Aperture

For a camera, there are basically two settings that will determine how much light from a scene will reach the sensor: Shutter speed and aperture. Shutter speed is relatively easy to understand; it is the time the shutter is open, typically measured in fractions of a second such as 1/125 s, 1/250 s or 1/500 s. But what is the aperture, and what exactly do these numbers such as 1.4, 2.0 and 2.8 mean?

Aperture is just another word for opening, it refers to the part of a lens through which the light rays pass from the scene to the sensor. Its size can be adjusted with a diaphragm or iris, which usually consists of a number of movable blades. Depending on the number and shape of the blades (straight or curved), the resulting aperture is more or less circular.

How can we best describe the aperture used to take a picture? A straightforward way 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 a2. For example, an image may be taken with a 50 mm lens and an aperture diameter of 35.4 mm, corresponding to an area of 982 mm2. But you have probably never heard anybody say that.

F-stop

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 per unit area. 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. Sounds much more familiar already. 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 change the focal length. According to equation (A1), for a fixed 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 per unit area on our sensor is thus proportional to the square of the focal length.

But the focal length is also closely related to the magnification. According to approximation (M12), for a fixed 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 per unit area on the sensor 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 exactly inverse, effectively cancelling out each other. The f-stop therefore describes how much light a lens captures per unit area on the sensor, regardless of its focal length. Results of a practical test are shown in figure 2.

For example, a 50 mm f/2 lens collects the same amount of light (per unit area) 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. If the longer lens had the same aperture diameter as the shorter one, it would be a considerably darker 100 mm f/4 lens.

photograph taken with 25mm lens at f/425 mm f/1.8, 1/3200 s, f/4.0, ISO 200
photograph taken with 45mm lens at f/445 mm f/1.8, 1/3200 s, f/4.0, ISO 200
photograph taken with 75mm lens at f/475 mm f/1.8, 1/3200 s, f/4.0, ISO 200

Fig. 2.

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

Step size

For shutter speed, most cameras offer a set of standard settings 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 approximately half (or twice) that of its neighbor, resulting in half (twice) the light reaching the sensor. This is enough to see a considerable difference in the exposure of your image.

It makes sense to define the same step size for the aperture. To half (or double) the aperture area, its diameter needs to be divided (multiplied) by the square root of 2, and therefore according to equation (A1), the f-stop needs to be multiplied (divided) by the square root of 2. Starting with an f-stop of 1, you get approximately 1, 1.414, 2, 2.828, 4, and so on. These are rounded to 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64.

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 certain 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/1.0. There are several camera lenses currently in production which are slightly faster, such as the Leica Noctilux-M 50 mm f/0.95 ASPH or the Voigtländer Nokton 25 mm f/0.95. Among the fastest photographic lenses ever used in practice is the Carl Zeiss Planar 50 mm f/0.7 developed for the NASA Apollo program. This huge lens which was mounted on a modified Hasselblad body is a whole stop faster than f/1.0 [Nasse 2011].