4.6. Focus on Math: The ultimate guide to f-numbers in digital cameras


A Digital Single-Lens Reflex (DSLR) camera is a photographic camera that has two parts: a digital imaging sensor in the body of the camera and an objective or lens, which is an optical system that gathers the light to the sensor. We will focus on the lens (sorry for the play on words!) But first, a couple of basic widely-known concepts by photography lovers.

First one: The focal length. It is a measure of how strongly the system converges light. The longer the focal length (e.g. 300 mm), the higher the magnification of the image. Conversely, shorter focal lengths (e.g. 24 mm) lead to lower magnification. When you zoom in and out you change the focal length.

The second concept to keep in mind is the entrance pupil of an optical system, which is closely related to the size of the physical aperture or diaphragm of the camera. In fact, the only difference between the entrance pupil and the physical aperture is the presence of lenses in front of the latter, which effectively modify the size of the image. The effective aperture is the apparent diameter of the diaphragm opening in the camera lens as seen through the front of the lens, and it is measured in mm.

The ratio of these two lengths —the focal length and the effective aperture— is the so-called f-number or fstop. It is represented by the letter N (not f, be careful!), while the letter f is used to represent the focal length. D stands for the diameter of the pupil entrance.

N = f/D

Note that the f-number is dimensionless. In DSLR cameras you can independently choose the values for N and f , so you are indirectly fixing the effective aperture D according to

D = f/N

For instance, if you have zoomed in to a focal length of f=200 mm and you have set the f-number to N=8 (which is labeled in cameras as f/8), then the diameter of the pupil entrance is D=f/N=200/8=25 mm.

One could argue that the diaphragm of a camera lens is not a circle of a diameter D, but a regular polygon delimited by blades. The area of a regular polygon is calculated by multiplying its perimeter (which is the sum of the lengths of all its sides) times the apothem (which is the distance from the center to the midpoint of one of the sides), and then dividing the result of the multiplication by 2. Both the apothem and the perimeter of a regular polygon are directly proportional to its radius (which is defined as the distance from the center of the regular polygon to a vertex), so the area itself is proportional to the square of the radius, thus to the square of the diameter –twice the radius–), as happens with the area of a circle. So... we are not going to make a fuss about whether the pupil entrance is a perfect circle or not, OK? It doesn't matter for our reasoning.

In the vast majority of camera lens you don't change the aperture continuously (you do it when it comes to the pupils of your human eyes). In fact, only certain values of the f-number are allowed in most cameras: f/2.8, f/4, f/5.6, f/8, f/11, f/16, and f/22. Why those weird values? Photography enthusiasts know the answer: they are chosen so that each value in the series allows half of the intensity of light pointing to the sensor as compared to the previous value. But... why is this (approximately) true? Because the area of the aperture corresponding to each of these numbers is half of the corresponding to the preceding value.

If we call A0 the area for a certain aperture, say f/2.8 for instance, and A1 the area for the succeeding f-number , say f/4, we impose the condition:

A0=2A1

In terms of the diameter D it would read

D02=2D12

and taking the square root:

D0=√2 D1

Therefore, the f-numbers form a geometric sequence with common ratio √2. If the first term is 1, next ones are

√2 ≈ 1.4, (√2)2 = 2, (√2)3 ≈ 2.8, (√2)4 = 4, (√2)5 ≈ 5.6,

(√2)6 = 8, (√2)7 ≈ 11, (√2)8 = 16, (√2)9 ≈ 22

Some digital cameras allow to change the f-number with more precision, being the possible values 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... Now, each step that halves the light is subdivided in three. In other words, the new common ratio k of the latter geometric sequence satisfies the condition k3=√2. Solving for k, the ratio is the sixth root of 2, i.e., , and the above numbers are approximations to the sequence {kn}, being n an integer number. For instance,

k12 = 4

k13 ≈ 4.5

k14 ≈ 5

k15 ≈ 5.6

In addition to the aperture, in DSLR cameras you can control the amount of light that reaches the sensor by varying the exposure time (sometimes called the shutter speed), that is, the seconds (or fractions of second, indeed) during which the sensor is exposed to light. Standard values in cameras for the exposure time are: 1/1000 s, 1/500 s, 1/250 s, 1/125 s, 1/60 s, 1/30 s, 1/15 s, 1/8 s, 1/4 s, 1/2 s, 1 s... (s stands for second.) Note that each fraction is roughly half of the previous fraction, so the amount of light that reaches the sensor doubles at each increment step of the exposure time.

Doubling the exposure time and the f-number at the same time does not alter the overall exposure of the photo. This is because the gain in light due to longer exposure time is compensated by the loss of half of the light due to a greater f-number (lower aperture). Thus, once the total amount of light is fixed to ensure a proper exposition (not too dark, not too burn), photographers play with the different pairs of values (f-number, shutter speed) that are equivalent in terms of light gathered, but different when it comes to effects such as sharpness, depth of field, diffraction, motion blur, noise, etc. These artistic effects can also be fully explained mathematically and physically... But that is out of our scope, sorry.