What's the relation between sensor size and maximum aperture

Question

Is there any relation between the maximum aperture of a lens and the size of the sensor?

I'm looking at the smartphone.

A smartphone has 4 mm focal length (equivalent to 28 mm) and f/1.6 aperture fixed. The lens diameter is of course 2.5 mm (4 mm / 1.6).

Why a manufacturer can't use a 3.3 mm lens (f/1.2) keeping 4 mm of focal length and the same sensor?

Answer 1

Technically, the manufacturer probably could have used a lens with a larger diameter, and thus a larger maximum aperture. Only they could answer definitively why they didn't, but it could be a result of any number of things, e.g.:

1. Larger diameter lens would also be thicker to maintain same focal length, which has implications on placing the lens at the right distance from the sensor, how far the lens extrudes from the body of the phone, and thus the design of the phone case itself, and also uses more real estate on the inside of the phone, which means other components may need to be moved around or reduced in size
2. Larger lens is probably more expensive to manufacture
3. Larger lens may be more difficult to manufacture to the same standards with regard to distortion and aberrations

That's just a few off the top of my head. There's probably quite a bit more. It comes down to design choices.

Oh, and with regard to your opening question - there isn't really any relationship between maximum aperture size and sensor size. Aperture is a characteristic of the lens - in the example of cameras with interchangeable lenses, you could take the same lens with the same focal length and aperture, and attach it to different camera bodies with different sensor sizes. If sensor size and aperture were somehow dependent, that would be more difficult...

Answer 2

Lenses are fitted based on the diagonal measure of the image sensor (or film format). This measure is the diameter of the required “circle of good definition”. Camera lenses project a circular image even though this image circle will be cropped by an image mask to craft a rectangular format.

As a rule-of-thumb the focal length fitted will be about equal to this measurement. Such a lash-up delivers a “normal” perspective. A lens with a focal length longer than the diagonal measure is in the realm of telephoto. A lens 70% of this value or shorter is classified as wide-angle”. Most cellphone cameras are fitted with a moderate wide-angle.

A wide-angle lens, with its shorter focal length, tends to deliver a circle of good definition that is too small. It takes special care to construct a wide-angle with an adequate circle of good definition. Naturally this runs up the cost.

A long focal length lens is far easier to make than a short focal length lens. A short focal length lens must have a steeper curve (figure). This translates to a lens that is fatter in the middle. It is this feature that gives the lens the power to refract and focus in such a short distance. All lenses suffer from aberrations. These are misdirected light rays that degrade the image. To mediate, the lens must be a sandwich of lenses, some with negative power (concave), and some with positive power (convex). Making a small multi element lens is no easy task.

If the lens is made 3.3mm in diameter, to achieve an f/1 lens the focal length must be also 3.3. That’s difficult! If it’s a 3.3mm diameter operating at f/1.2, it must have a focal length of 2.96mm. We are talking the laws of optics. In short a longer lens is better and they cost less to make.