Lens Aperture

When you take a picture, there are basically two settings that control the exposure, i.e. how much light reaches the sensor: Shutter speed and aperture.* Shutter speed is simply 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. 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 7 and 9) and shape (straight or curved) of the blades, the resulting aperture is more or less circular.

It is obvious that a larger aperture will capture more light from a scene than a smaller one. This is really what lenses are all about—to capture more light than a simple pinhole camera with its tiny opening. So 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 (a/2)2. 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. Even though this a perfectly valid description, you have probably never heard anything like this.

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 collects, or more precisely, how much light from the scene reaches the sensor. This is where the f-stop comes in.

The f-stop (also known as f-number, relative aperture or focal ratio) A is defined as the ratio of focal length f of the lens and the aperture diameter 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 equivalent to 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. Also note that a large aperture diameter corresponds to a small f-number, and vice versa.

To understand why the f-stop is so powerful, consider what happens if we use lenses of different focal lengths. 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 from the scene at any point of the sensor is therefore 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 fixed focusing distance d, the magnification m is approximately proportional to the focal length f:

m f / d (A3)

The area of the sensor on which the light from the scene is spread is thus proportional to the square of the focal length. The light received at any point of the sensor is therefore 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 each other out. In other words, for an increased focal length, the larger aperture compensates for the light loss due to a higher magnification. Therefore, the f-stop describes how much light a lens captures, regardless of its focal length. Results of a practical test are shown in figure 2.

photograph taken with 25mm lens at f/425 mm | 1/3200 s | f/4 | ISO 200
photograph taken with 45mm lens at f/445 mm | 1/3200 s | f/4 | ISO 200
photograph taken with 75mm lens at f/475 mm | 1/3200 s | f/4 | ISO 200

Fig. 2.

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

Lens speed

While the aperture can be adjusted with the iris, the maximum aperture or lens speed, i.e. the largest aperture (and therefore smallest f-number) available, is a major characteristic of any camera 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 between f/5.6 and f/8 are considered slow. However, these classifications also depend on sensor size, focal length and purpose of the lens.

From a construction point of view, a 50 mm f/2 lens collects the same amount of light as a 100 mm f/2 lens, resulting in the same brightness of the scene. 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.

Standard stop scales

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 (A1), 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. One of 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 to capture images of the far side of the moon. This huge lens which was mounted on a modified Hasselblad body is a full f-stop faster than f/1 [Nasse 2011].

 

*OK, there is also ISO. While shutter speed and aperture control the exposure, i.e. how much light reaches the sensor, ISO sets the sensitivity (or amplification) of the sensor and thereby controls how the light received by the sensor translates to brightness of the picture. As with shutter speed and aperture, double the ISO to get one stop brighter.