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,[3] 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.

Human eye

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. [4] 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.

Working f-number

The f-number accurately describes the light-gathering ability of a lens only for objects an infinite distance away.[5] 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.[5] (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.

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:[6]

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:[7]

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 [4], which was made widely available by Siegfried Czapski in 1893,[8] Dallmeyer knew that his working aperture was not the same as the physical diameter of the aperture stop:[7]

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.[8] 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."[9]

J. H. Dallmeyer's son, Thomas Rudolphus Dallmeyer, inventor of the telephoto lens, followed the intensity ratio terminology in 1899.[10]

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." [11] 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." [12]

Here is the situation as seen in 1899:

Piper in 1901[13] 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.[14] 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.

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 f/#.[15] Notations for f-numbers were also quite variable in the early part of the twentieth century. They were sometimes written with a capital F,[16] sometimes with a dot (period) instead of a slash,[17] and sometimes set as a vertical fraction.[18]

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

* Circle of confusion
* Depth of field
* Exposure value
* Optical telescope
* Pinhole camera
* Printer points
* Telescope

References

1. ^ Smith, Warren Modern Lens Design 2005 McGraw-Hill
2. ^ Eastman Kodak, "H2: Kodak Motion Picture Camera Films", November 2000 revision. Retrieved 2007-09-02.
3. ^ Michael John Langford (2000). Basic Photography. Focal Press. ISBN 0240515927.
4. ^ Hecht, Eugene (1987). Optics, 2nd ed., Addison Wesley. ISBN 0-201-11609-X. Sect. 5.7.1
5. ^ a b Greivenkamp, John E. (2004). Field Guide to Geometrical Optics, SPIE Field Guides vol. FG01, SPIE. ISBN 0-8194-5294-7. p. 29.
6. ^ Thomas Sutton and George Dawson, A Dictionary of Photography, London: Sampson Low, Son & Marston, 1867, (p. 122).
7. ^ a b John Henry Dallmeyer, Photographic Lenses: On Their Choice and Use—Special Edition Edited for American Photographers, pamphlet, 1874.
8. ^ a b Siegfried Czapski, Theorie der optischen Instrumente, nach Abbe, Breslau: Trewendt, 1893.
9. ^ Henry Crew, "Theory of Optical Instruments by Dr. Czapski," in Astronomy and Astro-physics XIII pp. 241–243, 1894.
10. ^ Thomas R. Dallmeyer, Telephotography: An elementary treatise on the construction and application of the telephotographic lens, London: Heinemann, 1899.
11. ^ C. H. Bothamley, Ilford Manual of Photography, London: Britannia Works Co. Ltd., 1891.
12. ^ John A. Hodges, Photographic Lenses: How to Choose, and How to Use, Bradford: Percy Lund & Co., 1895.
13. ^ 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.
14. ^ Conrad Beck and Herbert Andrews, Photographic Lenses: A Simple Treatise, second edition, London: R. & J. Beck Ltd., c. 1902.
15. ^ Google search
16. ^ Ives, Herbert Eugene (1920). Airplane Photography (Google), Philadelphia: J. B. Lippincott, p. 61. Retrieved on 2007-03-12.
17. ^ Mees, Charles Edward Kenneth (1920). The Fundamentals of Photography. Eastman Kodak, p. 28. Retrieved on 2007-03-12.
18. ^ Derr, Louis (1906). Photography for Students of Physics and Chemistry (Google), London: Macmillan, p. 83. Retrieved on 2007-03-12.

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