# Maximum resolution per lens size

This question is more practical than theoretical, but I am interested in the theoretical considerations as well.

My wife just bought a Samsung S3 phone with a 8 MP image sensor hiding behind a tiny lens. In daylight the pictures come out fine, but it suffers horribly in low-light conditions. Is there a theoretical limit as to how fine an image sensor can be behind a lens of a specific aperture, given a reasonable amount of ambient light and a reasonable shutter speed? Will increasing the sensor resolution beyond this limit decrease the actual resolution (the ability to resolve two points as individual points) of the final image?

Thanks.

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The resolution is controlled by diffraction at the smallest part of the lens system. The Wikipedia article on angular resolution goes into this in some detail. To quote the headline from this article, for a camera the spatial resolution at the detector (or film) is given by:

$$\Delta \ell = 1.22 \frac{f\lambda}{D}$$

where $f$ is the distance from the plane of the lens to the detector, $\lambda$ is the wavelength of the light and $D$ is the camera aperture. Making the pixel size smaller than $\Delta \ell$ won't do any harm, but it won't make the pictures any sharper.

I don't know if smartphone cameras contain a variable aperture. With conventional cameras larger apertures produce less diffraction so the picture quality should actually improve in low light. However larger apertures expose a larger area of lens and optical aberration dominates the quality. The end result is that there is an optimum aperture below which diffraction dominates and above which optical aberration dominates.

Incidentally, the poor performance at low light probably isn't due to diffraction. I'd guess it's just that the signal to noise ratio of the detected light falls so far the pictures get very noisy.

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I see, thanks. You've given me the keywords that I do need to continue research. I want to see how far they are pushing this poor lens. However, how is quality reduced as lens size is reduced? All the professional camera have large lenses, surely there is some advantage to a larger lens. –  dotancohen Oct 24 '12 at 16:25
A large lens allows in more light so you can use faster shutter speeds and/or shoot in poorer light. However spherical lenses all suffer from spherical aberration because rays far from the centre come to a focus at a different place to rays near the centre. The expensive way to fix this is to use lenses with a parabolic profile, but this is very expensive. The cheap way is to restrict rays to near the centre of the lens by using a small aperture. See en.wikipedia.org/wiki/Spherical_aberration for details. –  John Rennie Oct 24 '12 at 17:01
Thanks John! I did realize that a larger lens lets in more light, but I was unaware of spherical aberration. –  dotancohen Oct 24 '12 at 17:04