Now things get a little complicated……….

You should first read about crop factors and a thing called “equivalence”.

Equivalence is what we use to compare different camera formats (sensor sizes) on an equivalence basis. We can say that a compact camera is fitted with a '28-120mm lens' but the unspoken word in this description is tit has a 28-120mm “equivalent” lens. We do this as it is an easy It's a way to describe the range of fields-of-view that the lens offers, cancelling out the effect of sensor size by using a common reference point, the lens focal length on THIS particular camera.

A 100mm equivalent lens on a small-sensor camera will gives the same composition and (because that means shooting from the same position) the same perspective as an actual 100mm lens does on a full frame camera, regardless of sensor size, because they are equivalent.

This logic that the idea of 'crop factors' are based upon. The 'Four Thirds' sensor format has a diagonal very close to half that of a 'full frame' sized sensor. And if you calculate the angle-of-view of a 50mm lens on a system with a crop factor of 2, it's the same as for a full frame camera with a 100mm lens.

However, it is not just focal lengths that should be thought of in “equivalence” terms. There is also an 'equivalent' aperture (f stop) value as apertures and sensor sizes interact in more ways than simply a field of view equivalence.

In the film days people did not have to directly compare quality and characteristics across different formats as people did not use vintage lenses on full frame cameras. We just knew that 35mm was better than 110 format, medium format was even better than 35mm and large format was even better still. In the digital era, the significance of (and difference between) sensor sizes meeds to be appreciated, and this is not helped by a poor naming system. I had no idea why a f0.75 lens had exactly the same exposure settings as a f1.6 lens when I swiped them on my camera to take the same image UNTIL I realised one was for a small sensor area and the other for a large format giving them both an equivalence of f0.9!

Apertures and F-numbers

The effective aperture diameter is not necessarily the same as the diameter of the part of the front lens element in use. The aperture in a lens is placed at a point in the lens array where the image is as defocused as possible so as to minimise vignette. There is also an equivalence between sensor size and aperture and that turns out to be very similar to that between focal length and sensor size.

Depth of field also has an equivalence. It is the physical size of the aperture that defines depth-of-field, not its F-number. Two lenses offering the same angle-of-view with 25mm diameter apertures will give the same depth-of-field of shot at the same shooting distance and when viewed at the same output size.

Think about it this way to get the same amount of light projected per square mm of a bigger film plane too will need more light. Medium and large format lenses are also normally longer in order to get the same angle on view at the film so they actually have to let in even more "total" light through the lens to give equal film exposure.

If you have a room in your apartment or home and want to let in more light you have bigger windows or open the curtains or blinds (aperture) up more. Remember also that the most effective part of the lens is its centre lessening towards the edges. So a bigger diameter lens will be generally, all other things like the optical quality of the glass equal, give a better quality image.

When you see something called a speed booster, all it does is focus the larger area of the lens on a smaller sensor, this has the effect of reducing the size of the projected image circle AND thus increasing the light hitting those parts of the sensor. It is just adding optics to do this. Of course when we add optics we also add optical light reduction due to the density of the glass.

The F-number is focal length/aperture diameter. A 25mm aperture on a 100mm lens would be considered to be f/4 (100mm/25mm), whereas the same-sized aperture in the Micro Four Thirds lens would be an f/2 (50mm/25mm).

Helps to do the math in real numbers to understand the f number.

If the lens focal length is 50mm (~2") then the entrance pupil (entrance to the lens) needs to be 36mm (1.4" assuming there isn't anything else restricting the light path): 50/36=1.4 so f1.4.

If the lens focal length is 100mm (~4") and the entrance pupil remains 36mm (1.4"): 100/36=3.8 so f3.8. Since the pupil remained the same the total light stays the same but the focused projection now covers 4 times the area. A lot dimmer per square inch of film, hence the f3.8.

If the lens focal length remains 100mm (~4") then we need to increase the entrance pupil to 72mm (2.8"): 100/72=1.4 to get back to f1.4.

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.

The difference in sensor area, which is ultimately what equivalence is about, is the square root of the ratio of sensor areas (it's essentially the same number). And this effects the depth of field too. Cameras (and their lenses) end up blurring the background based on their sensor size.

Initial Setup

Let's assume that we have a 20mm × 20mm sensor with 1000×1000 pixels, we put a 30mm f/4.0 lens in front of it, and set the focus and exposure to get a well exposed image. The lens’s absolute aperture will be 7.5mm.

If we simply crop out pixels keeping only the center 10mm × 10mm region (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.). Each pixel receives the same amount of light as in the 20 x 20 sensor.

If we now take this cropped 10 x 10 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 the 20 x 20 but the view is more 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 the original 20 x 20 sensor 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.

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 before. 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 the original 20 x 20. The relative aperture will now be 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).

In 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 before: each pixel receives a quarter of the light compared to 20 x 20. 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. And a f2.8 large frame lens on full frame camera will inversely provide more light. More light per pixel also means lower image noise – because of the inherent quantum shot noise of photons and various background noise from the sensor electronics.

My experimentations bear-out these expectations to a pretty good degree - when set to 'equivalent' apertures, the background blur is very similar

We have now 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).