects at one distance in focus, with the rest of the image (nearer and farther elements) out of focus. This is frequently used for nature photography and portraiture because background blur (the aesthetic quality of which is known as 'bokeh') can be aesthetically pleasing and puts the viewer's focus on the main subject in the foreground. The depth of field of an image produced at a given f-number is dependent on other parameters as well, including the focal length, the subject distance, and the format of the film or sensor used to capture the image. Depth of field can be described as depending on just angle of view, subject distance, and entrance pupil diameter (as in von Rohr's method). As a result, smaller formats will have a deeper field than larger formats at the same f-number for the same distance of focus and same angle of view since a smaller format requires a shorter focal length (wider angle lens) to produce the same angle of view, and depth of field increases with shorter focal lengths. Therefore, reduced–depth-of-field effects will require smaller f-numbers when using small-format cameras than when using larger-format cameras. Image sharpness is related to f/number through two different optical effects: aberration, due to imperfect lens design, and diffraction which is due to the wave nature of light.[12] 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 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[citation needed]. 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. Even if aberration is minimized by using the best lenses, diffraction creates some spreading of the rays causing defocus. To offset that use the largest lens opening diameter possible (not the f/ number itself). 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. Photojournalists have a saying, "f/8 and be there", meaning that being on the scene is more important than worrying about technical details. Practically, f/8 allows adequate depth of field and sufficient lens speed for a decent base exposure in most daylight situations.[13]

Human eye Computing the f-number of the human eye 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 f/8.3 in a very brightly lit place to about f/2.1 in the dark.[14] Note that computing the focal length requires that the light-refracting properties of the liquids in the eye are taken into account. Treating the eye as an ordinary air-filled camera and lens results in a different focal length, thus yielding an incorrect f-number. Toxic substances and poisons (like atropine) can significantly reduce the range of aperture. Pharmaceutical products such as eye drops may also cause similar side-effects. Tropicamide and phenylephrine are used in medicine as mydriatics to dilate pupils for retinal and lens examination. These medications take effect in about 30–45 minutes after instillation and last for about 8 hours. Atropine is also used in such a way but its effects can last up to 2 weeks, along with the mydriatic effect; it produces cycloplegia (a condition in which the crystalline lens of the eye cannot accommodate to focus near objects). This effect goes away after 8 hours. Other medications offer the contrary effect. Pilocarpine is a miotic (induces miosis); it can make a pupil as small as 1 mm in diameter depending on the person and their ocular characteristics. Such drops are used in certain glaucoma patients to prevent acute glaucoma attacks.

Focal ratio in telescopes 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 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 (f/1.2) is made possible by an error correction system which includes secondary and tertiary mirrors, a three element refractive system and active mounting and optics.[15]

Working f-number The f-number accurately describes the light-gathering ability of a lens only for objects an infinite distance away.[16] 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. A practical example of this is, that when focusing closer, the lens' effective aperture becomes smaller, from e.g. f/22 to f/45, thus affecting the exposure. The working f-number Nw is given by: N w ≈ 1 2 N A i ≈ ( 1 + | m | P ) N {\displaystyle N_{w}\approx {1 \over 2\mathrm {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 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 "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, John Henry Dallmeyer called the ratio 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] Aperture numbering systems 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. 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."[24] Here is the situation as seen in 1899: 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 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: f, or f.[27] Notations for f-numbers were also quite variable in the early part of the twentieth century. They were sometimes written with a capital F,[28] sometimes with a dot (period) instead of a slash,[29] and sometimes set as a vertical fraction.[30] 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.

See also Physics portal Film portal Circle of confusion Group f/64 Photographic lens design Pinhole camera Preferred number

References ^ 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 Camera Films". Eastman Kodak. November 2000. Archived from the original on 2002-10-02. Retrieved 2007-09-02.  ^ Marianne Oelund, "Lens T-stops", dpreview.com, 2009 ^ Michael John Langford (2000). Basic Photography. Focal Press. ISBN 0-240-51592-7.  ^ Levy, Michael (2001). Selecting and Using Classic Cameras: A User's Guide to Evaluating Features, Condition & Usability of Classic Cameras. Amherst Media, Inc. p. 163. ISBN 978-1-58428-054-5.  ^ Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. ISBN 0-201-11609-X.  Sect. 5.7.1 ^ Charles F. Claver; et al. (19 March 2007). "LSST Reference Design" (PDF). LSST Corporation: 45–50. Retrieved 10 January 2011  ^ a b Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. ISBN 0-8194-5294-7.  p. 29. ^ Thomas Sutton and George Dawson, A Dictionary of Photography, London: Sampson Low, Son & Marston, 1867, (p. 122). ^ a b John Henry Dallmeyer, Photographic Lenses: On Their Choice and Use – Special Edition Edited for American Photographers, pamphlet, 1874. ^ Southall, James Powell Cocke (1910). "The principles and methods of geometrical optics: Especially as applied to the theory of optical instruments".  ^ a b Siegfried Czapski, Theorie der optischen Instrumente, nach Abbe, Breslau: Trewendt, 1893. ^ Henry Crew, "Theory of Optical Instruments by Dr. Czapski," in Astronomy and Astro-physics XIII pp. 241–243, 1894. ^ Thomas R. Dallmeyer, Telephotography: An elementary treatise on the construction and application of the telephotographic lens, London: Heinemann, 1899. ^ C. H. Bothamley, Ilford Manual of Photography, London: Britannia Works Co. Ltd., 1891. ^ John A. Hodges, Photographic Lenses: How to Choose, and How to Use, Bradford: Percy Lund & Co., 1895. ^ C. Welborne Piper, A First Book of the Lens: An Elementary Treatise on the Action and Use of the Photographic Lens, London: Hazell, Watson, and Viney, Ltd., 1901. ^ Conrad Beck and Herbert Andrews, Photographic Lenses: A Simple Treatise, second edition, London: R. & J. Beck Ltd., c. 1902. ^ Google search ^ Ives, Herbert Eugene (1920). Airplane Photography (Google). Philadelphia: J. B. Lippincott. p. 61. Retrieved 12 March 2007.  ^ Mees, Charles Edward Kenneth (1920). The Fundamentals of Photography. Eastman Kodak. p. 28. Retrieved 12 March 2007.  ^ Derr, Louis (1906). Photography for Students of Physics and Chemistry (Google). London: Macmillan. p. 83. Retrieved 12 March 2007.

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