I wrote the below aricle for MAV, and online magazine for Sony Mavica owners. MAV is now apprently defunct and that URL is being used by a porn site. The article is reproduced here as it was originally submitted to MAV.
24 Feb 00
F-stops Simplified (?)
Introduction:
Some time ago I placed a post on several forums asking for a clarification of the terms "increasing" or "decreasing" the f-stop. Jim Bullard provided an answer to the question by stating that increasing the f-stop means increasing the size of the aperture, not moving to a higher f-number (Bullard). The best course is to avoid confusing terminology and use "stop down" or "open up" to indicate what action is being taken. A number of people who replied to my post provided numerous explanations of the f-stop system as it relates to photography. Unfortunately, there were considerable differences in explanations from various sources. This piqued my curiosity and I began to research a variety of photography texts and online photography definition listings. Much to my surprise, I also found a great deal of inconsistency between supposedly authoritative sources, and even inconsistencies within single sources. Since photography as a field of professional endeavor has been around for more than 150 years, one would assume that definitions and terminology would have been set in stone a long time ago and would be strictly adhered to by photography instructors and practitioners of professional photography. Such seems not to be the case; therefore I have attempted to develop my own list of f/stop terminology using what appears to be the most commonly agreed upon definitions among various authorities that were readily available to me. Several of the definitions may refer to the same topic. I have also stepped through the math involved in a way that I hope clarifies the subject in an easy to follow manner.
DEFINITIONS:
Aperture. Lens opening. The opening formed by a fixed or adjustable diaphragm through which light passes to expose the film (MIR). Aperture size is usually calibrated in f -numbers--the larger the number, the smaller the lens opening (Kodak). Related terms that may be seen in the literature include:
Effective aperture. The diameter of the bundle of light rays striking the first lens element and which actually pass through the lens at any given diaphragm setting (MIR). Focal length (FL) divided by a specific f-number determines the effective aperture diameter (Agfa:7).
Effective diameter of the lens. Diameter of the diaphragm opening (Davis and Binau:5).
Entrance pupil diameter. Aperture diameter. D (or d) = f/f-stop (Graflex).
Limiting aperture. The actual size of the aperture formed by the iris diaphragm at any setting. Determines, but usually differs from the effective aperture (MIR).
Relative aperture or relative aperture ratio. Also known as f/number = focal length divided by diameter of effective aperture (Bruce:40).
Diaphragm or Iris. A group of thin metal sheets placed between the lens elements. The opening in the diaphragm is variable and limits the amount of lens surface actually used. The diaphragm has three purposes (Davis and Binau:5):
1. To regulate the amount of light entering the camera through the lens.
2. To reduce lens defects to a minimum.
3. To vary the depth of field.
Diaphragms may be manual or automatic.
f-number, f-stop, f/stop : This is the number which appears on the aperture adjustment ring on the lens barrel and which indicate the relative size of the lens aperture opening. The f-number series is a geometric progression based on changes in the size of the lens aperture as it is opened and closed. As the scale rises, each number is multiplied by a factor of 1.4. The standard numbers for calibration are 1.0,1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, etc., and each change results in a doubling or halving of the amount of light transmitted by the lens to the film plane. Basically, calculated from the focal length of the lens divided by the diameter of the bundle of light rays entering the lens and passing through the aperture in the iris diaphragm (MIR). f-number = FL/d where FL = focal length of the lens and d = diameter of the diaphragm opening. Also called f-stops, they work in conjunction with shutter speed to indicate exposure settings (Kodak). May be written as 1.4, f/1.4, f-1.4, 1:1.4, etc.
Effective f-number. f-number corrected for bellows factor (Graflex).
F/stop (f/8, etc.)
(a). Most common notation used to indicate the f-stop. For example, f/8 indicates that the diameter of the opening is 1/8 of the focal length of the lens (Bullard). This notation is used as being synonymous with other notations such as f-8, 1:8, etc.
(b). A fraction used to determine the actual diameter of the aperture: the "f" represents the lens focal length, the slash means "divided by," and the word "stop" is a particular f-number; for example, with a 50mm f/1.4 lens, the actual diameter of its maximum aperture is 50mm divided by 1.4 or 35.7mm; at f/2, the diameter becomes 50mm/2 or 25mm; at f/2.8, the aperture is 50mm/2.8 or 17.9mm across; as the numerical value of the f-number increases, the aperture decreases in size (MIR).
f/value of the lens. The ratio of its focal length to the diameter of the opening of the lens.
f/value system. System of f-numbers used to calibrate diaphragm openings that was devised to standardize the markings on a wide variety of cameras. Being a ratio of the lens' focal length and opening diameter, the same f/number can be expected to transmit the same amount of light on any camera, whatever the focal length of the lens, provided that other factors remain constant (Bruce:42).
Focal length of a lens (FL). Distance back of the lens at which an infinite object is brought to a sharp focus (Davis and Binau:5). Distance between the film and optical center of the lens when the lens is focused on infinity (Kodak). A focal length lens of approximately the diagonal of the film frame is considered a "normal" lens which means that the image will be neither wide-angle nor enlarged. A focal length significantly larger than the diagonal of the film frame will result in image enlargement, and a focal length significantly smaller than the lens focal length will result in image reduction from normal, that is, a wide-angle image. Long focal length lenses provide a narrow angle of view and short focal length lenses provide a wide angle of view. (MIR).
Lens size. Some sources use the term "lens size" when they really mean lens focal length.
Stops. Various openings of the diaphragm, f-numbers, f/32, etc. Derived by dividing the focal length of the lens by the diameter of the diaphragm opening. The term is a hold over from the days when metal disks with various sized holes were inserted into the lens barrel to change its effective diameter. The metal disks were known as stops. Stop is a particular f-number (Davis and Binau:5). Also known as Waterhouse stops ( Adams :46).
Speed of a lens (maximum light gathering power). Proportional to the diameter of the lens such that f-number = FL/d where FL = focal length and d = diameter of the lens (Davis and Binau:52). Once the lens in mounted, the speed of the lens will be dependent upon the maximum achievable opening of the diaphragm. Lenses with low f-numbers in relation to their focal lengths are commonly referred to as "fast" lenses. On most lenses, and all American lenses, the lowest f/number represents the value of the maximum aperture. Usually written as 1:1.2, 1:3.5, etc. on the front of the lens barrel (Agfa:7).
CALCULATIONS:
1. The f-stop is determined by dividing the focal length of the lens by the diameter of the aperture. Knowing any two of the three factors, it is possible to mathematically determine the remaining factor. f-stop or f-number = FL/d.
2. Since the diameter of a circle is twice the radius, the formula for the f-stop could be written as f-stop = focal length/2r.
3. The area of a circle is determined by the equation a = p (d/2) 2 or a = p r 2 where r is the radius and d is the diameter.
4. For convenience, the f-stop system was originally set up so that one change in shutter speed either doubled or halved the amount of light entering the camera, and one change in f-stop did the same. This procedure allows the photographer to maintain the same exposure for a variety of f-stop and shutter combinations so long as a one click change in shutter speed is combined with the appropriate one stop change in aperture (Davis and Binau:5). Shutter speeds commonly used (in seconds) are 1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/125, 1/250, etc. Each change of one click doubles or halves the amount of light entering the camera. An additional benefit of this system is that it works for all cameras and lenses when selecting combinations of f-stops and shutter speeds. The relative value of the opening of the diaphragm causes the amount of light on the film to be a constant value between lenses regardless of focal length, in effect, as the focal length of the lens increases, the light reaching the film is kept constant by increasing the diameter of the opening (Bullard). Therefore, it isn't necessary to adjust a light meter for different lenses or the type of camera format being used when determining shutter speed and f-stop combinations.
6. Because each click in shutter speed either doubles or halves the amount of light entering the camera, the area of the aperture opening must compensate by halving or doubling the aperture area in order to maintain the same exposure or equal amount of light entering the camera.
7. Halving or doubling the area of the aperture opening in a consistent way is related to the formula for the area of a circle, a = p r 2 . That is, the radius (and diameter) of the aperture must be changed appropriately so that the area of the aperture is either doubled or halved as the f-stop is opened or closed by one click.
8. Since r is squared in the equation, the question becomes: what change in r must take place so that when the new r is squared the result will be double or half the value of the previous r 2 ? That is, 2 p (r a ) 2 = p (r b ) 2 (opening/aperture is doubled in area where the original radius is 'a' and new radius is 'b'), or 1/2 p (r a 2 ) = (r b 2 ) (aperture area is cut in half) .
Another way of writing these equations would be to remove p from both sides and use 'xr' to indicate the size of the new radius (some number multiplied by the previous radius). Thus 2r 2 = (xr) 2 = x 2 r 2 (double the area), and 1/2r 2 = (xr) 2 = x 2 r 2 (half the area). Removing r 2 from each side of these two equations, we have 2 = x 2 (double the area) and 1/2 = x 2 (half the area). Thus x = 1.414 in the first case and 1/1.414 in the second case. This means that to double the area of a circle the radius or diameter must be multiplied by 1.414, and to reduce the area by 1/2 the radius must be multiplied by 1/1.414 (that is, divided by 1.414).
To summarize, if you multiply the radius (or diameter) of any particular aperture opening by 1.414, you will be doubling the area of the aperture and doubling the exposure. If you divide the radius of any particular aperture opening by 1.414, you will be halving the area of the aperture and halving the exposure.
9. How does all this result in the familiar f-stop notations we have on our lenses? To see how this works, merely make a table beginning with an arbitrary aperture radius of 1 and increase each additional radius by multiplying by 1.414 as below:
r a = 1 (inch, cm, mm, whatever), r b = 1.4, r c = 2, r d = 2.8, r e = 4, r f = 5.6, r g = 8,
r h = 11, r I = 16, r j = 22, etc. Note that when you multiply a number by the square root of 2 and then do so again it is the same as multiplying by 2. This means than every other f/stop in the series is a whole number multiple of 2, that is, f/2, f/4, f/8, f/16, f/32 f/64, etc. The intermediate f/stops, f/2.8, f/5.6, etc. are not whole numbers and have been rounded off to one decimal place. The reason they are not whole numbers is that the square root of 2 is not a whole number. For example, multiplying f/4 by 1.4142, or dividing f/8 by 1.4142, results in an f/stop of 5.6. Multiplying f/8 by the square root of 2 results in an f/stop of f/11, and so on ( Adams :47).
As a historical note, an old European aperture sequence consisted of f/4.5, f/6.3, f/9, f/12.7, f/18, f/25, f/36, etc. That ratio between the stops was the same as the current system, but the whole number stops were f/9, f/18, f/36, etc. ( Adams :47).
10. Why does a larger f-stop indicate a smaller aperture opening? Remember: f-stop = focal length/2r. That is, the diameter of the aperture is being divided into the focal length, thus as the diameter of the aperture gets smaller, the f-number which appears on the camera barrel will increase in size and vice-versa. To see how this works, set up a table with any particular focal length lens such as 50mm. As you move to the right, decrease each aperture radius (or diameter) by dividing by 1.414 (since the diameter is twice the radius, changing the radius by 1.414 will also change the diameter by 1.414).
In this example, start with an aperture diameter of 50mm which provides an f-stop of f/1, then divide each successive aperture diameter by 1.414. As you move to the right, compute the new f-stops (approximate numbers in each case) by dividing the diameter into the FL.
FL: 50mm 50mm 50mm 50mm 50mm 50mm 50mm 50mm 50mm
Aperture
Diam.: 50mm 35mm 25mm 18mm 12.5mm 9mm 6.3mm 4.5mm 3mm
f-stop: f/1 f/1.4 f/2 f/2.8 f/4 f/5.6 f/8 f/11 f/16
Note also that the light admitted is inversely proportional to the squares of the f/values. For example, f16 admits only one-fourth as much light as f8 (8 2 /16 2 = 64/256 = 1/4) (Davis and Binau:22).
11. As previously noted, a significant advantage of this system is that under similar lighting conditions a particular combination of shutter speed and aperture will result in the same amount of light falling upon each square inch or centimeter of a frame of film/sensor regardless of the size camera being used. This means that if you are using a light meter you do not have to tell the meter what kind of camera you are using or the focal length of your lens. That is, a 4 X 5 (102mm X 127mm) and 2 1/4 X 2 1/4 (57mm X 57mm) format camera photographing the same scene at f/8 and with the same ISO film would also be set for the same shutter speed. For example, a 4 X 5 camera with a 165mm (normal) lens and set at f/8 would have an aperture diameter of about 20.6mm (165/8 = 20.6). The 2 1/4 camera might have a 82mm lens for normal image size (a little larger than the frame diagonal) and at f/8 would have an aperture diameter of about 10.25mm (82/8 = 10.25).
If the above statements are true, the ratio between the two film sizes and the areas of the diaphragm openings at f/8 should be approximately the same. This would indicate that each is receiving the same amount of light per square mm of film (note that light enters the camera as a circle, but the image is restricted to a rectangular portion of that circle by internal camera design). The area of the 4 X 5 film frame is about 12,954mm 2 (102mm x 127mm) and the area of the 2 1/4 film frame is about 3,249mm 2 (57mm x 57mm). This is an area ratio of about 4:1 between the larger film size and the smaller. The area of the aperture for the 4 X 5 camera at f/8 is = p r 2 or p (10.3) 2 = 106 p mm 2 . The area of the aperture for the 2 1/4 camera would be p (5.125) 2 or 26.3 p mm 2 . This is also a ratio of about 4:1; thus the ratio of aperture areas is approximately equal to the ratio of the film areas meaning that each would receive equal exposures at f/8 with equal shutter settings. This is only a rough comparison because the light beam is circular and the film frame is rectangular, but it is close enough to illustrate the point. Also, all equations in this list of definitions are true only for subjects which are at infinity. When the subject is closer than infinity, or when a bellows extension is used, correction factors must be referred to if an extremely accurate exposure is required. Also, zoom lenses are a different ball game as concerns f-stops.
12. Although the f/stop and shutter speed system is designed to control exposure by doubling or halving the amount of light admitted, nature does not provide light variations in such convenient stair-step fashion. As a result, a light meter reading may show that a particular f/stop admits too much light, and that the next full stop admits too little light. To provide more precise exposure control between full stops, some professional cameras have f/stop markings in between the standard f/stop notations noted above. Cameras with built-in light meters and automatic aperture and shutter controls may allow an infinite variety of aperture openings so as to precisely control exposure in accordance with the light present. Intermediate positions for manually adjustable apertures are in increments of either one-half stop or one-third stop. One-third stop intervals may correspond to a change in film speed from one ISO index number to the next ( Adams :47). For example, changing from ISO 100 film at f/8 to ISO 200 film would call for a change to f/11, but ISO 125 film might best be matched by f/8 plus 1/3rd stop. Also, exposure bracketing may sometimes best be done with partial stop changes because a full stop up or down might be excessive.
Another case where a marked f/stop on a lens may not correspond to a full stop is the
maximum aperture or "speed" of a lens. For example, markings may begin at f/3.5 with the next marking being f/4 (Bruce:43).
In summary, full f-stop notations indicate the relationship in sizes of circle radii or diameters which will double or halve the area of a circle (aperture in this case). A smaller f-stop number indicates a larger opening because the diameter of the opening (aperture) is on the bottom of the fraction in the equation (d = FL/f-number) and is divided into the focal length. A larger diameter opening thus equates to a smaller f-number and vice versa.
If you are not sufficiently confused at this point, you may wish to refer to the Graflex Photographic Lenses Tutorial which may be found at the Graflex web site: http://www.graflex.org/, for an in-depth explanation of a variety of topics concerning lenses.
For more on f/stops, exposure, lenses and related topics, I highly recommend Jim Bullard's excellent online tutorial. Jim was a key source of material for this article and was more than patient in answering my many questions.
Another helpful person is Peter iNova, Vice President of Metavison and an expert in media display technologies. Although Peter's Metavison duties keep him quite busy, he takes the time to monitor certain digicam forums and provide informative posts on a variety of topics.
One of the better digicam forums for obtaining camera information is INFO-SHARE operated by Arthur Bleich. Art is a nationally known expert on digicams and writes columns for a number of magazines and web sites. He monitors the posts placed on INFO-SHARE and personally answers questions from amateurs such as myself. A highly recommend site.
An interesting site is www.allexpets.com . A number of individuals with expertise in a wide variety of areas, including photography, have volunteered to answer questions concerning their area of specialty free of charge. There are currently 8-10 photographers listed who will attempt to answer questions that may be puzzling you.
Many thanks to Arthur Bleich and Peter iNova for reviewing this article and supplying numerous helpful suggestions.
Adams, Ansel. The Camera . New York Graphic Society. Little, Brown and Company. 1980.
Agfa. A Guide to Digital Photography . Mortsel , Belgium . Agfa-Gevaert N.V. 1996.
Bleich, Arthur. INFO-SHARE forum. http://www.dpcorner.com/cgi-bin/share/index.cgi or arthur@dpcorner.com
Bullard, Jim. Photography Share Lessons: Why f/8 is f/8 . Internet. http://www.northnet.org/jimbullard/lessons.htm , or bullard@northnet.org . 1 Feb 00.
Davis , F.W. and H.G. Binau. Basic Photography . Columbus , OH . The Department of Photography of the Ohio State University . 1956.
Graflex. Photographic Lens Tutorial . Internet. http://www.graflex.org/lenses/photographic-lenses-tutorial.html . 2 Feb 00.
Bruce, Helen Finn. Your Guide to Photography: A Practical Handbook . USA . Barnes & Noble, Inc. 1965.
iNova, Peter. http://www.metavision.com/ or inova@metavision.com .
Kodak. A Glossary of Photographic Terms . Internet. http://www.kodak.com/global/en/consumer/glossary/glossaryContentsText.shtml . 1 Feb 00.
MIR (Malaysian Internet Resources). A Glossary of Photographic Terms . http://www.mir.com.my/rb/photography/glossary/index.htm . 1 Feb 00.