One interesting point is why is the far-field amplitude of light diffracted by the lens a spatial Fourier transform? The best explanation I know of is in Chapter 21 of the Feynman Lectures on Physics (vol 2), accessible online (thanks to CalTech) at http://www.feynmanlectures.caltech.edu/II_21.html#Ch21-S3
In short: Moving charges create electromagnetic radiation (light). Since light travels at non-infinite speed, when a wiggle of light reaches us, we're actually seeing the imprint of the motion of the source charges at an earlier point in time. This earlier time is clearly related to how far the observer is from the source. Hence, if the distance is r then at time t we would be seeing the wiggle of the source charge at an earlier time = t - r/c. For a collection of source charges we need to integrate the delayed source potential over the volume of the source, so it's intuitively clear why that would be a Fourier transform.
There's technicalities of far-field and 1/r fall-off, which I've glossed over, you can find full details in Chapter 21. The final line is a gem: 'You will not, then, be surprised to find that the laws of electricity and magnetism are already correct for Einstein's relativity. We will not have to “fix them up,” as we had to do for Newton's laws of mechanics.'
In short: Moving charges create electromagnetic radiation (light). Since light travels at non-infinite speed, when a wiggle of light reaches us, we're actually seeing the imprint of the motion of the source charges at an earlier point in time. This earlier time is clearly related to how far the observer is from the source. Hence, if the distance is r then at time t we would be seeing the wiggle of the source charge at an earlier time = t - r/c. For a collection of source charges we need to integrate the delayed source potential over the volume of the source, so it's intuitively clear why that would be a Fourier transform.
There's technicalities of far-field and 1/r fall-off, which I've glossed over, you can find full details in Chapter 21. The final line is a gem: 'You will not, then, be surprised to find that the laws of electricity and magnetism are already correct for Einstein's relativity. We will not have to “fix them up,” as we had to do for Newton's laws of mechanics.'