optics - Why can't we perfectly focus light-abberations aside

I don't understand why there is necessarily a diffraction limitation on optical systems. Where does this limitation in focusing light come from?

Answer

That's a good question, and one that looks simple but has a complicated answer. Here's my attempt at an answer with no maths - as usual in physics you'll only really understand it by getting stuck into the mathematics.

It's commonly believed that lenses work by bending the light. You see diagrams like:

showing the light ray bending as it passes through the lens. This is one way of looking at it, but a more fundamental explanation is that the lens changes the phase of the plane wave passing through it. Specifically the phase change produced by the lens varies with distance away from the centre line. So on the left side of the lens we have a plane wave of constant phase, while on the right side we have a plane wave with the phase varying with distance. The result is that on the right side we get an interference pattern - we generally call the interference pattern the image, but it is an interference pattern.

Incidentally, this is why a Fresnel lens can focus light even though it's a completely different shape to your usual convex lens. The Fresnel lens produces the same phase changes as a convex lens, so it focuses light in the same way.

But back to your question: the reason that the image isn't perfect is that it's formed by interference of only a finite portion of light wave i.e. the portion passing through the lens. The bigger the lens the more of the light wave forms the interference pattern and the better the image. But to get a perfect image you need all the light, i.e. an infinitely big lens, to contribute to the interference pattern.

Mathematically, the light intensity at the focal plane is the Fourier transform of the incoming light. The integration limits of a Fourier transform are from $-\infty$ to $+\infty$, but a finite size lens restricts the integration limits and changes the intensity in the focal plane away from the perfect case. It actually convolves the light intensity in the focal plane with the Fourier transform of the aperture through which the light passes. For a round lens this means your image is convolved with an Airy disk, and this smears out the image slightly.