Understanding F-stop

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F/Stops

F/stops are a bit confusing to new photographers because the numbers seem so arbitrary. The standard sequence of f/stops ranges from f/1.4 to f/22. Although it may not seem obvious at first, the f/1.4 setting in this sequence lets in the most light while the f/22 setting lets in the least. Also, each of these f/stops has precisely the same halving/doubling relationship as the shutter speed sequence.

At first look, going from f/4 to f/5.6 doesn’t sound like it is half the amount of light. What’s more, 5.6 is a larger number and sounds like it ought to be more light, not less. Neither does f/4 to f/2.8 sound like doubling the amount of light. In fact, each of the numbers in this sequence is a halving/doubling of the amount of light from its immediate neighbours, just like the shutter speed settings are. Not only that, but it makes sense, as I shall show below.
The reason that both the halving and doubling and the smaller numbers mean more light things make sense is that the f/stop is a ratio. The ratio is between the diameter of the aperture in the lens and the focal length of the lens. The focal length is generally measured in millimeters, so we’ll stick with those as our unit of measure. On a 50mm lens, f/2 is saying that the diameter of the aperture is 25mm. The ratio is: 50/25 = 2. That seems pretty straightforward. A good question might be, what is the area of that aperture? Well, the aperture is usually a set of five to fifteen blades which form a roughly circular hole, so we’ll use the formula for the area of a circle, which as I’m sure you’ll recall is pi * radius^2. For π I’ll use 3.14159265. On our 50mm lens, the aperture at f/2 has a diameter of 25mm which is a radius of 12.5mm. The area of the aperture is thus π X 12.52, or 3.14159265 X 156.25, or 490.9 square millimetres.
This fact by itself isn’t all that useful. It is useful in relation to the adjacent f/stops. What is the area of the aperture at f/2.8? Well, because the f/stop is a ratio of the focal length to diameter, our 50mm lens at f/2.8 would have a diameter of 50/2.8 = 17.86mm. Remember, we have to divide that by 2 to get the radius of 8.93mm, so the area of the circle thus formed would be π X 8.932, or 250.5 square mm. Rounding off a bit, that’s about 250 sq. mm at f/2.8 and 500 sq. mm at f/2, a double/half relationship. Aha! So that’s it! The area of the hole doubles and halves, it’s just represented by a ratio on the lens! No wonder it’s so confusing.
Here’s a table of the aperture areas for the common f/stops for a 50mm lens:

If you look down the column of figures on the right, you can see the (more or less) doubling/halving going on up and down the column. You can see also how the big numbers make for smaller areas since the f/stop number is being divided into the focal length, then halved, then squared, then multiplied by π. It’s no wonder this seems obscure.
Why not just call for the area of the aperture directly? A couple of reasons. First of all, if you have a 50mm lens on and say “I shot this with my 50mm at 1/125th of a second and an aperture area of 63 square millimeters” you will impart correct and exact information that precisely zero people will understand. It’s much easier to say “I shot this at 1/125th at f/5.6”. Also, 63 square millimeters is f/5.6 only with a 50mm lens; if your lens is a 35mm, or an 85, or a 300, the ratio changes and the exposure is different. That 63 sq. mm is about f/4 on the 35mm, f/9.5 on the 85mm and f/32 on the 300. Knowing only the area of the aperture requires also knowing the length of the lens to be informative as to the amount of light coming through the lens. The f/stop figure incorporates both of these in one useful if initially confusing measure and the lens length is immaterial. It’s shorthand. When you say f/8, you mean for this focal length (the f?), give me an aperture whose area is such that diameter of the resulting circle goes eight times into my focal length. Fortunately, the lens makers figure out all these things and just mark the f/stops on the lens for us. They’re doing us a big favor.
So, does that mean f/8 is the same brightness of illumination on the film/sensor regardless of focal length of the lens?
Yes. That’s why they’re so handy! It’s also why handheld light meters work–they have no idea what sensor size or camera body or lens you’re using, but they can tell you that, for your selected ISO, 1/125th at f/8 is the correct exposure (or at least one of a whole host of equivalent exposures, we’ll get to that later). It doesn’t matter if you set that f/8 on a 20mm or a 400mm, or if the camera was made in 1954 or last July.

When people talk about a fast lens, what does that mean?
Lenses are referred to by their maximum aperture (that’s the biggest hole, the smaller number). Thus, Nikon made (at least) four 28mm lenses at one point, a 28 f/1.4, a 28 f/2.0, a 28 f/2.8 and a 28 f/3.5. All four of these lenses would stop down to f/4, f/5.6, and so on up to f/16; they were distinguished by the maximum amount of light they could let in. The 28mm f/3.5, one of which I own, when set to its maximum aperture of f/3.5, lets in one third less light that the 28 f/2.8. The 28 f/2.8, in turn, at its maximum aperture, lets in only half the light of the 28 f/2.0 at its maximum aperture. And that 28 f/2.0 lets in only half the light of the 28 f/1.4 at its maximum aperture. Lenses which have wide maximum apertures and let in lots of light are called fast lenses. Lenses which let in comparatively less light at their maximum apertures are called slow lenses. The 28 f/1.4 is a very fast 28mm; 28 f/2.0 would be a fast lens; the 28 f/2.8 would be sort of regular, for which there isn’t really a name; the 28 f/3.5 would be kind of slow. That 28 f/1.4 sold for enough new that Nikon didn’t make many; once discontinued, their price in the used market went up! You can find them on eBay from time to time going for $2,500 or more. Nikon in 2010 introduced a new 24 f/1.4 lens, just a bit wider, so you don’t have to buy used but you still have to pay around $2,500.
Why wouldn’t you always use a fast lens?
Weight and expense. To get those larger diameter apertures means you need larger pieces of glass mounted in correspondingly larger lens barrels. They’re harder to manufacture, the lens barrel keeps getting heavier to hold all that glass in alignment so it all gets weighty in a hurry, they’re harder to autofocus and they’re more challenging optical designs. In the rangefinder world there start to be concerns about the accuracy of the focusing since depth of field gets to be very limited at large apertures. Also, there have been very fast lenses made which have the reputation of being really nice wide open but kind of doggy stopped down. If you normally do not use the fast lens at its widest settings, if you are mostly at, say, f/8, then you are carrying around an unnecessarily heavy and expensive lens which may be underperforming its cheaper brethren stopped down.
The size penalty is really obvious in the long lenses. The weight balloons and the cost skyrockets. For instance, I used to own a Nikon 300mm f/4.5 ED-IF lens. The IF is internal focus, the ED had to do with the Extra-low Dispersion glass used. It was a sweet lens, 300mm in length, with silky smooth (manual) focusing and weighed in at 2 lbs. 2.9 oz. (989g). If I stepped up to the 300 f/2.8 lens the weight went to 5 lbs. 8 oz (2500g). Not fast enough? How about Nikon’s 300 f/2? It weighed in at 15 lbs. 6.9 oz. (7000g). The 300 f/2 picks up 2 1/3 stops over the 300 f/4.5 I owned, but it takes an eminently hand-holdable telephoto that fits in the camera bag and makes it into an unwieldy unit needing a tripod, requiring its own suitcase and weighing seven times as much. The 300 f/2s still sell used in the $20,000+ range.
You also see the size in the zoom lenses. Most modern consumer zooms are handy, light and slow. They are also have a variable maximum f/stop across their zoom range. The professional level lenses tend to be fixed f/stop (the Nikon 70-200 f/2.8, for instance, is f/2.8 across the whole range, is a standard pro lens, but it weighs a lot and costs $2,500). In everything there are tradeoffs.
Even on shorter lenses the difference is noticeable; my brother-in-law’s Nikon 55 f/1.2 is much heavier than my 50 f/1.8. His viewfinder sure is bright and that last fraction of a stop can be handy sometimes, but the camera weighs a lot on the neckstrap and you start to question its value if you’re shooting at f/11 anyway. If you do decide you want the fastest possible lenses, go buy yourself a Leica M6 (film) or M9 (digital), for which you can buy a 50mm f/1.0 lens and a 75mm f/1.4. And before you think that it’s modern technology that allows these wonders, recall that Canon made a 50mm f/0.95 for their rangefinder cameras back in the 1950s, and still make an 85 f/1.2 for their SLRs.
My Zoom lens says it’s f/4.5 to 5.6. What’s that mean?
This relates to the size versus lens speed issue. I own a Canon 24-85 f/2.8-4 zoom lens, as an example. It’s not exactly small and light, but it tries. If you think about it, the size of the elements needed to get f/2.8 at 24mm is much smaller than the size needed to get f/2.8 at 85mm. To keep lens sizes and costs down, the lens manufacturer accepts that the lens wil be faster at the wide end of its focal length range and slower at the long end. As a result, you can open up to f/2.8 at the 24mm end of the range, but only to f/4 at the 85mm end. This is particularly noticeable in small lenses that do, say, 70-300mm where at the 300mm end the maximum f/stop is f/5.6 or worse. It also shows up in some compact point and shoot superzooms, which can have unspeakably slow long focal lengths. You’d better have a sunny day!
I hear stops referred to a lot. Are these always f/stops?
No. A source of confusion is that “stops”, as in f/stops, has become a handy shorthand for other doubling/halving relationships when referring to exposure. Thus, when someone says they “stopped down”, they probably did change the aperture from, say, f/8 to f/11. However, if someone says they wish they had a stop more light, they mean they wish they had twice as much. If they say they got some ISO 400 film which is two stops faster than their Sensia II, it means it is four times as sensitive and you can infer that the Sensia was ISO 100 (from 400, 200 would be one stop, one halving, and 100 would be the second stop, the second halving). Even experienced photographers get confused sometimes; Long ago I had a guy tell me he “pulled his film 6 stops, from ISO 100 to ISO 6”. Well, that’s not six stops, it’s four. Here, count along: 100 to 50 is one, 50 to 25 is two, 25 to 12 is three, 12 to 6 is four.
Note that stops always refer to exposure things. You would not say a 100mm lens is a “stop longer” than a 50mm because it was twice as long! You would say it was twice as long, or just that it’s a 100mm.
What is stopping down?
When you stop down a lens, you are going to a larger number/smaller aperture and therefore less light. Going from f/8 to f/11 is stopping down. Going the other way is called “opening up”, such as changing from f/11 to f/8 which is moving towards the smaller number/larger aperture and therefore more light.

What About my weird f/stops?
The f/stop sequence I listed is the full stops. Most things in photography work in 1/3 or 1/2 stop increments, and you will find lenses with maximum apertures at other-than-full f/stops. In fact, among the lenses I own or have owned, there are maximum apertures are f/1.4, f/2, f/2.8, f/4 and f/8, all right on the full stops, and others in between at f/1.8, f/2.5, f/3.2, f/3.5, f/3.8 and f/4.5. You see these much more in digital cameras which tend to show half or third-stop increments (or your choice!: my Nikon D700 allows me to select 1/2 or 1/3 stop increments) in their LCD displays, fine gradations you never saw on physical f/stop rings.
You Say Most things Double and Halve?
Yep. Shutter speeds do the 1/15  1/30  1/60  1/125 thing referred to earlier. The f/stops we have referred to extensively in their f/2.8  f/4  f/5.6 etc. sequence. Film speeds do the same thing. The doubling goes like this in the common range of film speeds:

25   50   100   200   400   800   1600   3200   6400

Each step here is a doubling/halving of the film’s sensitivity to light. Thus, ISO 100 requires twice as much light to be correctly exposed as an ISO 200 but only half as much as ISO 50. You would say it was a stop slower than the 200, a stop faster than the 50. In film days, the ISO was for the whole roll of film; now on digital cameras we get to change the ISO from shot to shot, and as you change this you’ll see the half or third-stop increments go by as you choose.
There are third-stop intervals in ISOs as well. Here are the third stop increments of film speed with the full-stops in bold.
25  32  40  50  64  80  100  125  160  200  250  320  400
That was a common range of film speeds just ten years ago. Now they seem amazingly slow, don’t they? There were lots of still films made some intermediate speeds, like Kodachrome 64 slide film, Plus-X Pan Professional black and white at ISO 125, and Fuji NPS and some Kodak Portra color negative film at ISO 160.
How do you refer to exposures between full f/stops?
Generally, I just say f/5.6 and a third, or halfway between f/5.6 and f/8, or something. I have a Sekonic L-308B handheld light meter that reads full f/stops plus a fraction in between expressed in tenths. If I took a reading that said 1/125th of a second at f/5.6 plus four of these ten segments, I could go through the machinations to figure out exactly what f/stop that is (f/6.25) but that’s not all that handy, to tell you the truth. No lenses are incremented in tenths of stops and tenth-stops are a needless amount of precision anyway given all the sources of slop in photography. Half and third stops are about as fine a distinction as matters.

Why are they called f/stops?
• In the late 1800’s there were a variety of aperture systems which all operated more-or-less the way we are all familiar with, which is to say that the concept of letting more or less light through a lens via an aperture system was being used extensively, even if the different systems used different naming conventions. (Well-known systems at the time included Sutton and Dawson’s ‘apertal ratio,’ and Dallmeyer’s ‘intensity ratio’)
• In 1858 John Waterhouse invents a system of metal discs with different sized holes to act as the aperture which literally get dropped into a slot in a lens. He called them ‘Waterhouse Stops’ because the discs were literally stopping light from entering the camera. This is likely the first instance of the use of the word ‘stop’ as it relates to aperture.
• In 1895 John A. Hodges first champions the ‘fractional number’ system (which he abbreviated to ‘F-number’) in defiance of the Photographic Society of Great Britan’s use of the ‘Uniform System (U.S.)’ This is the first recorded instance of the ‘fractional number’ and is likely the original meaning of the ‘f’ in F-stop.
• In 1901 C. Welborne Piper first proposes a unified system of describing aperture marking called the ‘f-diamater’ (or fractional diameter) after observing similarities between a half-dozen of the more popular methods of the day.
• From the early 1900’s through about 1920, the most common way to refer to Piper’s unifying system was as the ‘f-number.’
• In 1961 the American national Standards Institute (ASA) officially adopted the ‘f-number’ as the specification for photoelectric photographic meters. Essentially codifying the term and making ‘f-number’ the common phrase used to describe aperture for camera manufacturers of the day.
• While the phrase ‘f-number’ has morphed somewhat since 1961 to the now more commonly used ‘f-stop’ we use today, there is currently no standardized and generally accepted definition for the ‘f’ in f-stop. In fact, it has variously been referred to as focal, factorial, fractional, and many other designations by a wide variety of sources over the last 120ish years to the point where even though the system itself has been standardized, there simply is no singular recognized designation for the ‘f’ in f-stop.

So What’s Important in all this?
You need to know the doubling/halving relationship and how it works with shutter speeds in exposure. You need to understand that for a given amount of light and ISO, there are many combinations of shutter speeds and f/stops that give the same amount of light on the film/sensor (see next paragraph). You should understand that, with digital cameras, if you find yourself in a bad spot in terms of shutter speed and f/stop, you may be able to mitigate matters by changing your ISO. This is key since the shutter speeds and f/stops you choose have implications in how your final photograph will look in ways other than purely the amount of light on the film. You need to know that as you stop down you get more depth of field. You do not need to go around calculating aperture areas for your lenses and f/stops. If you’re like me, it’s worth doing it once to see that it works, then forgetting about.

How a Range of Settings Gives the Same Amount of Light
Now, to bring this all together, we know that the shutter speeds and f/stops both double and halve. Thus, we know that we can open up an f/stop (letting in twice the light) and move the shutter speed one step faster (cutting the time in half) and have the same amount of light on the sensor. It’s like that bucket of water; run the water twice as fast for half the time and the bucket is still full. As an example, if we meter a scene and it tells us that, at our current ISO, 1/125th at f/8 is the correct exposure, any of the following combinations would work:

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