f-number

Diagram of decreasing apertures, that is, increasing f-numbers, in one-stop increments; each aperture has half the light-gathering area of the previous one.

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").^[1]^[2]^[3] It is also known as the focal ratio, f-ratio, or f-stop, and is very important in photography.^[4] It is a dimensionless number that is a quantitative measure of lens speed; increasing the f-number is referred to as stopping down. The f-number is commonly indicated using a lower-case hooked f with the format f/N, where N is the f-number.

The f-number is the reciprocal of the relative aperture (the aperture diameter divided by focal length).^[5]

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In this video we're going to cover a concept called "f-stop". In brief, f-stop is simply one of the settings on your camera that uses a set of numbers to determine how much light is coming into the camera through the lens. If we look at a top-down view of a camera here, we have a lens up here at the front, we have our film or sensor here at the back, and in between these two is a mechanism called the "iris". The iris is basically a circular shield with a hole in the center that can open or close, depending on how much light you want to let in the camera. The hole itself is called the "aperture". The mechanism that's making this hole is called the iris. This is good to know because most of the time you'll see the aperture referred to, but sometimes you'll see the iris referred to, and it's just good to know that usually we're just interested in the aperture. So the effect of this is that with an aperture open wide, you're going to be letting in a lot of light into your camera through the lens. But with a small aperture, you're not really going to be letting in that much light. The f-stop numbers are basically a scale of numbers for talking about how much light we want coming into the camera with regard to how wide open the aperture needs to be to that amount of light. With the f-stop scale of numbers, every major point on the scale either doubles or halfs the amount of light, depending on if you're going up or down the scale. So if we have these different aperture sizes here every time we go to the next aperture size, we're doubling the amount of light. But we need to pay attention to one interesting thing here about the size of the aperture. Let's use this one for example and say it's got a diameter of twenty millimeters. That means the area of the aperture is 314 millimeters. This is important because, basically, the area of the aperture is saying, "this is how much light were letting through this aperture." So if we go up to the next size aperture in the f-stop scale, we're doubling the area of the aperture. Which means, were saying we're letting in double the amount of light into this aperture. But what about the diameter of the aperture? Is that doubling too? No. It turns out, in order to double the area of the aperture, we need to multiply the diameter of the aperture by the square root of two. This will end up doubling the area of the aperture. So then if we want to go up to the next aperture size, we're going to double the area again. Because this is going to let in, again, double the amount of flight as the previous aperture size. So again, in order to double the area, we're going to need to take the diameter size and multiply it by the square root of two. And now we're going to have a diameter of forty millimeters. So this number, the square root of two, this is an important number because every time we multiply the diameter by the square root of two, we're doubling the amount of light. Just the same, if we were to do the opposite and divide the diameter by the square root of two, we're going to get half the amount of light. So remember when I mentioned before that the f-stop scale is a way to talk about the amount of light coming in with regard to the size of the aperture. So why don't we just make a scale of aperture sizes? And say, "Oh, well if you want to let in, you know, a certain amount of light, just open up your aperture to forty millimeters." Well, the reason is that not all lenses are the same, and this same aperture size is going to let in different amounts of light depending on the lens you use. I'll show you why. So here we have two different setups with two different amounts of light we want to get. Here on the top we're getting a lot of light. And here on the bottom we're getting not so much light. In the top here, in order to get this amount of light, we would need an aperture size of, say, twenty millimeters. And here in the bottom, in order to get this amount of light, we would need an aperture size of, say, ten millimeters. In both of these examples, the lens is the same distance from the film. This distance is called the focal length, and in this example the focal length is forty millimeters. Okay, that's fine, but what if we have a lens way out here, say at a focal length of eighty millimeters? Well now, in the top example, we're goint to need a aperture size of, say, forty millimeters in order to let in the same amount of light, and in the bottom example we're gonna now need an aperture size of, say, twenty millimeters in order to let in this amount of light. Do you see the problem yet? The size of the aperture is not the only thing determining how much light is coming in. For example, here's two same-size apertures letting in different amounts of light, and here's two different size apertures letting in the same amount of light. So in order to really think about the amount of light coming in, We also need to think about the focal length as well as the size of the aperture. So if we're going to talk about the amount of light coming in by considering both the diameter and the focal length, let's look at how those would relate. So in the top example here when we had our forty millimeter focal length and our twenty-millimeter aperture, that was letting in the same amount of light as when we had our eighty millimeter focal length and our forty millimeter aperture. These are letting in the same amount of light. Just as down here on the bottom example, when we had a forty millimeter focal length and a ten millimeter aperture, that was letting in the same amount of light as an eighty millimeter focal length with a twenty millimeter aperture. Calculating these down, we see that in the top example these are both equal to two, and in the bottom example, these are both equal to four. These don't mean anything in terms of units, like millimeters or units of light or anything like that. All they really represent is the proportion of focal length and aperture diameter that will let in a certain amount of light. That's it. That's all that represents. But this is so important, because it means that no matter what lens setup you're using, if the focal length and the diameter come out to this proportion, then you're going to be letting in this amount of light. And this number here, that's the number used for the f-stop scale. This is why you can see you can have different size lenses and know that, despite being different size or shape, as long as you set it to the same f-stop number, you'll be getting the same amount of light. So here we have these three numbers here. We have the focal length, we have the diameter, and we have, you know, X, representing the proportion between these two (our f-stop number). How would we put these into a formula that we can use to see how they're all related? So in this example here, we would have our focal length divided by diameter is going to equal our X-number, our f stop number. Well, remember, f-stop's all about being able to know how wide open to make the diameter, so we need a formula will tell us the diameter. So let's rewrite this formula like this: f divided by x equals d. Now, if you remember your high school math, we can swap these two and these equations will still mean the same thing. They will still be balanced out with each other. So now we can say, "at this focal length, if we want this amount of light, we need a diameter of this size. This side of the question here might look a little bit familiar because the f-stop or f-number you may often see written like this: f slash x, or with an actual number, f slash two, just as an example. So now we're mostly there. We know what the f-stop number is and what it represents (the amount of light coming in), but we haven't learned how the f-stop scale works and why it uses such specific numbers. In order to understand how the f-stop scale works, we need to go back to the scale of apertures we're working with before. So remember these aperture sizes we were dealing with. Also remember that these aperture sizes are very specific. They're specific because we multiplied or divided the diameter by the square root of the two, depending on whether we wanted double the amount of light or half the amount of light. So let's put our f-stop formula here and see how these relate. So we have our focal length divided by our x number (our f-stop number) gives us a diameter. So remember, if we're multiplying or dividing the diameter every time we go up or down the f stop scale, in order to keep this equation balanced, we're going to need to do the opposite to x in this equation. So, for example, let's say we're going to double the amount of light. Well, in order to do that we need to multiply the diameter by the square root of two. Now, in order to keep this question balanced, we're going to need to divide the f-stop number by the square root of two. Let's plug in a real example and see how this would work. So with these apertures, let's say were dealing with the focal length of forty millimeters. So let's take our twenty-millimeter aperture size here, for example, and see what the f-stop number would be for this. Well, if we have our forty millimeter focal length here, and we're ending up with a twenty-millimeter diameter, that means our f-stop number is obviously going to be two. So let's put two here as our f-stop number. Now, we want to go to the next size aperture and it get double the amount of light, so we're going to have to multiply our diameter by the square root of two to double the amount of light. Well, that means, remember, we're going to have to divide our f-stop number by the square root of two in order to keep this balanced. So now we have a new f-stop number of one point four. Now, you see, every time we go up a number in the f-stop scale, we're dividing our f-stop number by the square root of two. And every time we go down the scale, we're multiplying our f-stop number by the square root of two. So even though the f-stop numbers seem kind of random, they're really not. They're just being divided or multiplied by the square root of two, depending on if you want double the light or half the light. So now let's take a look at the f-stop scale itself. So here's the f-stop scale and every point on the scale represents double the amount of light compared to the f-stop number below it. And remember, as we multiply the size of the diameter, and it gets bigger and bigger, we're dividing the f-stop number and it gets smaller and smaller. So the small f-stop numbers here at the top represent larger apertures, and the larger f-stop numbers here at the bottom represent smaller apertures. The scale still looks confusing though, and I can imagine it'd be hard to remember the sequence of numbers, so I'm going to show you a small trick here that will make the scale easy to remember. Let's start at one and look at every other number. We have one, two, four, eight, sixteen, thirty-two. Starting to notice the pattern? Every time we skip a number, we're doubling the number. Let's start at one point four. One point four, two point eight, five point six, and then rounding when we double again, we get eleven, and then twenty two, and then forty four... See the same pattern? We're just doubling the number every time we skip a number. So as long as you remember these first two numbers, one and one point four, if you only remember these two numbers, you can build out the entire rest of the scale quickly and easily in your head just by skip-counting and doubling each time, starting with one and one point four. It makes it a lot easier to remember. Some last bits of information about f-stop... If someone says they're going up a stop, or down a stop, that just means they're either going to a larger aperture size or a smaller aperture size. Going up a stop means you're doubling the amount of light, going down a stop means you're halving the amount of light. If someone says they have a fast lens or a slow lens, that's just referring to the size of the aperture on that lens that it can do. A fast lens can have a wide open aperture, which means you can let in a lot of light; a slow lens wil only be able to have a much smaller aperture and let in a lot less light. This means that on a fast lens, your shutter speed can be fast because you're letting in a lot of light all at once. You don't need it open that long. Same with a slow lens, you have a slower shutter speed because you need your shutter open longer in order to let in the right amount of light. Shutter speed is its own topic and i'm going to be covering than a separate video because it's too big of a topic to cover here. Depth of field is a subject that f-stop plays a crucial role in. Depth of field talks about the area of your photo that's in focus. So for instance if you have a photo where you need a nice blurry background or you need a landscape photo where you need everything in focus, that's what depth of field covers. And f-stop, as I said, plays a key role, so if you want understand how to control this, I suggest you watch this video. Then we have lens ratings and in lens ratings you'll often see the maximum aperture size listed. So, for instance, you might have an eighty millimeter lens with an f/4 as its maximum aperture size, or you might see something like a fifty five millimeter to two hundred millimeter lens where the focal a can vary between these two, but at this focal length you might have an aperture size of four, where at this focal length you'd have an aperture size of five point six. This is due to the construction of the lens where it will only allow a certain maximum aperture size at different focal lengths. Lastly, when we say we have an aperture size of twenty millimeters, we don't mean that the physical iris inside the camera, where we have our lens, and our film, and our iris here, it doesn't mean that iris has a physical opening of twenty millimeters. It may be actually much much smaller than that. What we're talking about is if you were to look at the front of your camera, straight into the entrance of your lens, you would see the aperture size looking like it's twenty millimeters. That's what's twenty millimeters, and that's what you're actually calculating with your f-stop, not the physical size itself. I want to mention this because if you happen actually see the iris in your lens, you would be quite surprised to find that it's much, much smaller, usually, than the aperture size that you're gonna find with your f-stop rating. So that's the end of this video. I hope I've answered a lot of questions for you and help you understand what f-stop is and also an easy way to remember the whole f-stop scale. If you have any more questions feel free to leave them in the comments, and thank you for watching.

Notation

The f-number $N$ is given by:

N={\frac {f}{D}}\

where $f$ is the focal length, and $D$ 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 were 10 mm and its entrance pupil diameter were 5 mm, the f-number would be 2. This would be expressed as "f/2" in a lens system. The aperture diameter would be equal to $f/2$ .

Most lenses have an adjustable diaphragm, 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) decreases with the square of the f-number. A 100 mm focal length f/4 lens has an entrance pupil diameter of 25 mm. A 100 mm focal length f/2 lens has an entrance pupil diameter of 50 mm. Since the area varies as the square of the pupil diameter,^[6] the amount of light admitted by the f/2 lens is four times that of the f/4 lens. To obtain the same photographic exposure, the exposure time must be reduced by a factor of four.

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 f/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 is an f-number adjusted to account for light transmission efficiency.

Stops, f-stop conventions, and exposure

A Canon 7 mounted with a 50 mm lens capable of f/0.95

A 35 mm lens set to f/11, as indicated by the white dot above the f-stop scale on the aperture ring. This lens has an aperture range of f/2 to f/22.

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) 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 $1/{\sqrt {2}}$ 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 of numbers that corresponds to the sequence of the powers of the square root of 2: f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32, f/45, f/64, f/90, f/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}{({\sqrt {2}})^{0}}},\ f/1.4={\frac {f}{({\sqrt {2}})^{1}}},\ f/2={\frac {f}{({\sqrt {2}})^{2}}},\ f/2.8={\frac {f}{({\sqrt {2}})^{3}}},\ \ldots

In the same way as one f-stop corresponds to a factor of two in light intensity, shutter speeds 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 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

Computer simulation showing the effects of changing a camera's aperture in half-stops (at left) and from zero to infinity (at right)

Most twentieth-century cameras had a continuously variable aperture, using an iris diaphragm, 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 (1⁄3 EV) are the most common, since this matches the ISO system of film speeds. 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 f/2.8 is f/3.2, two-thirds smaller is f/3.5, and one whole stop smaller is f/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 {2}})^{0},\ ({\sqrt {2}})^{1},\ ({\sqrt {2}})^{2},\ ({\sqrt {2}})^{3},\ ({\sqrt {2}})^{4},\ \ldots

The steps in a half stop (1⁄2 EV) series would be

({\sqrt {2}})^{\frac {0}{2}},\ ({\sqrt {2}})^{\frac {1}{2}},\ ({\sqrt {2}})^{\frac {2}{2}},\ ({\sqrt {2}})^{\frac {3}{2}},\ ({\sqrt {2}})^{\frac {4}{2}},\ \ldots

The steps in a third stop (1⁄3 EV) series would be

({\sqrt {2}})^{\frac {0}{3}},\ ({\sqrt {2}})^{\frac {1}{3}},\ ({\sqrt {2}})^{\frac {2}{3}},\ ({\sqrt {2}})^{\frac {3}{3}},\ ({\sqrt {2}})^{\frac {4}{3}},\ \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 (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 is often not an integral power of √2 (i.e., √2 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 √2.

Modern electronically controlled interchangeable lenses, such as those used for SLR cameras, have f-stops specified internally in 1⁄8-stop increments, so the cameras' 1⁄3-stop settings are approximated by the nearest 1⁄8-stop setting in the lens.^{[citation needed]}

Standard full-stop f-number scale

Including aperture value AV:

N={\sqrt {2^{\text{AV}}}}

Conventional and calculated f-numbers, full-stop series:

AV	−2	−1	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
N	0.5	0.7	1.0	1.4	2	2.8	4	5.6	8	11	16	22	32	45	64	90	128	180	256
calculated	0.5	0.707...	1.0	1.414...	2.0	2.828...	4.0	5.657...	8.0	11.31...	16.0	22.62...	32.0	45.25...	64.0	90.51...	128.0	181.02...	256.0

Typical one-half-stop f-number scale

AV	−1	−1⁄2	0	1⁄2	1	1+1⁄2	2	2+1⁄2	3	3+1⁄2	4	4+1⁄2	5	5+1⁄2	6	6+1⁄2	7	7+1⁄2	8	8+1⁄2	9	9+1⁄2	10	10+1⁄2	11	11+1⁄2	12	12+1⁄2	13	13+1⁄2	14
N	0.7	0.8	1.0	1.2	1.4	1.7	2	2.4	2.8	3.3	4	4.8	5.6	6.7	8	9.5	11	13	16	19	22	27	32	38	45	54	64	76	90	107	128

Typical one-third-stop f-number scale

AV	−1	−2⁄3	−1⁄3	0	1⁄3	2⁄3	1	1+1⁄3	1+2⁄3	2	2+1⁄3	2+2⁄3	3	3+1⁄3	3+2⁄3	4	4+1⁄3	4+2⁄3	5	5+1⁄3	5+2⁄3	6	6+1⁄3	6+2⁄3	7	7+1⁄3	7+2⁄3	8	8+1⁄3	8+2⁄3	9	9+1⁄3	9+2⁄3	10	10+1⁄3	10+2⁄3	11	11+1⁄3	11+2⁄3	12	12+1⁄3	12+2⁄3	13
N	0.7	0.8	0.9	1.0	1.1	1.2	1.4	1.6	1.8	2	2.2	2.5	2.8	3.2	3.5	4	4.5	5.0	5.6	6.3	7.1	8	9	10	11	13	14	16	18	20	22	25	29	32	36	40	45	51	57	64	72	80	90

Sometimes the same number is included on several scales; for example, an aperture of f/1.2 may be used in either a half-stop^[7] or a one-third-stop system;^[8] sometimes f/1.3 and f/3.2 and other differences are used for the one-third stop scale.^[9]

Typical one-quarter-stop f-number scale

AV	0	1⁄4	1⁄2	3⁄4	1	1+1⁄4	1+1⁄2	1+3⁄4	2	2+1⁄4	2+1⁄2	2+3⁄4	3	3+1⁄4	3+1⁄2	3+3⁄4	4	4+1⁄4	4+1⁄2	4+3⁄4	5
N	1.0	1.1	1.2	1.3	1.4	1.5	1.7	1.8	2	2.2	2.4	2.6	2.8	3.1	3.3	3.7	4	4.4	4.8	5.2	5.6

AV	5	5+1⁄4	5+1⁄2	5+3⁄4	6	6+1⁄4	6+1⁄2	6+3⁄4	7	7+1⁄4	7+1⁄2	7+3⁄4	8	8+1⁄4	8+1⁄2	8+3⁄4	9	9+1⁄4	9+1⁄2	9+3⁄4	10
N	5.6	6.2	6.7	7.3	8	8.7	9.5	10	11	12	14	15	16	17	19	21	22	25	27	29	32

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 discs 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). A lens with a T-stop of $N$ projects an image of the same brightness as an ideal lens with 100% transmittance and an f-number of $N$ . A particular lens's T-stop, $T$ , is given by dividing the f-number by the square root of the transmittance of that lens:

T={\frac {f}{\sqrt {\text{transmittance}}}}.

For example, an f/2.0 lens with transmittance of 75% has a T-stop of 2.3:

T={\frac {2.0}{\sqrt {0.75}}}=2.309...

Since real lenses have transmittances of less than 100%, a lens's T-stop number is always greater than its f-number.^[10]

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.^[11]^[12] T-stops are sometimes used instead of f-numbers to more accurately determine exposure, particularly when using external light meters.^[13] Lens transmittances of 60%–95% are typical.^[14] 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.^[13] 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 lenses by Minolta and Sony.

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 f/16 and the shutter speed closest to the reciprocal of the ISO speed of the film; for example, using ISO 200 film, an aperture of f/16 and a shutter speed of 1⁄200 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

Comparison of f/32 (top-left half) and f/5 (bottom-right half)

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 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 (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 which is due to the wave nature of light.^[15] 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. At small apertures, depth of field and aberrations are improved, but diffraction 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) 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 (in 35 mm and larger formats) allows adequate depth of field and sufficient lens speed for a decent base exposure in most daylight situations.^[16]

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.^[17] 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

$Diagram of the focal ratio of a simple optical system where f {\displaystyle f} is the focal length and D {\displaystyle D} is the diameter of the objective$

Diagram of the focal ratio of a simple optical system where

f

is the focal length and

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 $N$ . 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:

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 (about 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.^[18]

Camera equation (G#)

The camera equation, or G#, is the ratio of the radiance reaching the camera sensor to the irradiance on the focal plane of the camera lens:^[19]

G\#={\frac {1+4N^{2}}{\tau \pi }}\,,

where $τ$ is the transmission coefficient of the lens, and the units are in inverse steradians (sr⁻¹).

Working f-number

The f-number accurately describes the light-gathering ability of a lens only for objects an infinite distance away.^[20] 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 $N w$ is given by:^[20]

N_{w}\approx {1 \over 2\mathrm {NA} _{i}}\approx \left(1+{\frac {|m|}{P}}\right)N\,,

where $N$ is the uncorrected f-number, $NA i$ is the image-space numerical aperture of the lens, $|m|$ is the absolute value of the lens's magnification for an object a particular distance away, and $P$ 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. 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.^[21]

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; i.e., 1⁄3 is the intensity ratio.^[22]

Although he did not yet have access to Ernst Abbe's theory of stops and pupils,^[23] which was made widely available by Siegfried Czapski in 1893,^[24] 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.^[22]

This point is further emphasized by Czapski in 1893.^[24] 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."^[25]

J. H. Dallmeyer's son, Thomas Rudolphus Dallmeyer, inventor of the telephoto lens, followed the intensity ratio terminology in 1899.^[26]

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."^[27] 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."^[28]

Here is the situation as seen in 1899:

Piper in 1901^[29] 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.^[30] The R.P.S. had changed their name and moved off of the U.S. system some time between 1895 and 1902.

Typographical standardization

Yashica-D TLR camera front view. This is one of the few cameras that actually says "F-NUMBER" on it.

From the top, the Yashica-D's aperture setting window uses the "f:" notation. The aperture is continuously variable with no "stops".

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.^[31]

Notations for f-numbers were also quite variable in the early part of the twentieth century. They were sometimes written with a capital F,^[32] sometimes with a dot (period) instead of a slash,^[33] and sometimes set as a vertical fraction.^[34]

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 ƒ/ or ƒ: 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.

References

^ ^a ^b Smith, Warren Modern Optical Engineering, 4th Ed., 2007 McGraw-Hill Professional, p. 183.
^ Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. p. 152. ISBN 0-201-11609-X.
^ Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. Bellingham, Wash: SPIE. p. 29. ISBN 9780819452948. OCLC 53896720.
^ Smith, Warren Modern Lens Design 2005 McGraw-Hill.
^ ISO, Photography—Apertures and related properties pertaining to photographic lenses—Designations and measurements, ISO 517:2008
^ See Area of a circle.
^ 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. (2007-03-19). "LSST Reference Design" (PDF). LSST Corporation: 45–50. Archived from the original (PDF) on 2009-03-06. Retrieved 2011-01-10. {{cite journal}}: Cite journal requires |journal= (help)
^ Driggers, Ronald G. (2003). Encyclopedia of Optical Engineering: Pho-Z, pages 2049-3050. CRC Press. ISBN 978-0-8247-4252-2. Retrieved 2020-06-18.
^ ^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". Macmillan: 537. theory-of-stops. {{cite journal}}: Cite journal requires |journal= (help)
^ ^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. ISBN 9780598722225. Retrieved 2007-03-12.
^ Mees, Charles Edward Kenneth (1920). The Fundamentals of Photography. Eastman Kodak. p. 28. Retrieved 2007-03-12.
^ Derr, Louis (1906). Photography for Students of Physics and Chemistry (Google). London: Macmillan. p. 83. Retrieved 2007-03-12.

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