When you take a photograph, there are basically two settings that will control the *exposure*, i.e. how much light will reach the sensor: Shutter speed and aperture.^{*} *Shutter speed* is simply the time that the shutter is open, typically measured in fractions of a second such as 1/125 s, 1/250 s or 1/500 s. The longer it takes, the more light you get. But what is the aperture, and what exactly do these numbers such as 1.4, 2 and 2.8 mean?

*Aperture* is just another word for opening. It describes how much the lens is opened to let the light rays pass from the scene to the sensor. Its size can usually be adjusted with a diaphragm or iris, which consists of a number of movable blades. Depending on the number (typically between 5 and 15) and shape (straight or curved) of the blades, the resulting aperture is more or less circular.

How can we best describe the aperture used to make an image? A straightforward answer would be to measure its diameter **a** in millimeters, inches, or any other unit of length. The incoming light is thus proportional to the area of a circle with diameter **a**, given by the well-known formula π/4 **a**^{2}. For example, an image may be taken with a 50 mm lens and a diameter of the aperture of 35.4 mm, corresponding to an area of 982 mm^{2}. Even though it's an accurate description, you have probably never heard anybody say that.

Why isn't the diameter used in practice? This being photography, we would like to have an absolute measure of how much light a lens captures, or, more precisely, how much light reaches the sensor from any point in the scene. This is where the f-stop comes in.

The *f-stop* (also known as *f-number* or *focal ratio*) **A** is defined as the ratio of focal length **f** of the lens and the diameter of the aperture **a** as

A | = | f / a | (A1) |

Note that the f-stop is dimensionless, i.e. it does not have a unit. For a 50 mm lens, an aperture diameter of 35.4 mm as in the example above is the same as an f-stop of 1.4. Instead of just the number 1.4, you will often find notations involving an f such as f/1.4 or simply f1.4.

To understand why the f-stop is so powerful, consider what happens if we use lenses of different focal lengths (or a zoom lens, for that matter). According to equation (A1), for a given f-stop **A**, the aperture diameter **a** is proportional to the focal length **f**:

a | = | f / A | (A2) |

The aperture area and thus the light received from any given point in the scene is thus proportional to the square of the focal length.

On the other hand, the focal length is closely related to the magnification. According to approximation (M12), for a given focusing distance **d**, the magnification **m** is proportional to the focal length **f**:

m | ≈ | f / d | (A3) |

The area of the sensor on which the light from our scene is spread is thus proportional to the square of the focal length. The light received from any given point in the scene is thus inversely proportional to the square of the focal length.

For a fixed f-stop and a fixed focusing distance, these two effects are obviously exactly inverse, effectively cancelling each other out. In other words, for an increased focal length, the increased aperture diameter compensates for the light loss due to a higher magnification. Therefore, the f-stop describes how much light a lens captures on the sensor, regardless of its focal length. Results of a practical test are shown in figure 2.

Images taken with three lenses of different focal lengths, but with the same focusing distance, shutter speed, f-stop, and ISO. Despite fairly different magnifications, exposure of the scene is almost identical.

While the aperture can be adjusted with the iris, the *maximum aperture* or *lens speed*, i.e. the largest aperture (and smallest f-number) available, is a major characteristic of any lens. Together with the focal length, it is usually printed on the lens barrel. Prime (fixed focal length) lenses with a maximum aperture of f/1.4 or zoom lenses with f/2.8 are generally considered fast (allowing for fast shutter speeds), while lenses with a maximum aperture of 5.6 or 8 are considered slow. However, these classifications also depend on sensor size, focal length and purpose of the lens.

A 50 mm f/2 lens collects the same amount of light as a 100 mm f/2 lens. For this, the longer lens needs an aperture with twice the diameter and 4 times the area of the shorter lens - more glass, more weight and higher costs. If the longer lens had the same aperture diameter as the shorter one, it would be a considerably slower 100 mm f/4 lens.

For shutter speed, most cameras offer a set of standard settings or *stops* such as 1 s, 1/2 s, 1/4 s, 1/8 s, 1/15 s, 1/30 s, 1/60 s, 1/125 s, 1/250 s, and 1/500 s. The important thing here is that every value is (more or less) half that of its predecessor, resulting in half the light reaching the sensor. This is enough to see a considerable difference in the exposure of your image. A shutter speed of 1/125 s is said to be one stop faster than 1/60 s, and one stop slower than 1/250 s.

It makes sense to define the same step size for the aperture. In order to halve the aperture area, its diameter **a** needs to be divided by the square root of 2. According to equation (A2), the f-stop **A** therefore needs to be multiplied by the square root of 2. Starting with an f-stop of 1, you get a sequence of (rounded) 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64, and so on. Most camera lenses provide a subset of these, e.g. between 2.8 and 22. In analogy to the shutter speed, an f-stop of 2 is said to be one stop faster than 2.8, and one stop slower than 1.4.

As a consequence, an exposure of 1/125 s and f/4 is the same as 1/250 s and f/2.8, or 1/500 s and f/2. This makes it very easy to balance speed and aperture, e.g. for a fast shutter speed in sports photography, or to obtain a deeper or shallower depth of field. Modern cameras also offer half or third steps; 1/160 s and f/3.5 or 1/200 s and f/3.2 are two more examples of the same exposure.

Interesting to note, there are no physical limitations to the lens speed at f/1. There are several camera lenses currently in production which are slightly faster, such as the Leica Noctilux-M 50 mm f/0.95 ASPH. Among the fastest lenses ever used in practice is the Carl Zeiss Planar 50 mm f/0.7, which was developed for NASA in the 1960s. This rather big lens which was mounted on a modified Hasselblad body is a whole stop faster than f/1 [Nasse 2011].

^{*}OK, there is also ISO. However, it is not related to the exposure, i.e. how much light reaches the sensor. Instead, it sets the sensitivity (or amplification) of the sensor, i.e. how the amount of light captured translates to brightness of the picture. Double the ISO to get one stop brighter.