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Absolute and relative lens apertures

When choosing lenses and operating a camera, what do the aperture f-numbers really mean? How does the focal length and aperture affect the exposure of the image? How do lens aperture sizes compare across cameras with different sensor sizes? To answer these questions and more, we will examine a model of how a lens-camera system works and draw observations from the model.

Contents

Simplified lens model

A basic optical scene consists of a sensor, a single lens, and an object at a far distance from the lens. All three components are flat and perpendicular to the central axis, and the lens has negligible thickness. We can assume that the object emits light or reflects light from some external source. An image of the object (possibly sharp or blurry) is formed on the sensor.

Some of the light coming from the object head toward the lens. The rays are refracted by the lens depending on where it hit the surface and at what angle. In an ideal lens, two rules determine how the rays are bent: A ray that intersects the center of the lens proceeds unchanged, and all rays emanating from one point of the object will steer toward one point on the focal plane (which is where the sensor is placed).

Basics of absolute apertures

The amount of light a lens receives from the environment is proportional to the area of the lens’s front surface. In the diagrams on this page, the diameter is the height of the lens. For example, the diameter of a lens might be 36 mm. We will assume that all lenses are circular. For an ideal lens, the rate of light entering the lens equals the rate of light exiting the lens. This principle will be important shortly.

An aperture is not a part of the simplified lens model here, but many real lenses in have apertures. It is a variable-sized disc within the lens assembly that blocks light from the outer parts of the lens, effectively making the lens behave like a smaller one. For our purposes, we will assume that the aperture is fully open (blocking nothing), so we can refer to the lens size and the aperture size interchangeably to mean the same thing.

Focal lengths

The focal length of a lens is a property of a lens itself (unrelated to the scene), determined by the curvature of its surfaces and the refractive index of its material. It is defined as the distance where parallel rays (from an object infinitely far away) converge to a single point.

Lenses with longer focal lengths produce a larger image for the same object at the same distance. For objects that are sufficiently far away, the size of the image is approximately proportional to the focal length. For example, a 100 mm lens will produce an image that is twice as large as compared to a 50 mm lens.

However, the law of conservation of energy comes into play here. The amount of light gathered by a lens is proportional to its area. If the outgoing projected image is enlarged, then the image must be dimmer at each point because the total energy of the image must remain the same. This situation is analogous to moving a lamp farther from a wall or moving a video projector farther from the screen, both of which make the lit surface dimmer.

Therefore, if two lenses have the same aperture diameter but different focal lengths, then the longer focal length lens will produce a dimmer image. This effect can be counterbalanced by making the aperture diameter bigger in proportion to the focal length, which brings us to the topic of relative aperture sizes.

Notation for relative apertures

If we divide the aperture diameter by the focal length, we get a dimensionless number. For example, a lens with focal length f = 50 mm and aperture diameter 25 mm has a relative aperture size of 25 mm ÷ 50 mm = 0.5 = 1/2 (the aperture is half as large as the focal length). We can take this ratio and say that the aperture is equal to f/2, because 50 mm / 2 = 25 mm.

This is why we must write relative apertures in the notation like f/2, not F2, F/2, or otherwise. The lowercase italic f is a variable that denotes the physical quantity called focal length. The slash indicates division – e.g. f/2 is mathematically equivalent to f ÷ 2, f × 0.5, and 0.5f, though these other notations are unrecognizable to photographers. (If italics are unavailable, it’s acceptable to write f/2. Do not use the lookalike character ƒ (U+0192), which has a different meaning.)

This also explains why the aperture number seems to get bigger as we make the physical aperture size smaller – because the aperture size is the focal length divided by this number. A notation like f/5.6 makes this division operation clear whereas a notation like F5.6 does not. (In real-world products, Nikon gets the notation right whereas Canon gets it wrong.)

Properties of absolute and relative apertures

We have seen that if we hold the absolute aperture size constant while increasing the focal length, the image becomes dimmer. In particular, due to the inverse square law for radiation in 3D space, the image brightness is inversely proportional to the square of the focal length. For example, doubling the focal length will make each point a quarter as bright.

At the same time, doubling the absolute aperture size will quadruple the area of the lens. Putting these two facts together, if we double the absolute aperture size and double the focal length then there will be no change in the image brightness. Since the relative aperture size is the ratio of the two quantities, it’s clear that it does not change in this example scenario.

Therefore, we conclude that the square of the relative aperture size is proportional to the image brightness. For example, an f/2.8 lens shooting a uniformly lit white wall will deliver the same image brightness to the sensor no matter what the focal length of the lens is. This is why relative aperture sizes are so useful to the photographer. But it is absolute aperture sizes that explain the physics going on and justify the need to express in terms of relative apertures.

A consequence of these observations is that “constant-aperture” zoom lenses like the popular 24–70mm f/2.8 are actually not constant in terms of absolute aperture size. As you zoom the lens in, the aperture opening as seen from the front appears to increase in size, just as predicted by the equation: absolute aperture = focal length / 2.8.

Another consequence is that we can estimate the physical size of a lens package based on its focal length and aperture specifications. For example, an expensive 200mm f/2.0 telephoto lens has an absolute aperture of 100 mm, which means its front element must be at least 10 cm (4 inches) in diameter!

As for teleconverters, we can see why they make the image dimmer. A teleconverter fits behind the lens, magnifying the image and thus increasing the effective focal length. The lens remains unchanged and thus the absolute aperture and the amount of incoming light don’t change. As a result, the relative aperture becomes smaller. For example, a 2× teleconverter doubles the focal length, thus the relative aperture becomes half (e.g. a 300mm f/2.8 lens becomes 600mm f/5.6, “losing” 2 stops of speed for the magnification).

Comparing different-sized sensors

Step Sensor size Sensor resolution Focal length Absolute aperture Relative aperture Field of view Light per pixel
Step 0 20 mm × 20 mm 1000 × 1000 30 mm 7.5 mm f/4.0 37° 20 units
Step 1 10 mm × 10 mm 500 × 500 30 mm 7.5 mm f/4.0 19° 20 units
Step 2 10 mm × 10 mm 1000 × 1000 30 mm 7.5 mm f/4.0 19° 5 units
Step 3 10 mm × 10 mm 1000 × 1000 15 mm 7.5 mm f/2.0 37° 20 units
Step 4 10 mm × 10 mm 1000 × 1000 15 mm 3.75 mm f/4.0 37° 5 units

Step 0: Suppose we have a 20mm × 20mm sensor with 1000×1000 pixels, put a 30mm f/4.0 lens in front of it, and set the focus and exposure to get a decent image. The lens’s absolute aperture is 7.5mm.

Step 1: If we simply crop out pixels keeping only the center 10mm × 10mm region of 500×500 pixels, then we get a smaller viewport of the world while everything else stays the same (focal length, proper focus, proper exposure, etc.). Note that each pixel receives the same amount of light as in step 0.

Step 2: Next, if we take this cropped region and re-engineer the sensor resolution back to 1000×1000 pixels in the smaller 10mm × 10mm region, this gives us the same image resolution as in step 0 but the view is like a 2× magnification. Now because each physical sensor pixel is half the size on each dimension, it is a quarter of the area compared to steps 0 and 1 and thus receives a quarter of the light under the same exposure settings. To fix the exposure back to normal, we must boost the ISO sensitivity, enlarge the aperture, and/or lengthen the shutter-open time by a total of 2 stops.

Step 3: Because the view is still magnified, we need to reduce the focal length of the lens to get the same viewing angle of the world as in step 0. In particular, we halve the focal length to 15 mm. If we keep the absolute aperture the same at 7.5 mm, then the image will become 4× brighter as a result of the focal length shrink. In fact, each pixel will receive the same amount of light as in step 0. The relative aperture is now f/2.0. Therefore we can conclude that if we shrink the sensor size and the focal length in the same proportion but keep the absolute aperture size unchanged, then the viewing angle and light per pixel will be unchanged but the relative aperture will be bigger (i.e. smaller fraction).

Step 4: But real-world cameras try to keep the relative aperture the same instead of maintaining the absolute aperture. If we shrink our 15 mm lens’s aperture back to f/4.0, then we are back in the same situation as step 2: each pixel receives a quarter of the light compared to step 0. Hence, this is why an f/2.8 lens on a full-frame camera will deposit far more light on the sensor (and each pixel) than a similarly-rated f/2.8 lens on a smaller-sensor camera framing the same scene. More light per pixel means lower image noise – because of the inherent quantum shot noise of photons and various background noise from the sensor electronics.

More info

This article only begins to scratch the surface of the physical and engineering principles behind photography. You can read more elsewhere on the web, and here are some topics as starting points: