## Photography Notebook: F Number

The f number is used to control the size of the circular opening (entrance pupil) that allows light to reach your camera's sensor. The f number affects picture sharpness and plays a part in the exposure of the picture. In every photograph the camera takes, either the camera sets the f number automatically or the photographer sets it manually, so the f number is a key concept in photography.

Emphasis Focus can be placed on one part of the scene. | Low Light A clear shot might be possible in low light. |

Learning about the f number requires understanding its mathematical basis as well as the language used to describe it. The f number involves terms, settings, concepts, and technology dating back to the definition of "apertal ratio" in 1867, and current photographers use the f number and associated terms like f stop in their work regularly.

I first want to set out a notation for discussing f numbers. An f number is sometimes written with the letter f and the number. Sometimes there is a slash or sometimes there is not a slash. For example, f/8 refers to an f number of 8. However, on the Rebel camera and in other literature, the f numbers are displayed on the controls without the slash, so I am going to use the convention of referring to an f number of 8 as f8.

A user of a camera encounters the f number in camera settings.
The basic concept is that the f number is the
lens focal length divided by the diameter of the lens entrance pupil (the
opening allowing light to reach the sensor).
Lower f numbers (such as f2, f2.8, f4) correspond to a larger entrance pupil for the lens. Higher f numbers (such as f16 f22 f32) correspond to smaller entrance pupil for the lens.
The fact that *increasing* the f number *decreases* the pupil opening is because the pupil diameter and f number are inversely related to each other.

The characteristics of the image vary with f numbers. At low f numbers, a larger pupil is exposed and the depth of field decreases and may be shallow so that only part of the subject may be in focus. At larger f numbers, a smaller pupil is exposed and the depth of field increases so that more of the scene may be in focus. A user can choose f numbers by using the Aperture Value (Av) setting for the camera or the manual (M) setting and setting an f number. The camera itself will set an f number in the case of other modes such as program (P) or time value (Tv).

f2.8 for 1/3200 sec
Note that the f2.8 setting gives a focus to the center of the frame where the camera was focused, and the foreground flowers and background wall are blurred. |
f22 for 1/50 sec
The f22 setting gives a focus to the whole field of view. The wavy lines in the background wall are clear. |

Note that, in this example, the time value of the exposure for Av of f22 is 1/50 sec, 64 times the time value for the scene at Av of f2.8 (1/3200 sec). This is because the pupil opening at f2.8 has 64 times the area than the pupil opening at f22. The camera adjusted this time value automatically so that the photo would remain exposed properly.

The optics of a particular lens have a specific range of f numbers possible.
For example, a Canon EF-S 10-22mm f/3.5-4.5 USM SLR
lens has the range its *lowest possible* f numbers shown in the name of the lens itself (3.5 to 4.5), corresponding to the minimum (10 mm) and maximum
focal length (22 mm).

As a new user, I was highly confused by the f number. The associated terms like "f-stop" and "stop down the lens" are used so quickly by photographers, with associated jargon and a mysterious sequence of numbers (f1.4, f2, f2.8, f4, f5.6, f8, f11, f16, f22, f32) that I was not able to figure out what was being talked about until I understood the mathematical basis of f numbers.

### The Mathematical Basis of F Numbers

The mathematics behind f numbers arises from the definition of the f number, N:

N = F/Dwhere F = the focal length of the lens and D is the diameter of the pupil. Note that the same units would be used to measure F and D, so that N is dimensionless (the f number is not a distance measurement, but a ratio). Don't get confused between F and N! The quantity F is the focal length of the lens, such as 50 mm; N is the f number, such as 2.8, written f2.8. I'm representing the focal length as a capital F to help alleviate this possible confusion of F and N--which confused me on first reading about this. We can express D in terms of F and N:

D = F/NWe can illustrate the relationship among these numbers by an example. We can use camera settings to set the focal length F (with zoom lenses for example, or fixed values with prime lenses) and the f number N (with the Aperture value (Av) setting of the camera, for example). Let's say we set a focal length of 50 mm on lens with a Av setting of f4. We have F = 50 mm and N = 4, and so the pupil diameter, D, is

D = F/N = 50 mm / 4 = 12.5 mmSay we change the f number on this same lens to an Av setting of f5.6. We then have

D = F/N = 50 mm / 5.6 = 8.93 mm (rounded)We can see our increase in the f number resulted in a decreased pupil diameter. The pupil is circular, so that the area of the opening, A, is

A = π (D/2)So our area with the Av setting of f4 is^{2}

A(f4) = π ((50/4)/2)Our area with the Av setting of f5.6 is^{2}= 122.72 mm^{2}(rounded)

A(f5.6) = π ((50/5.6)/2)So that while our diameter was changed from about 12.5 mm to about 8.93 mm, our area was approximately halved in going from f4 to f5.6.^{2}= 62.61 mm^{2}(rounded)

f4 for 1/1600 sec | f5.6 for 1/800 sec |

Note that in our example, the camera (on its Av mode) adjusted the time exposure so that the f5.6 photo took twice as much time as the f4 photo. This was to compensate for the fact that the area of the f5.6 pupil opening was double that of pupil opening in the f4 photo. Note that the depth-of-field for the f5.6 photo is slightly larger (more area is clear) than in the f4 photo.

#### The F stops in cameras

The mathematical relationships among D, F, N, and A
are continuous (for N > 0 and D > 0 and F > 0).
However, photographers and camera makers wanted to note special
points in these relationships.
These special points are the f stops.
The f stops are set up so that the area of the lens pupil openings are
in a sequence where the area doubles with each successive f stop down
(reduced f number) on the camera controls and changes by half with
each successive f stop up (larger f number).
We saw this in our previous example: if we change the Av setting of f4 to
f5.6 for our 50 mm lens, we increase the area of the circle of the lens pupil
by about two.
In other words, we changed "one stop" from f5.6 to f4, and
this doubled our lens pupil area.
Here is an important point: our "one stop" went down
from f5.6 to f4. As a new user, I wondered why "one stop" was not a
difference of 1, such as from f5.6 to f4.6.
The answer is because the relationship of one stop to another is in terms
of successive *areas*
which double or halve based on reducing or increasing
the f number in the camera controls, not f numbers separated by 1.

Our area for the f stop k, A(k), and the area for the f stop k-1, A(k-1), are in this relationship:

A(k-1) = 2 * A(k)That is, going down one f stop means we get a doubling of the lens pupil area in relation to the original pupil area. We can express A(k) in terms of the diameter, D(k), for the f stop k and A(k-1) in terms of the diameter D(k-1) for the f stop k-1:

A(k) = π (D(k)/2)And so given A(k-1) = 2*A(k), we have^{2}

A(k - 1) = π (D(k-1)/2)^{2}

π (D(k-1)/2)Simplifying this expression by dividing both sides by π, we have:^{2}= 2 * π (D(k)/2)^{2}

(D(k-1)/2)Taking the square root of both sides^{2}= 2 * (D(k)/2)^{2}

(D(k-1)/2) = (D(k)/2) * sqrt(2)Multiplying both sides by 2 gives us:

D(k-1) = D(k) * sqrt(2)Thus, the f stops have diameters in successive ratios of the square root of 2 (approximately 1.414). So our stop changed from f5.6 to f4, and this corresponds to this ratio (5.6/4 = 1.4). Since D = F/N, we have, where N(k) is the f number for the kth f stop:

F/N(k-1) = F/N(k) * sqrt(2)Dividing each side by F, simplifying, and collecting terms:

1/N(k-1) = 1/N(k) * sqrt(2)Thus, as we go up in stops, each f number is in a ratio of sqrt(2) to the previous one.

1 = N(k-1) * 1/N(k) * sqrt(2)

N(k) = N(k-1) * sqrt(2)

Where do f stops start and stop? That seems to be dependent on the camera and what you want to define as the "first" f stop. In charts that I have seen, there are different markings for f stops, although they all share the mathematical relationships outlined here (with variations for rounding and truncating the f numbers for display purposes). Of course, the lowest f number for a lens is limited by the optics of the lens itself--as mentioned before--and a lens may not be capable of f1.4 or even f2 settings.

In a theoretical chart of possible f numbers, it seems that one logical limit for an f number is where the diameter of the pupil, D, is equal to the focal length, so since N = F/D, an N of 1 seems to be the "base" f stop. If we number this f stop as 0 (for the origin, we can make a chart:

f stop | f number (N) | f number(N) (approximate) |
---|---|---|

0 | 1 | 1 |

1 | sqrt(2) | 1.4 |

2 | (sqrt(2))**2 | 2 |

3 | (sqrt(2))**3 | 2.8 |

4 | (sqrt(2))**4 | 4 |

5 | (sqrt(2))**5 | 5.6 |

6 | (sqrt(2))**6 | 8 |

7 | (sqrt(2))**7 | 11 |

8 | (sqrt(2))**8 | 16 |

9 | (sqrt(2))**9 | 22 |

10 | (sqrt(2))**10 | 32 |

... | ... | |

k | (sqrt(2))**k | (1.414)**k |

This is why the settings for a camera's f stops run in this (strange) sequence!

### Lingo

Once you know the mathematical basis of the f number, the next step is understanding how people refer to f numbers. Some points to review about the language related to f numbers:

- The f in f number stands for focal.
- The pupil is often referred to by the more generic term aperture, which means opening.
- Many times, people say "f stop" when they mean f number. (Note that sometimes they do mean f stop, though).
- For a given lens, its speed is the minimum f number possible for that lens. A lens with a lower minimum f number is said to be "faster" than a lens with a higher minimum f number. For example, the Canon EF 50mm f/1.8 II lens is faster than the Canon EF-S 10-22mm f/3.5-4.5 USM SLR lens. Note that the lens speed is right in the lens name. Note that for the zoom lens, the lens speed is given as a range because the numbers correspond to the minimum f numbers at the minimum and maximum focal lengths of the lens.
- Many times, people write or talk about "opening up a few stops," or "stopping down the lens" to describe changes in the f number.
- When people say use the lens "wide open," they mean an Av setting at its minimum f number.
- You can drive yourself crazy by continually qualifying a reference to an f number by stating what it means in relation to the lens pupil size. For example, "Use an Av setting of f1.8, opening up the lens." I find that just thinking in terms of the f number alone suffices: "Use an Av setting of f1.8." The pupil size and f number are inversely related by definition--so it is unnecessary to continually point that out.
- Using the f number to describe photos makes it easier for me to understand what is going on. The f number is used in controls I can see and touch. The f number is a quantity visible and controllable on the camera display and dial. The lens pupil diameter is not shown, so saying "open it up" or "stop it down" makes no sense to me.

### But there is more!

F stops, f numbers, and other terms, slang, and jargon and associated scales, cameras, optics, and mechanics change over the centuries and seem to vary by practitioner. But the mathematics underlying the relationships haven't changed. I will therefore leave this discussion at this point and refer you to the Sources Consulted at the bottom of you want to read more.

### Summary

Preparing these notes helped me understand the key ideas behind f numbers.- F numbers represent different size lens pupil openings.
- Decreasing the f number increases the size of the lens pupil opening because N = F/D (the f number N is equal to the focal distance of the lens F divided by the diameter D of the lens pupil).
- F stops are camera settings used to mark off successive points where the area of the lens pupils vary by a factor of 2. The resulting f numbers change between successive stops by a factor of the square root of 2, approximately 1.414.
- The varying lens openings give different visual effects through changing the depth of field.
- A particular camera can be set at a range of f numbers based on the optics of the lens and the controls of the camera.
- Setting the f number in the camera is a key practice in photography.

### Sources Consulted

- "f-number." Wikipedia. Wikipedia, 2008. Answers.com 01 Jul. 2009. http://www.answers.com/topic/f-number
- "lens-speed." Wikipedia. Wikipedia, 2008. Answers.com 20 Jul. 2009. http://www.answers.com/topic/lens-speed