A **DSLR (Digital Single-Lens Reflex) 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 concepts widely known 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 closed related to the the size of the physical aperture or **diaphragm** of the camera. In fact, the entrance pupil and the physical aperture itself differ only because of the presence of lenses in front of the latter that effectively modify the size of the image. The apparent diameter of the diaphragm opening in the camera lens as seen through the front of the lens is called the **effective aperture** and is measured in mm.

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

*N = f/D*

Note that the f-number is dimensionless. In DSLR cameras you can choose independently 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 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 can 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 multiplying the perimeter –the sum of the lengths of all its sides– times the apothem –the distance from the center to the midpoint of one of the sides– and dividing by 2. Both the apothem and the perimeter of a regular polygon are directly proportional to its radius –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 (and then 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 a camera lens you don't change the aperture continuously** (you do when it comes to the pupils of your human eyes). In fact, **only certain values for the f-number are allowed** in most cases: 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: Those numbers are chosen so that each one implies half the amount of light pointing to the sensor than the previous one. But... why this is (approximately) true? Because these numbers give an area for the aperture which is half than the preceding value.

If we call A_{0} the area for a certain aperture, say f/2.8 for instance, and A_{1} the area for the succeeding f-number, say f/4, we impose the condition:

A_{0}=2A_{1}

In terms of the diameter D it would read

D_{0}^{2}=2D_{1}^{2}

and taking the square root:

D_{0}=√2 D_{1}

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

**In some digital cameras you can 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 *k*^{3}=√2. Solving for *k*, the ratio is the sixth root of 2, i.e. *k*=^{6}√2. And the above numbers are approximations to the sequence {*k ^{n}*}, being

*n*an integer number. For instance,

(^{6}√2)^{12 }= 4

(^{6}√2)^{13 }≈ 4.5

(^{6}√2)^{14 }≈ 5

(^{6}√2)^{15 }≈ 5.6

Apart from the aperture, in DSLR cameras you can control the amount of light that reaches the sensor varying the **exposure time** (sometimes called the **shutter speed**), that is, the seconds (or fractions of second, indeed) that 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 the half of the previous, so at each increment step of the exposure time, the amount of light that reaches the sensor doubles.

**Doubling the exposure time and the f-number at the same time does not alter the overall exposure of the photo** because the gain in light due to longer exposure time is compensated by the loss of a half of light due to the greater f-number (lower aperture). Thus, once the total quantity 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.

*REFERENCES*

https://en.wikipedia.org/wiki/F-number

https://en.wikipedia.org/wiki/Focal_length

https://en.wikipedia.org/wiki/Entrance_pupil

https://en.wikipedia.org/wiki/Diaphragm_(optics)

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